Finsler Metrics - A Global Approach

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich E Takens, Groningen Subseries: Scuola Norrnale Superiore, Pisa Advisor: E. Vesentini

1591

Marco Abate Giorgio Patrizio

Finsler Metrics A Global Approach with applications to geometric function theory

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Authors Marco Abate Dipartimento di Matematica Universit~t di Ancona Via Brecce Bianche 60131 Ancona, Italy Giorgio Patrizio Dipartimento di Matematica Universit~t di Roma Tor Vergata 00133 Roma, Italy

Mathematics Subject Classification (1991 ): 32H 15, 53C60

ISBN 3-540-58465-X Springer-Verlag Berlin Heidelberg New York

CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera ready by author SPIN: 10130166 46/3140-543210 - Printed on acid-free paper

Contents

Preface

......................................................................

vii

1. Real Finsler geometry 1.0 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 N o n l i n e a r c o n n e c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 P r e l i m i n a r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 H o r i z o n t a l a n d v e r t i c a l b u n d l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 L o c a l c o o r d i n a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 G e o d e s i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2

1.3

1.4

3

3 5 11 14

1.2.1 D e f i n i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 C h a n g e of c o o r d i n a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 L i n e a r c o n n e c t i o n s o n M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 17 18

Torsion and curvature ................................................. 1.3.1 C o n n e c t i o n s a n d differentials o n t e n s o r b u n d l e s . . . . . . . . . . . . . . . . . . 1.3.2 T h e t o r s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 T h e c u r v a t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 21 22

The Cartan connection ................................................

24 24 26 30

Vertical connections ...................................................

1.4.1 F i n s l e r m e t r i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 T h e C a r t a n c o n n e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 T h e h o r i z o n t a l flag c u r v a t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 F i r s t 1.5.1 1.5.2 1.5.3 1.6

1

and second variations ............................................

The setting ...................................................... T h e first v a r i a t i o n of t h e l e n g t h i n t e g r a l . . . . . . . . . . . . . . . . . . . . . . . . . . T h e s e c o n d v a r i a t i o n of t h e l e n g t h i n t e g r a l . . . . . . . . . . . . . . . . . . . . . . .

The exponential

1.6.1 1.6.2 1.6.3 1.6.4

map ..................................................

Parallel transport ................................................ Geodesics ........................................................ The Gauss lemma ............................................... G e o d e s i c s a r e locally m i n i m i z i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.6 J a c o b i f i e l d s a n d A u s l a n d e r ' s t h e o r e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 J a c o b i fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 T h e M o r s e i n d e x f o r m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 T h e C a r t a n - H a d a m a r d a n d B o n n e t t h e o r e m s . . . . . . . . . . . . . . . . . . . . .

32 32 36 38 39 39 40 45 46 51 51 55 59

2. Complex Finsler geometry 2.0 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

2.1

64 64 67 68 74

Comp]ex non-linear connections .......................................

2.1.1 2.1.2 2.1.3 2.1.4

Preliminaries .................................................... C h a n g e of c o o r d i n a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Horizontal and vertical bundles .................................. Local coordinates ................................................

vi 2.2

C o m p l e x vertical connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 D e f i n i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 G o o d v e r t i c a l c o n n e c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 C o n n e c t i o n s o n c o m p l e x t e n s o r b u n d l e s . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 T h e t o r s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 T h e c u r v a t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76 76 77 79 80 81

2.3

T h e C h e r n - F i n s l e r connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 C o m p l e x F i n s l e r m e t r i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 T h e C h e r n - F i n s l e r c o n n e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 S o m e c o m p u t a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 T o r s i o n s a n d c u r v a t u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 K~ihler F i n s l e r m e t r i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 84 87 89 92 93

2.4

Variations 2.4.1 T h e 2.4.2 T h e 2.4.3 T h e

o f t h e l e n g t h integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . setting ...................................................... first v a r i a t i o n f o r m u l a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . second variation formula ...................................

2.5 H o l o m o r p h i c c u r v a t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 S y m m e t r y p r o p e r t i e s of t h e h o r i z o n t a l c u r v a t u r e . . . . . . . . . . . . . . . . 2.5.2 T h e h o l o m o r p h i c c u r v a t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6

T h e C a r t a n connection vs. the Chern-Finsler connection . . . . . . . . . . . . . . 2.6.1 T h e r a d i a l v e c t o r fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 T h e h o r i z o n t a l b u n d l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 T h e c o v a r i a n t d e r i v a t i v e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 T h e h o r i z o n t a l c u r v a t u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96 96 99 102 105 105 108 111 111 115 119 123

3. M a n i f o l d s w i t h c o n s t a n t h o l o m o r p h i e c u r v a t u r e 3.0 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1

Geodesic c o m p l e x curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 D e f i n i t i o n s a n d p r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 T h e e x i s t e n c e t h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 A c h a r a c t e r i z a t i o n of t h e K o b a y a s h i m e t r i c . . . . . . . . . . . . . . . . . . . . . .

3.2 M a n i f o l d s w i t h c o n s t a n t n o n p o s i t i v e h o l o m o r p h i c curvature . . . . . . . . . . . 3.2.1 A c o m p l e x C a r t a n - H a d a m a r d t h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 T h e M o n g e - A m p b r e e x h a u s t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3

C h a r a c t e r i z a t i o n o f C n and o f convex circular d o m a i n s . . . . . . . . . . . . . . . . 3.3.1 G e o m e t r y of t h e M o n g e - A m p ~ r e f o l i a t i o n . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 A c h a r a c t e r i z a t i o n of C n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 A c h a r a c t e r i z a t i o n of convex circular d o m a i n s . . . . . . . . . . . . . . . . . . .

127 128 128 133 142 147 147 154 161 161 166 167

References ..................................................................

171

List o f s y m b o l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

Index .......................................................................

179

Preface

When, some years ago, we started working on a differential geometric study of the structure of strongly convex domains in C", we did not expect to end up writing a book on global Finsler geometry. But, along the way, we found ourselves needing several basic results on real and complex Finsler metrics that we were unable to find in the literature (or at least not in the form necessary to us). So we felt compelled to provide proofs - - and this is the final result of our work. Our exposition is very much in the vein of the work of Cartan [C], Chern [Chl, 21, Bao-Chern [BC] and Kobayashi [K]; in particular the latter gave us the preliminary idea for our approach to smooth complex Finsler metrics. We would also like to say that we would have been very happy to know earlier of [BC], which, although only marginally related to our work, would have been of great help in solving questions which we treated independently. Our starting point was the study of the existence and global behavior of complex geodesics for intrinsic metrics in complex manifolds. Our goal was to look at this question from a differential geometric point of view, with the hope of possibly reproducing in a wider class of complex manifolds what Lempert [L] was able to prove for strongly convex domains in C". The idea was to treat complex geodesics through a point as images of disks through the origin in the tangent space at the point via the exponential map of a complex Finsler metric; thus we were led to study the local and global theory of geodesics of a Finsler metric. As in Hermitian (and Riemannian) geometry, the local theory of geodesics means the study of the first variation of the length integral, and of the associated Euler-Lagrange equation. The global theory, on the other hand, involves the accurate control of the second variation and hence of the curvature, together with Jacobi fields, conjugate points, the Morse index form and the like. In particular, we needed a Finsler version of the Cartan-Hadamard theorem (originally proved by Auslander [Aul, 2]), and a way to apply it in a complex situation. The main difficulty at this point was that the problems we were interested in involved complex Finsler metrics, and whereas there is a clear understanding of the relationship between the complex geometry and the underlying real geometry of a Hermitian manifold, nothing of the kind was available to us in Finsler geometry. We then started following the tradition of "linearizing" the questions by passing from the study of Finsler metrics on the tangent bundle (real or complex) to the study of the associated Hermitian structure on the tangent bundle of the tangent bundle. At this level it is also possible to describe the correct relationship between the complex and the corresponding real structure of objects like connections and curvatures. But our approach is different from the traditional one for two main reasons. First of all, we everywhere stress global objects and global definitions (in fact, we are interested in global results), using local coordinates almost uniquely as a computational tool (in a way not too far from the first chapter of Bejancu [B]). But the main difference is another one. Possibly because of our motivations, working in this area we discovered that there might be a danger of carrying out the llnearization program previously described too far. In fact, the formal setting naturally leads to

viii very general definitions which make proofs of theorems easier, but do not give much geometrical insight: we had the feeling that working only at the tangent-tangent level was too restrictive, too formal, too far away from the actual geometry of the manifold. For this reason, our point of view now is to stick to notions which really provide informations about the geometry of geodesics on the manifold, and about the curvature of the manifold. This approach leads to "minimal" definitions, which are probably more complicated to state and surely more difficult to handle, but nevertheless more effective and really conveying the geometry of the manifold. For instance, there are many ways of generalizing the notion of K~hlerianity to Finsler metrics, but not all of them have non-trivial examples and applications. We shall show how the notions we singled out can be effectively used by illustrating their applications in complex geometric function theory. The first two chapters of this book are devoted to the exposition of our approach to real and complex Finsler geometry. In the first chapter, after setting the stage introducing the necessary general definitions and objects, we define in a global way the classical Cartan connection, and we discuss the variation formulas of the length integral, the exponential map, Jacobi fields, conjugate points and the Morse index form up to provide a proof of the Cartan-Hadamard and Bonnet theorems for Finsler metrics suitable for our needs in complex geometry. In the exposition we stress the similarities with the standard Riemannian treatment of the subject, as naturally suggested by our global approach. In the second chapter we study the geometry of complex Finsler metrics. After having adapted the general definitions of chapter 1 to the complex setting, we introduce (following Kobayashi [K]) the Chern-Finsler connection, which is our main tool. We discuss at some length several K~b_ler conditions, and we introduce the notion of holomorphic curvature of a complex Finsler manifold, showing the equivalence of the differential geometric definition with a variational definition previously used in function theory. Finally the third chapter contains the results and applications that motivated our work. From a differential geometric point of view, it is devoted to the study of the function theory on K~hler Finsler manifolds with constant nonpositive holomorphic curvature; from a complex analysis point of view, it is devoted to the study of manifolds where there is a Monge-Amp~re foliation with exactly the same properties as the one discovered by Lempert in strongly convex domains. In particular we prove the existence of nice foliations and strictly plurisubharmonic exhaustions satisfying the Monge-Amp~re equation on K~rler Finsler manifolds with constant nonpositive holomorphic curvature. ~ r t h e r m o r e we prove that the only complex manifold admitting such a metric with zero holomorphic curvature is C n, and we describe a characterization of strongly convex circular domains in terms of differential geometric properties of the Kobayashi metric. Of course, this book is not intended as a definitive treatise on the subject; on the contrary, it is just the description of an approach to Finsler metrics that we found reasonable and fruitful, but still leaving a lot of open problems. Just to mention a couple of them: the comparison between the complex Finsler geometry and the underlying real one carried out in section 2.6 seems to suggest that the Cartan connection contains terms which have no direct influence on the geometry of the manifold - - and so maybe it is not the correct connection to use even in real Finsler

ix geometry. Or: in the third chapter we give a fairly complete description of the complex structure of K/~hler Finsler manifolds of constant nonpositive holomorphic curvature, which is satisfying from a geometric function theory point of view, but it still leaves completely open the problem of classifying the metrics with these properties (we remark that there are many more such manifolds and metrics than in the Herrnitian case: there are at least all the strongly convex domains in C" endowed with the Kobayashi metric, thanks to Lempert's work ILl) - - and in fact it is even still far from being completed the classification of simply connected real Finsler manifolds with constant (horizontal flag) real curvature. Or: it follows from chapter 3 that the only part of Lempert's results actually depending on the strong convexity of the domain is the smoothness of the Kobayashi metric. It would be then interesting to construct directly a smooth weakly K/ihler Finsler metric of constant holomorphic curvature - 4 on any strongly convex domain; then this metric would automatically be the Kobayashi metric of the domain, and we would have recovered the full extent of Lempert's work. So we hope that the possibly new perspectives on Finsler geometry introduced in this book will eventually lead to new results in this field; and in particular in geometric function theory of complex Finsler manifolds, where all this work started.

CHAPTER1

Real Finsler geometry

1.0. I n t r o d u c t i o n As already discussed in the preface, this book is mainly devoted to the study of complex Finsler geometry; but of course such a study cannot leave out of consideration real Finsler manifolds. So this first chapter is devoted to a discussion of real Finsler geometry, starting from the very basics and ending with a proof of the appropriate versions of the Cartan-Hadamaxd and Bonnet theorems for Finsler manifolds, obtained using global Riemannian-style techniques. Let M be a real manifold endowed with a Finsler metric, that is with a positively homogeneous function F: T M --* R + smooth outside the zero section of T M a n d strongly convex on each tangent space. Roughly speaking, our main idea is to replace the given Finsler metric on T M by a Riemannian metric on a suitable subbundle of T ( T M ) - - in a certain sense we linearize the Finsler metric going one step upstairs - - and then use the standard tools of Riemannian geometry there. A canonically defined isometric embedding of T M (outside the zero section, actually) into this bundle will then allow us to transfer information back and forth, thus giving geometrical results about the original manifold. For instance, we shall be able to recover for Finsler manifolds more or less all the results described in the first chapter of [CE] for Riemannian manifolds. We also refer to [C], [Rdl], [M], [Call and [B] for a description of the standard theory of real Finsler metrics, and to [Ch2] and [BC] for a recent approach akin in spirit to ours. To be more precise, let M be a manifold, and let ~r: T M --* M denote the tangent bundle of M; 2~/will stand for T M \ {zero section}. The v e r t i c a l brindle is ]) = kerdTr, a sub-bundle of T2~/. Take a Finsler metric F: T M ~ R + on M, and set G = F 2. Then it is easy to see (section 1.4) that using the Hessian of G it is possible to define a Riemannian metric on ]) in such a way that a canonically defined section ~: _~/--* ]) of ]) (see section 1.1) turns out to be an isometric embedding of .~/ into ]). But this is not yet the setting mentioned before. The point is that to such a Riemannian metric on the vertical bundle it is possible to associate two objects: a linear connection D on ]) with respect to which the given Riemannian metric is parallel; and a horizontal bundle, that is a sub-bundle 7-/of T M such that T . ~ / = 7-/0]). The general theory of horizontal bundles yields a bundle isomorphism O: ]) --* 7"/; using O we can define a Riemannian metric and a connection on 7-I - - and hence

on TAT/. This is the Cartan connection, the exact analogue in the Finsler setting of the Levi-Civita connection (e.g., its torsion is almost zero - - in a very definite sense; furthermore, the torsion is identically zero if and only if the Finsler metric actually is Riemannian, and in this case the Cartan connection coincides with the Levi-Civita connection). The bundle H with this structure (and with its section X = O o L) provides the aforementioned setting, where one can use Riemannian tools to get Finsler statements. The main examples of this assertion are provided by the first and second variations of the length integral, derived in section 1.5; we get formulas formally identical to the Riemannian ones, just replacing the curvature of the Levi-Civita connection by (a suitable contraction of the horizontal part of) the curvature of the Cartan connection. Then we shall be able to recover the Hopf-Rinow theorem for Finsler manifolds (this is not too surprising, since it holds in much more general settings; see [Ri D and the theory of Jacobi fields and of the Morse index form, in a way exactly parallel to the one presented in standard Riemarmian geometry texts. In particular, in section 1.7 we shall be able to prove the generalizations (originally due to Auslander [Aul, 2]) to Finsler manifolds of the classical Cartan-Hadamard and Bonnet theorems. In detail, the content of this chapter is the following. In section 1.1 we discuss at some length the general theory of horizontal bundles, horizontal maps (i.e., maps like our @ above) and non-linear connections on M. In section 1.2 we introduce the concept of vertical connection (i.e., of linear connection on the vertical bundle), and we show how to associate to certain vertical connections (we call them the good ones) a horizontal bundle, and hence a non-linear connection on M and a linear connection on/~/. In section 1.3 we define and discuss the torsion and the curvature of a good vertical connection. In section 1.4 we define Finsler metrics, and we show that to any Finsler metric F is canonically associated a good vertical connection, the Cartan connection mentioned before. Section 1.5 is devoted to prove the first and second variation of the length integral for Finsler metrics; section 1.6 to parallel transport, geodesics, the exponential map and the Hopf-Rinow theorem for Finsler metrics. Finally in section 1.7 we shall define Jacobi fields and the Morse index form in this setting, and we shall prove the Finsler versions of the Cartan-Hadamard and Bonnet theorems.

1.1. Non-linear

connections

1.1.1. Preliminaries In this subsection we fix our notations and collect a few formulas concerning change of coordinates. We choose symbols and notations so to be compatible with the complex case we shall discuss in chapters 2 and 3; this is the reason behind some apparently slightly unusual choices (u instead of v to denote tangent vectors, and the like). Let M be a real manifold of dimension m; we shall denote by T M its tangent bundle, and by ~r: T M ~ M the canonical projection, as usual. The cotangent bundle will be denoted by T * M . If ( x l , . . . ,x m) are local coordinates on M about a point P0 9 M, a vector u 9 T p M (with p close to P0) is represented by U~-- Ua

0

P,

where we are using the Einstein convention, and lowercase roman letters run from 1 to m. In particular, local coordinates on T M are given b.y ( x l , . . . , x m, u l , . . . , urn), and so a local frame of T ( T M ) is given by { 0 ~ , . . . , 0m, 0 ~ , . . . , (~m}, where 07 O~ = Oz-------

and

0 bb - Ou b .

We shall denote by o: M ~ T M the zero section of T M , that is o(p) = or is the origin of T p M , and we set M = T M \ o ( M ) , the tangent bundle minus the zero section. 2~/ is naturally equipped with the projection 7r: _~/ ~ M , the restriction of the canonical projection of T M . Correspondingly, T2~/ C T ( T M ) comes equipped with a natural projection ~: T.~/--~ M, the restriction of the natural projection ~: T ( T M ) ---* T M . We shall use uppercase roman letters to denote different coordinate patches. A coordinate patch (UA, ~ A ) in M determines a coordinate patch (UA, @A)in T M (and _/tT/) setting UA = rc-I(UA) and V'a 9 UA

~ A ( u ) ~- d~A(U).

If ~A = ( x ~ , . . . , z ~ ) , then {(O/OxJA)lp} is a basis of T p M for any p 9 UA. Writing (OlOx ), then

u = u

A(u)

On UA n U s we have

:

m XA, U 1A , . . ' , Un~).

4

,,,here ax'~la, h = a ( ~ o ~]l),laz ~. By the way, we set

clearly,

JAB = j ~ l

0

(~A 0 ~B1).

Taking tt E UA I~l UB a~D,dexpressing it in local

coordinates, we

find

~'~9 = ~x'~(~) = ax~ ~ ( uk~ 0 ) = OX'B a ~ ~'j ozjA

that is UB ~ J B A ?2A"

Therefore

(XB,UB)=~BO~AI(XA,UA)=(~BO~AI(XA),JBAUA).

(1.1.1)

Up to now everything was quite standard. But now something different: change of coordinates in

T(TM).

Define

~

(UA,~A) by setting

uA = ~ - 1 ( 5 ~ ) = (~ o ~ ) - ~ ( u A )

and ~A(X) = d~A(X) for any X e UA. A vector X E T(TM) in local coordinates is expressed by XA = X~(Oi)A + X~(Oj)A = h(XA) + v(XA). Taking derivatives of (1.1.1), we find the Jacobian matrix for

(

Ox~ Ox~

Y~A =

~ (JBA)' =

"

O~x~

0 l

T(TM): ) "

Setting

ax~a~' we find (XB,UB, h(XB), v(XB)) = 4~ o 4A' (XA, UA, h(XA), v(XA))

( ~ o ~1(~ ~), s . ~ ~ , s . ~ h(x ~), (H~)'~, ~ h(X ~) k + S~A v(X ~)). (1.1.2)

Now {Oi,Oi) is a local frame for T ( T M ) ; let {dxi,du j } be the dual coframe (note that dxi[~, is not the same as dxi[p). First of all, (1.1.2) yields

~

(1.1.3) = 04 ~

+

o~4

h ! k ~'~dx~ = (J~a)~a~ + (H~A)~,~,a~,.

Recalling that {dx i, du j } is the dual frame of {Oi, Oi), we get

--1 j --1 k " = (J~a)i (0,)a -- (a~a)i ( H ~ a ) kh , ( J ~- 1) h r " l. (&)a,

(1.1.4)

- 1 k (0k)A. " = O x k ( b k ) A = (Y~a)h

1.1.2. H o r i z o n t a l and vertical bundles Now we may introduce our first main actor. DEFINITION 1.1.1: The vertical bundle of a manifold M is gr: ~ -+ T M of rank m = dim M given by

the vector bundle

= ker&r C T ( T M ) . In local coordinates, ~ A O 71" O ~ A I ( X A ,

UA) = XA,

and so

@A 0 dr o (~AI(X A, UA, X A) = d~oA o dr o (d@A)-I(x A, UA,X A) = d ( ~ . o ~ o ~?4')(x., u . , h(Xa), v(XA)) = (xa, h(Xa)). This means that {C~h} is a local frame for I~. We get charts restricting ~n, and in particular (1.1.2) yields

(xs, us, vB) = ~

o ~;'(xA, ua, Ym) = (~B o ~;~(xa), J e . ua, J B . V~)

Let Jn: TpM ~ T M be the inclusion and, for u E TriM, let ku: TriM ~ Tu(TpM) denote the usual identification. Then we get a natural isomorphism

t~, = d(j,~(u))~, o k~,: T,~(u)M ---* l~,.

(1.1.5)

DEFINITION 1.1.2: T h e radial vertical vector field is the n a t u r a l section t: T M --* V given by = clearly, e(u) 9 Vu. In local coordinates,

in particular, if u = ua(O/Ox a) then

Note t h a t t h e derivatives with respect to x (coordinates in M ) become derivatives with respect to u (coordinates in TM). The vertical bundle is canonically defined; this is not the case for a horizontal bundle. We m a y describe horizontal bundles using three different points of view, each with its own advantages and disadvantages. The first two are easily introduced: DEFINITION 1.1.3: A horizontal bundle is a subbundle 7-/of T ( T M ) such t h a t

T ( T M ) = ~ @ V. DEFINITION 1.1.4: A horizontal map is a bundle m a p O: ~3 --~ T ( T M ) such t h a t

Vu E T M

(dTr o 0)~ = ~:1.

(1.1.6)

We defined horizontal bundles (and horizontal maps) on T M , b u t it turns out t h a t t h e y are interesting only over 3~/. In fact, let o: M ~ T M denote the zero section. It is easy to check t h a t dop(O/Ox j) = Oj; therefore we have the n a t u r a l splitting

To~(TM) = ~o~ @ Yop, where 7-/op = dop(TpM). We shall then assume t h a t all our horizontal bundles coincide with dop(TpM) over the zero section, and, analogously, that all our horizontal m a p s satisfy Oo (&lo ) = 0hlop. Clearly, this m a y cause problems with the smoothness at the origin. We shall henceforth assume t h a t our horizontal bundles and maps will be smooth over IVI, but t h e y m a y be not smooth over the zero section. The reasons behind this approach will become clear in section 1.4, when we shall define the concept of Finsler metric. As mentioned before, there is a third approach to horizontal bundles, via the notion of non-linear connection. But to describe it we need a digression on linear connections. If p: E --* M is any bundle over M , we shall denote by X ( E ) the space of sections of E.

7 DEFINITION 1.1.5: A linear connection on a manifold M is a R-linear m a p

D: X ( T M ) --o X ( T * M | T M ) satisfying the derivation property

(1.1.7)

V~ e X ( T M ) Vf E Coo(M) b ( f ~ ) = df | ~ + f b ~ .

As a consequence, b ~ at a point p E M depends only on the value of ~ and d~ at the point p. Indeed, let ~' E X ( T M ) be another vector field with ~(p) = ~'(p) and d~p = d~v. Then ~' = ~ + f~? for suitable 7/E X ( T M ) and f E C~176 with f(p) = 0 and dry = O. So

D~' = b~ + D(frl) = b~ + df | r1 + fD~l,

and b g = b~p. There is another way of expressing this. Let ~, ~' E X ( T M ) be such that ~(p) = ~'(p) = u; then ~' = ~ + f r / w i t h f(p) = 0. In particular,

b~v = D~v + dfn | rl(p). Now, for any v E TpM, writing v = v"(O/Ox") and ~ = ~b(O/Oxb), one has o 0~

b

9

d~v(v ) = vaOalu + v 0--~(p)0b[~.

(1.1.8)

So d(~, - d~p maps TpM into I]u; furthermore, (1.1.8) also yields

Vv e TvM d~(v) - d~v(v ) = v ( f ) ~ (r/(p)) = tu (v(f)rl(p)), and thus

,-~1 o (d~'v - d~v ) = dfn | 71(p).

(1.1.9)

Summing up, if ~(p) = ~'(p) = u we get b ~ 'v - b~v

= t~-a o

(d(v - d~v),

(1.1.10)

which we m a y consider as an intrinsic way of saying that b~v depends only on ~(p) a n d d~p.

There is another easy consequence of (1.1.7) worth remarking. (1.1.7) to the zero section o with f -- 0 we get Do = 0,

If we apply (1.1.11)

i.e., Dop(u) = Op for all p E M and u E TvM. We are now ready to introduce the third incarnation of horizontal bundles. DEFINITION 1.1.6: A non-linear connection is a m a p D: X ( T M ) --* X ( T * M @ T M ) satisfying (1.1.10) and (1.1.11). b ~ is called the covariant differential of the vector field ~ e X ( T M ) , and b~v(u ) (which we shall denote by V~,~) is the covariant derivative of ~ in the direction of u E TpM.

Before stating the smoothness assumption we shall need, we remark that the covariant differential of a vector field is uniquely determined at its zeroes: indeed, if ~(p) = 0 then (1.1.10) and (1.1.11) applied with ~' = o force

9 ~ --_ Q1 o (d~ - dop). So we shall say that a non-linear connection/9 is smooth if for any ( 9 X ( T M ) and p 9 M with ~(p) r 0 the s e c t i o n / ) ( is smooth in a neighborhood of p. Equation (1.1.10) is enough to show that all non-linear connections enjoy two of the main properties of a differential operator: locality, and the dependence of ~7,,~ only on the restriction of ~ to any curve with u as tangent vector. Indeed: LEMMA 1.1.1: Let D be a non-linear connection on a manifold M . Then: (i) /f~, ~' 9 X ( T M ) are such that ~ - ~' in an open subset U of M , then D~ ==_D~' in U; (ii) fake p 9 M , u 9 TpM and ~: ( - e , r --. M a curve in M with a(O) = p and ~(o) = ~. Let ~, ~' be ~wo vector ~elds with ~ o ~ = ~' o ~; then r

-- r

Proof: (i) Take p 9 U; then ~(p) -- ~'(p), d~p = d ~ and (1.1.10) yields/)~p = / ) ~ . (ii) Let v = ((p) = ('(p); therefore (1.1.10) yields ~.(~r

- V~r

d~(u)(~)

= u(g o ~) = ~?(9 o ~ o

Since

= dr

d

- dr

~) t=0

for all g 9 C~176

we get d~p(u) = d ~ ( u ) and we are done.

[]

Now we may go back to horizontal bundles. We anticipated earlier that nonlinear connections are just another way to look at horizontal bundles; let us now explain this assertion. Since, as already remarked, everything is determined over the zero section, we shall work in dlT/. Let 7-/ be a horizontal bundle; since T/17/ = 7"/@ ld, we can associate to 7-/ a vertical projection ~: TitS/ --, Y. Then we may define a non-linear connection b~/ on M by setting

for any p 9 M and ~ 9 X ( T M ) . First of all, it is clear that / ) n ~ is a linear map from X ( T M ) into itself, and thus an element of X ( T * M | T M ) , smooth outside the zeroes of ~, as required. Furthermore, if ~(p) = ~'(p) = u then

because, as remarked before, d ~ - d~p maps TpM into Yu. Finally, since by assumption we have "Ho~ = d%(TpM), it follows that ~o, o dop --- 0 and s o / ) n o --- 0. Summing u p , / ) ~ is a non-linear connection as claimed.

Now let D: X ( T M ) ~ X ( T * M | be a non-linear connection. Take u e 2t~/p, and ~ e X ( T M ) such that ~(p) = u. Then define O~: Yn ~ Tn2tT/by O ~ = d~p o ~-i 1 - ~ o b~p o ~ 1 .

This definition does not depend on ~, but only on u, thanks to (1.1.10). Fhrthermore, O b is clearly smooth on 2t7/, and (dTr o O)~ = t~ 1 because dTrn o d~p = id and dTru o en = 0. So O b is a horizontal map. Finally, to a horizontal map O we associate a horizontal bundle 7-I~ simply by setting ~o = o n ( v . ) ; indeed (1.1.6) implies that 7-/~ N l) = (0) and that ~ o has rank m. In this way we have defined a correspondence among horizontal bundles, horizontal maps and non-linear connections. As announced before, this is one-to-one:

PROPOSITION 1.1.2: Let M be a manifold. Then the maps Tl ~ D x , b ~ 0 b and 0 ~ ~ o define a one-to-one correspondence among horizontal bundles, non-linear connections and horizontal maps. Proof: It suffices to prove three facts. (i) 7"/~

= 7"/. Indeed, if ~ E X ( T M ) is such that ~(p) = u we have o~(v.)

= (d~, o , j 1 _ ~n o b~,~p o , j 1 ) ( v n ) = ( d e , o , ~ ' - ,n o ( , ~ ' o ~ . o d e , ) o , ~ l ) ( v n ) = (dCv - n,, o d~v)(TvU ) = n , , .

(ii) b ~ e ~ = D. Take ~ E X ( T M ) , p E M and let u = ~(p). By definition, H o t~

b = O n ( V , ) = (d~ v - e~ o DCp)(TpM).

Therefore the vertical projection associated to ~ 0~ of d~p is tn o D~v, and so

(iii) Ob'* ~ = O. By definition, the vertical projection associated to H ~ is ~ = id - 0 ~ Then, taking any ~ E X ( T M )

o ~ o

dlrn.

with ~(p) = u, we get

O ~ ~| = d~p 0 tu I -- *u 0 ~)7.r

0 ~ 1 = d~p 0/,~1 -- $tt o ( i , j I o K:u o d ~ p ) o ~j1

= d~p o t j 1 _ ( i d --On o tu o dTru) o d~p o ~ 1

= 0,, o tn o (dnn

o

d~p )

o

Lj I

=

On. []

10 So we have shown that non-linear connections, horizontal bundles and horizontal maps are just different aspects of the same object; in the rest of this work we shall need all of them. Actually, we shall not always be interested in generic non-linear connections, but only in homogeneous ones. DEFINITION 1.1.7: A non-linear connection/9 is homogeneous if

b(,~) = .~b~ for all X E R and ~ e X ( T M ) . It is easy to check that homogeneous connections enjoy the derivation property (1.1.7): LEMMA 1.1.3: Let D be a homogeneous connection. Then

b ( f ~ ) = df | ~ + f b ~ for all f E C~176 and ~ E X ( T M ) . Proof: Take p E M and apply (1.1.10) to f~ and f(p)~. We find

b ( f ( ) p - f(p)bCp = b(f~)p - b(f(p)r

-~ = t I(p)~(p) o (d(f~p)

-

d(f(p)~)p)

= dfp | ~(p), where we used (1.1.9), and we are done.

[]

Accordingly to the philosophy expressed before, there should be a notion of homogeneity for horizontal bundles and maps. DEFINITION 1.1.8: For any A E R, let #~: T M --~ T M denote the multiplication map ~ ( p ; u) = (p; ~u). Since 7r o #x = rr, the differential d#x sends 12~ onto l,~,x(,) for all u E T M . Clearly, a non-linear c o n n e c t i o n / ) is homogeneous iff

b ( , ~ o ~) = ~ o b ~ for all ), E R and ~ E X ( T M ) . DEFINITION 1.1.9: We say that a horizontal bundle 7 / i s homogeneous if

~.~) for all u E T M and A E R.

= d(~).(~.)

11 DEFINITION 1 . 1 . 1 0 :

W e say that a horizontal map 0 is homogeneous if

d(#~,)u o O . o t,~ = 0 ~ ( , , )

o ,.~(.)

for all u E T M and A E R. The definition of homogeneous horizontal map seems slightly far-fetched, but it is the right one to get the following LEMMA 1.1.4: Let M be a manifold. Then the correspondences defined in Proposition 1.1.2 preserve homogeneity. Proof: The proof depends on two remarks. First of all, it is not difficult to check that ~,~(.) o #~ = d(#~)u o e, (1.1.12) for all u E T M and A E R. Next, let ,r be the vertical projection associated to a horizontal bundle 7"(. Then 7"( is homogeneous iff ~,~(,) o d(p~)= = d ( # ~ ) , o ,% (1.1.13) for all u E T M and A E R. Then 7-( homogeneous implies, by (1.1.12) and (1.1.13), that / ) ~ is homogeneous. I f / ) is homogeneous, then (1.1.12) yields d(/~:O,, o O .b o , . = d ( ~ ) u

o (d~p o , : 1 _ , . o b~p o ,~1) o ,~

= d(#,x o ~)v - '#~(~) o #;~ o/)~v = d ( ~ o ~)~ - , . ~ ( ~ ( b ( ~

o ~)~) = o ~ ( u ) o ~ ( u ) ,

where as usual { is any vector field with {(p) = u, and 0 D is homogeneous. Finally, if 0 is homogeneous we have

d(,r

~ = d(,~)u o O . o , . ( T ~ M ) = 0.~(.~ o ,.~(.~(T~M) = ~.~(.~,o

and 7"/~ is homogeneous.

1.1.3.

[]

Local coordinates

Now let /9 be a generic non-linear connection on a manifold M , with associated covariant differentiation V; we would like to express V in local coordinates. Take e X ( T M ) , p E M and set u = ~(p). Then (1.1.8) shows that d(p

~

O( 9 =Oh+o--~Ok.

12 By definition of non-linear connection, the quantity e,(Qo/oxh ~) - d ( v ( O / O x h) must depend only on u; being t~(Qo/o~h() vertical, there exist I'~ = r ~ ( u ) e N such that

~(90/0~)

- d6

(o) ~

= --Oh + r~(~)&.

Summing up we get

0( k

v.

(1.1.14)

The P~ are the Christoffel symbols of the non-linear connection. It is clear that the non-linear connection is smooth iff the Christoffel symbols are smooth on h:/(where defined, of course). Reading (1.1.14) f o r / ) , we get

t,,, =

[oz h

+

o,,]

|

o

ox~"

Now it is easy to get the expression in local coordinates for the horizontal m a p El D. Take u e -~/p, and ~ E X ( T M ) with ~(p) = u. Now ~(O/Oz h) = cqh; therefore ~. o b ~ o 4'(c~h) =

[k O(

k

O--~-(p) + r~(~)

]

c~,

O~k , ,a d6 o ~-l(&) = Oh + ~ p ) c , k ,

and so O,,b

9 = Oh - r ~ ( ~ ) & . (Oh)

(1.1.15)

If O is any horizontal map, it is clear that O in local coordinates is expressed as in (1.1.15). We set ,~jl,, = e , , ( o j l , , ) = Oil,, - r ] ( u ) & l , ,

e ~e;

(1.1.16)

then {61 . . . . ,$,~} is a local frame for "He, and { ~ l , . . . , ~ m , b l , . . . , ~ m } is a local frame for T M . We shall denote by { d x l , . . . ,dxm,r 1. . . . ,era} the dual coframe; clearly,

Ch = duh + r] dx "i. DEFINITION 1.1.11: We set X, = Ou o ~,:TpM ~ `H~; then the radial horizontal vector field X - - the horizontal analogue of ~ - - is x = O o ~ ~ ,V(`He).

Clearly,

x(Olaz j) = ,~.

13 Now let 7-I be a horizontal bundle, and {61,... ,6m} a local frame such that j = 1 , . . . , m . Then the 6j's may be written as in (1.1.16) for suitable Fi~' s. If

d~r(6i) = O/chxJ for

x = x'o{ + 23a~ ~ T~M, then the local expression for the vertical projection associated to ~ is

~ . ( x ) : (x~r~(~) + X~)a~,

(a) roe,, r~(~(p))]~,

and so

b~

~

: [~p~

+

completing a full circle. Clearly, 7~ is homogeneous iff d(#~),(6i) I E R. Since

6 7-l~,~(,,)

for i = 1,... ,m, u 6 / V / a n d

~ ( x ~ a , + x'~b,,) : IX ~ + x'r?(~)]b.l~,

d(#;,),,(b,,I,,) = ,x,~,,I,,~(,,), and

d(/~,O,,(Ofl,,)

=

a~l,,~(,,),

we find that 7"l (and the associated non-linear connection and horizontal map) is homogeneous iff F~(#~(u)) --- Ar~(u), (1.1.17) that is iff

Vi = 1,... ,m

d(#2,),,(,%1,,) = ,5,1,,~(,,).

So a horizontal bundle (horizontal map, non-linear connection) is locally identified by the coefficients Fhk. To see how they transform under a change of coordinates, we remark that

(~,)e = (o,)B - (r,h)B ( & ) a = e((b,)~) = e((cG]),~(&)A) =

(Jg])~(o~)A

-

(1.1.18)

(Jg])~(r~)A(0k)~ = (J;])~(~h)A.

Compm~g (1.1.4)= d (1.1.18)we ~nd (r~)B(J~A)h + (HBAb(J~)~(J~)~ ~ = (r~)~ (J~),, that is (F{)B

=

i k -I h (JBA)k(rh)~(J~A),

ox~ ~ o~ : 0-~ (rh)A0~

-

( HBA)J,( fl6])~ U~A

o2~ O~u, o;~-~ ox~

Note that (1.1.18) together with (1.1.4) and (1.1.3) yield

(r

=

c d ( ~ 7 eA)d(r )A-

(1.1.19)

14 Finally, for the sake of future references we collect here the following easy. computations: [5,, c3a]

[0o,&]

=

ba(F~)bb,

=

o.

(1.1.20)

1.1.4. G e o d e s i c s

In what follows, horizontal bundles and maps will play a fundamental role, whereas the non-linear connection will somewhat fade in the background. But there is one instance where non-linear connections are essential: in defining parallel transport and geodesics. If a is a curve in M and ~ is a vector field defined only along (the trace of) a, by Lemma 1.1.1 it makes sense to consider ~7~. DEFINITION 1.1.12: We say that ~ is parallel (or horizontal) if fY~ - 0; and a itself is a geodesic if

%e-0.

We may write the condition of parallelism and the equation of geodesics in local coordinates. Recalling (1.1.14), we see that a vector field ~ along a curve a is parallel iff d~~ ~ r~(~)~ b = o, dt for a = 1 , . . . , m; analogously, a curve a is a geodesic iff /~a + F~(&)/rb = 0

(1.1.21)

for a = 1 , . . . , m. These equations are quasi-linear ODE; hence the local theory of geodesics and parallel transport for non-linear connections is more or less identical to the theory known for linear connections. These equations may be interpreted in another way, as shown in the following LEMMA 1.1.5: Let D : X ( T M ) --* X ( T * M | T M ) be a non-linear connection on a manifold M . Then a vector field ~ along a curve er is parallel iff ~ is horizontal iff = X~(&). Analogously, the curve a itself is a geodesic iff?r is horizontal iff?r = X(&).

15 Proof: A vector field ~ along a curve a is a curve in TM; analogously, ~ is a curve in T ( T M ) . A computation in local coordinates yields

= x~(~) +

~(%~), []

and the assertions follow. We shall return to geodesics and parallel transport in sections 1.5 and 1.6.

1.2. V e r t i c a l

connections

1.2.1. D e f i n i t i o n s

DEFINITION 1.2.1: A vertical connection on a manifold M is a linear connection on the vertical bundle, that is a R-linear map D: Pd(Y) ~ X'(T*h?/| Y) such that D ( f Y ) = r o Y + df | V for every f e C ~ ( ~ / ) and V E X(Y). In other words, if V is a section of Y, then D V is a V-valued 1-form on _~/. We set, as usual, VxV = (DV)(X) for all X E T~2t:/and V E X(Y). Note that D is not defined over the zero section of TM; we could have it defined there too but, in accord with the previous section and for reasons we shall see in section 1.4, we would not ask for any smoothness over the zero section. Since anyway we shall not need it, all our vertical connections will be undefined over the zero section. In local coordinates, writing V = VaO~, we get D r = d r a | Oa + V~DO~, and D~ a

b " OJa ~ ~ b ,

where (w b) is a locally defined matrix of 1-forms:

b

IO a

for suitable coefficients F~b ; i and I'~r

r~;~ dx' + r~c dv c,

16 So

V x V = [X(V ") + w~(X)Vb]3,

(1.2.1)

or, writing X = XJOj + JCbOb,

The idea is that we may associate a non-linear connection (i.e., a horizontal bundle) to certain vertical connections. Let D be a vertical connection, and denote by A: T.AT/~ V the bundle m a p defined by A(X) = Vx~. In local coordinates, if X = XJOj + X~Oa we get a ( x ) = [X~ + w~(X)ub]3a. DEFINITION 1.2.2: We say that D is a good vertical connection if A[v: V ~ V is a bundle isomorphism. The importance of good vertical connections is explained by LEMMA 1.2.1: Let D : X ( V ) ~ X(T*Y/I | V) be a vertical connection.

Then D is

good iff T2~7/= (ker A) G V,

that is iff 7"l = ker A is a horizontal bundle. Proof: A[v is an isomorphism iff 7-/~ N V~ = (0) for all u E )17/; furthermore, if A[v is an isomorphism then A is onto, 7-t has rank m = dim M and the assertion follows. [] In local coordinates, A(bb) = [6~ + I'~buC]ba

and

A(0i) = Fc;iu-ac'0~,

where here 6~ is the Kronecker delta. Therefore D is good iff the matrix L~ = 6~ + F~bu ~~ ~ is invertible. If ((L-1)~) denotes the inverse matrix, a local frame for 7-/is given by 6i

=

Oi

-(L

-1

) b~ -F ~c ; i U r

b 9

where r~

b ~a = (L - 1 )arc;i

~c

axe the Christoffel symbols of the non-linear connection associated to 7"/.

17

1.2.2. Change of coordinates So let again D: X(V) ~ X(T*l~/l | V) be a good vertical connection on M. We shall denote by {dz i, r a } the local coframe dual to {~i, cSa}, where ~ba .--- du a + I"~ d x j .

Correspondingly, we may decompose w~ = F~;i dx j + r~r162 ~, where ~ F~c = F~

a

-~

~a

c

Fb; j = Fb; j -- FbcFj.

and

In particular, if X = X i g i + j(bob we have

=

r o .b~

xo [o~(v ~) + r~ov ~]

(1.2.2)

Later on we shall need to know how w~, F~r and F~;j behave under change of coordinates. We have (Wba)B |

(Jff~)kb(~k)A

= (~3b)B |

(~b)B =

--1 k

D(ba)B = D((Jg~)~(Oh)A)

"

--1 h

k

"

= d(YgA)a | (0k)A + (YgA)a(~h)A | (0k)A. Now --1 k d(JBa)"

=

O. ( j f f ~ ) k OXJa

adxjA

-1 h -1 k i dXJA, = --(J'BA)a(~.~BA)i (HBA)hj

and so = ((fBA)a(Wh)A(~BA)k

-- ( J B A ) a ( H B A ) h j

It follows that b

--~h

-~k

,

b

--1 k

d

b

-~h

-~k

b

(l"a;i)B = ( f f B A ) a ( f f BA)i (F h;k)A( f f BA)j -- ( J'BA)a(~.~BA)i ( H BA)hk, b (rac)B

--1 h

(1.2.3)

= (J~A)a(J[~A)o(rhk)A(JBA)d.

In the sequel it will be useful the following LEMMA 1.2.2: Let D: X(V) --* X(T*I~/I | V) be a good vertical connection on M. Then F ja

Fab;jU b .

Proof: Indeed ra

b;j •

b -- ~a . b _ ~a u b : L - l ~ c ~ t --

b;j t~

bc

k

)l

d;j

Ud = [

LL~-

~ c U d] / L - I , c ~ t . ~

)llb;j u

b

= F3" ~. []

18 1.2.3. Linear c o n n e c t i o n s on ~r

We have already seen that, if O: ~ ~ 7"l denotes the horizontal map associated to the horizontal bundle ~ associated to a good vertical connection D, in local coordinates we have o(3 ) = Using O, then, we may define a linear connection (still denoted by D) on ~ by setting

VII 6 Z ( ~ )

OH = O(DO-I(H)),

that is

VII 9 X('H)

V X 9 T2VI V x H = O [ V x ( O - I H ) ] .

In this way a good vertical connection gives rise to a linear connection D defined on .~r by setting

V Y 9 X(TJ~I)

D Y = D(~Y) + D ( Y - ~Y),

that is

VX, Y 9 X(TI~/I) V x Y = Vx(~Y) + V x ( Y - t~Y), where ~: T.~r --~ Y is the vertical projection associated to 7"/= ker A. It will turn out that this linear connection (and in particular its horizontal part) is the right geometrical object to study. For the moment, we just remark that, by definition, VHET-/

VVEF

VHX=0

and

VvX=OA(V).

Fhrthermore, by definition Clearly, the linear connection D on h:/and the non-linear connection/) on M associated to 7"/are closely related. To describe how, we need a definition. DEFINITION 1.2.3: Let ~ 9 X ( T M ) be a tangent vector field; then the vertical l/t'~ ~v 9 X(l;) and the horizontal lift ~H 9 X(~'~) are defined by

for all u 9 ]~/. Then PROPOSITION 1.2.3: Let D: X 0 ) ) --~ X(T*37/| );) be a good vertica] connection. Let 7"/be the horizontal bundle associated to D and let D: X ( T M ) --* X( T * M | T M ) be the non-linear connection associated to ~ . Take ~ E X ( T M ) and p E M. Then vue

xap (v

) =

19

that is X~(p) o L)~ = D~ H o X~(p). Proof: In local coordinates, (1.1.14) yields X~,(p)(~u~) = U h [ O~k" " +

ri(~(p))]

eklr

On the other hand, V

H

uh

ro~ ~, []

by (1.2.2) and Lemma 1.2.2, and the assertion follows.

1.3. Torsion

and curvature

1.3.1. C o n n e c t i o n s a n d differentials on t e n s o r b u n d l e s DEFINITION 1.3.1: Let D: X(V) ~ X(T*AT/| be a good vertical connection, and let D also denote the associated linear connection on M. We now define a linear connection on T'AT/, which we shall denote by V: X(T*Jf/I) --* X(T*_~-I | T'if,I), simply requiring that Vqo(X) + qo(DX) = d(qo(X)), for every ~0 e X(T*_~I) and X 9 X(Tlf.I). In particular, if u 9 A:/ and Y 9 T ~ / , then Vvqo 9 T * M is such that

(vr~)(x) + ~ ( v r x ) = y(~(x)). This defines uniquely Vq0. In local coordinates, if ~ = ~oidx i -t- ~ a r a, then Vyqo = [Y(~oi) - w i ( r ) ~ j ] d x i + [Y(~ba) -

02ba( y

9 a, )qo~]r

that is v ~ = [d~i - ~ j ~ i ] | dx ~ + [d#o - ebbS] |

r176

Now let T(r's)JtT/denote the bundle of (r, s)-tensors over ]tSir, that is T(r's)-Q = T'AT/| .:. | T*hT/| T . ~ / | r times

| T.~/. s t~mes

20 Then we can extend V to a linear connection V: X(T(~'s)f/I) ~ X(T*f/I | T(~,s)f/i) just by setting V(R| = ~TR| + R | V S for any pair of tensors R and S. For instance, if R 6 X(T(1J)]r

is locally given by

R = RIj 6i | dx j + R~ 5i | eb + R aj Oa | dx j + R~ Oa | eb, then

v n --_ [dni, - n ;h ~ ~j + "-; ~ j ' ~kJh,|

' b+ ~ Rbwk] ~ ' 6i | j + [Jn~ _ R~w

~

We now introduce an exterior differential for T(~,~)A~/-viued p-form coinciding with V on T(~,~)tY/-viued 0-form, i.e., on sections of T(~,~)A~/. DEFINITION 1.3.2: Let ~0 6 X ( A p T*JQ | T(~")f/I) be a T(~,')AT/-vlued p-form; locally, ~0 = %oa | e,, where {ea} is a l o c l frame for T(r's).lQ composed by tensor products of {6i,tga} and {dxJ,r Then we set D~0 = d~o" | e, + (-1)P~" AVe..

(1.3.1)

For instance, if ~ 6 X(A p T'A)/| T.g/), then

D~ = [d~ i + (-1)P~ i A w;] | 6i + [d~" + (-1)P~b b A w~] | Oa, where we have written

= ~i | 5, + ~" | 0..

(1.3.2)

We should check that D%0is well-defined. For the sake of simplicity (and because it is standard), here we write the proof for TA)/-valued forms only; the g e n e r i case is identical. Let ~o 6 X(AP T*/tT/| Tit:/) be expressed as in (1.3.2). It is then easy to see that

(~')~ = (J~a)~(~h)a

and

(~")B = (JBa)~(~')a,

and so we get

+ (--1)P(JSA)~(~k)A A ( J ~--1 ), h [(U;h)A(JBA) i

-

- 1 j 5 ,)~ (HB~)'~,dx'~]] | (Y~A)~(

= [d(~i)A + ( - - 1 ) ' ( ~ ) A ^ (W~)A] @ (~j)~. An analogous computation works for the vertici components, and so D%o is a globally well-defined T M - v i u e d (p + 1)-form. Finally, we explicitly remark that

D(f%o) = df A %o+ fD%o for all f E C~

21 1.3.2.

The torsion

DEFINITION 1.3.3: The tangent bundle T M (and hence _~/) is naturally equipped with a 1-form, the canonical form rl = dx i | Oi -[- du" | Oa 6 X ( T * I~/I | T~/I). r/is a well-defined T2tT/-valued global 1-form on iV/, as easily checked using (1.1.3) and (1.1.4). Now let D: X ( V ) ~ X(T*_f/I | V) be a good vertical connection, and let D also denote the associated linear connection on M. Then it is easy to see that = dx i | ~i + r

| ha;

in particular, yox=

and

71ot=t

X.

DEFINITION 1.3.4: The torsion 0 E X(A 2 T*/~7/| TJV/) of a good vertical connection D is the Tlt~/-valued 2-form 0 = D~ = ( - d x j hw~) |

+ (de a - cb Aw~) | 0,

= O' | ~, + #a | b. c X(A2T*~/| The/), where D is the exterior differential defined in (1.3.1). Locally, the Th~/-valued 2-form 0 is given by 0 i = F}; h d x h A d x j -k F jic r

1

i

i

c

A dx j

i r

= ~[Fj; h - Fh;j] dx h A dx j + Fjc ha

=

A dxi;

(1.3.3)

1 a a i " a a b i 1 a a b c ~[%(r,)-~,(r i)]dxi A d x +[0~(r,)-rb;,]r ^dx +~[r~b-r~o]r ^r (1.3.4)

The relationship between the torsion and the covariant derivative is the usual one: PROPOSITION 1.3.1: Let D: X ( T M ) ~ X ( T*.f/I | T.f/I) be the linear connection on 1~I induced by a good vertical connection. Then for any X , Y E 2d(T1VI) we have V x Y - V v X = IX, Y] + O(X, Y ) .

Proof: Write X = x i 6 i + XaO, and Y = YJ6j + Y'bOb. Then V x Y = [X(Y i) + wj(x)YJ]6i + [X(I~") + w~(X)yb]o,; V y X = [Y( X i) + w j ( r ) x J ] s i + Iv()(a) + w~(Y)Xb]ba;

e(x, Y) = [ ~ ( x ) v J - ~j(y)xJ]~, + [ ~ ( x ) ? b - ~ ( y ) . ~ b ] 0 o + [dr176 Y)]b.;

22

dr = X ( Y a) - Y(.,~") - r IX, Y] = x ( y i ) i f l - Y(Xi)tfi + r

Y]); y])cS.,

[]

where we used (1.1.20), and the assertion follows.

1.3.3.

The

curvature

DEFINITION 1.3.5: The curvature tensor R: X(T_~/I) ~ X ( A 2 T*.~/| T3~/) associated to a good vertical connection D is

R=DoD, that is

VX

9 X(T1VI)

Rx

=

D(DX).

We called R a tensor for a good reason: it is C~ g = g i ~ , E X(TI) and f E C~ we have

Indeed if we take

D(df | H) = D ( H i d f | ~i) = dH i A df | ~i - H i d f A w~ | 6i = - d r A (dHi § HJw~) | ~i = - d r A DH. Analogously one shows that D(df | X ) = - d r A D X for any X hence

R(fX) = D(D(fX))

= D(df|247

9 X(T_~I), and

= -dfADX§247

= fR(X).

Locally, R is given by

R( 6i) = D(w{ | 6i) = [dw{ - w .h, A w j] | 6j,

DEFINITION 1.3.6: Set

~)ba = dwba

_

b w ac A wc,

and define the curvature operator ~ E X ( A 2 T*.~/| T*.~/| TJtT/) by

= ~ | [dz b | L + cb | 0~]. The curvature operator i) is a global T*M| 2-form, that is gt(X, Y) is a global TJt~/-valued 1-form for any X , Y E X ( T M ) . Indeed, we have

f~(Z, Z ) Z = n z ( X , Y ) for any X, Y, Z E A'(T1Q), and so ~ is well-defined. The relationship between the curvature operator and the covariant derivative is as usual:

23 PROPOSITION 1.3.2: Let D: X ( T JVI) ~ X ( T* ~I | T IQ) be the linear connection on

induced by a good vertical connection. Then for any X , Y E X(TI(/I) we have

V x V y - V y V x = Vtx,Y] + fl(X, Y).

(1.3.5)

Proof: The assertion is proved as soon as we show that we get an identity both applying (1.3.5) to a vertical vector field V E X ( P ) and applying it to a horizontal vector field H E X(7/). So take V = Vac5= E X(I]); then

VxV =

IX(W)

~(x)v']&;

+

V x ( V y V ) = { X ( Y ( V ~) + w ~ ( r ) v b) + w'~(X)[Y(V') + w~(Y)V b] }Oa = { X ( Y ( V ~ ) ) + X(w'~(Y))V b + ~o~(Y)X(Vt) + w ~ ( X ) Y ( V ' ) + w~(X)w{(Y)Vb}ba; 7v(VxV) = {Y(X(Va)) + Y(w~(X))V b +r

+ w'~(Y)X(V ~) + w'~(Y)w~(X)Vb}b.;

7[x,v]V = { X ( Y ( V a ) ) - Y ( X ( V ~ ) ) +w~([X,Y])Vt}b=, and dw~(X, Y) = X ( w ~ ( r ) ) - Y ( w ~ ( X ) ) - o.,~'([X, Y]). So

V x V y V - Vy V x V - Vix,y] V = [ d ~ ( X , r ) - (w~ A w~ )( X, Y)]VbOa = ~(x,r)vbba

= ~(X,Y)V. lq

The same computation works for a horizontal vector field H E 2((7-/). We end this section recovering the Bianchi identities: PROPOSITION 1.3.3: Let D: X(T~/I) ~ X ( T * M | ]VI induced by a good vertical connection. Then

be the linear connection on

[ DO = q A f~, t Df~ = 0.

(,1.3.6)

Proof: Recalling (1.3.3) and (1.3.4) we get dOi + 0 j A wji = dx h A d w i h _ d x h A t J h9A w ji dO~ + ob A ~

= dx~ A f~i,

= - d e ~ ^ ~ + ~ ^ d ~ + d~ b ^ ~ - ~ ^ ~ ^ ~

= r

A f~,

24

a n d the first formula is proved. For the second one, (D~)~ = d~ =

+ ~ A

wea

A

+

-- f.d~A d~3ca

-

w~ h A

c -~- t.d b

Q ca +

Awc

d

A

-

A

A

a

Aw d

----0,

[]

a n d we are done.

1.4. The Cartan connection 1.4.1. Finsler metrics F i n a l l y we m a y start talking about Finsler metric. DEFINITION 1.4.1: A Finsler metric on a manifold M is a function F : T M --+ R + satisfying the following properties: (a) (b) (c) (d)

G = F 2 is smooth on AT/; f ( u ) > 0 for all u 9 37I; F ( # ~ ( u ) ) = ]A[F(u) for all u 9 T M and A 9 R; for any p 9 M the indicatrix IF(p) = {u 9 T p M [ F ( u ) < 1} is strongly convex.

A manifold M endowed with a Finsler manifold will be called a Finsler manifold. A couple of comments are in order. First of all, if (gab) is a Riemarmian metric on M , it is clear t h a t F : T M ~ R + given by

Vu 9 T p M

F ( u ) = (gab(p)uaub) 1/2

is a Finsler metric; we shall say t h a t F comes from a Riemanrdan met~c. In this case, though, G -- F 2 is smooth on the whole T M , and not only on M as required in (a). As we shall see in a moment, this is not accidental: a Finsler metric F is s m o o t h on T M iff it comes from a Pdemannian metric. Condition (a) is t h e reason b e h i n d t h e somewhat anomalous smoothness assumptions in the previous sections. T h e easiest example of Finsler metric not coming from a Riemannian metric is described in the following definition: DEFINITION 1.4.2: A t e a / M i n k o w s k i space is given by R m endowed with the Finsler metric F : R 'n x R 'n ~ T R m ~ R + defined by

Vp E R "~ Vu 9 TpR m -~ R'*

r ( p ; u ) = Ilull,

where ][. [[: R m --* R + is a norm with strongly convex unit ball on R m. If [[-[[ is not the n o r m associated to a scalar product, then F does not come from a R i e m a n n i a n metric.

25 A Pdemannian structure on M is obtained defining an inner product on any tangent space T p M varying smoothly with p. A Finsler structure, on the other hand, is obtained defining a norm on any T p M varying smoothly with p; this is the content of conditions (a), (b) and (c). As we shall see in the next section, condition (c) will allow the measurement of length of curves on the manifold - - and then to define geodesics. Finally, condition (d) is slightly stronger than convexity of the norm on each TpM. To be precise, we axe requiring that for every u E hTf the Hessian (with respect to the vector variables) of F 2 is positive definite. In symbols, we require that W 9M

(Gob(U)) > 0,

where (here and in the rest of the book) G = F 2 and subscripts denote derivatives. We shall use a semi-colon to distinguish between derivatives with respect to the point variables and derivatives with respect to the vector variables; for example, OG G;i = Oz i,

OG Ga = Ou a,

02G Ga;~ = OuaOzi,

and so on. The main (and almost unique) property of the function G is its homogeneity, that is V(p; Au) = A2V(p; u) (1.4.1) for all p 9 M , u 9 T p M and A 9 R. Differentiating with respect to A and setting A = 1 we get V(p; u) 9 T M

ea(p; u)u a = 2G(p; u).

(1.4.2)

On the other hand, differentiating (1.4.1) with respect to u a we get Ga(p; Au) = AGa(p; u);

(1.4.3)

differentiating again with respect to A and setting A = 1 we get Gab(p; u)u b = Va(p; u)

and

Gab(p; u)uau b = 2V(p; u).

(1.4.4)

Differentiating another time (1.4.3) with respect to u b we obtain Gab(p; Au) = Gab(p; u),

(1.4.5)

and hence Gabr

u)u c = 0.

(1.4.6)

This is enough to prove the claim about the smoothness of G: LEMMA 1.4.1: Let F : T M ~ R + be a Finsler metric on a manifold M . G = F 2 is smooth on T M itf F comes from a Riemannian metric on M .

Then

Proof: One direction is clear. Conversely, assume G smooth (C 2 is enough) on T M . Take p E M and u E TpM, u ~ op; then (1.4.4) yields 2G(p; u) = Gab(p; u ) u % b.

26 Take t > 0; by (1.4.5) Gab(p;tu) = Gab(p;u). Hence 2G(p; u) = Gab(P;tu)uau b.

Vt > 0

Letting t go to zero, by smoothness we obtain 2G(p; u) = Gab(p; % )u au b. It is then easy to check that setting gab(P) = 89 metric on T M , whose associated norm is exactly F.

op) one gets a Riemanniaa []

A consequence of condition (d) is that the Hessian matrix (Gab) is invertible. Later on we shall need derivatives of its inverse (Gab); they are given by the following formula: O(G ab) = -GarGsbO( Grs), (1.4.7) where 0 here denotes any first-order differential operator.

1.4.2. T h e C a r t a n c o n n e c t i o n The aim of this section is to associate to any Finsler metric a good vertical connection which is, in some sense, the generalization of the Levi-Civita connection. The first observation is that condition (d) allows us to introduce a Riemannian structure on V, by setting

VV, W e ]}u

(V I W)u = 89

b.

It is easy to check that ([) is well-defined, and it actually is a Riemannian metric on V. Note that (1.4.4) implies that G - (t It>.

(1.4.8)

In other words, embedding A~/in V by means of t we recover the Finsler metric. DEFINITION 1.4.3: The Riemannian structure ([) on V so defined is said to be in-

duced by the Finsler metric. The main result of this section is the construction of the good vertical connection associated to the Riemannian structure induced by F. THEOREM 1.4.2: Let F: T M ---* g(+ be a Finsler metric, and ([) the Riemannian structure on V induced by F. Then there is a unique vertical connection D: X(V) --* X(T*)(/I | V) such that (i) D is good; (ii) for all X E T)(I and V, W C X(V) one has

X ( V [W) = ( V x V [ W I + (V [ VxW);

(1.4.9)

27 (iii) O(V, W) = 0 for all V, W C V, where 0 is the torsion of the linear connection

on ~I induced by D; (iv) O(H, K) 9 V for all H, K 9 7"l.

Proof: Assume such a connection exists; we shall recover the connection forms w~, showing its uniqueness. First of all, (1.4.9) yields

= 2(w)(ba)0a t c~) + 2(c~r I wk(c~,)Vhk) = Gh~r)a + G~hF~a. Analogously we find a.o

= a,

rL +

Now, by (1.3.4) and (1.3.3) condition (iii) is equivalent to

so we obtain

P)~ = ~aGhk~ Lrrsk~

(1.4.10)

h r where (G hk) denotes the inverse matrix of (Ghk). In particular, Frsu = 0; so AIv = id Iv (where we recall that A(X) = Vxt), and (i) turns out to be a consequence of (ii) and (iii). Using again (ii) we get

~i(a,,) = 26i(c5~ 10,) = 2(xYr,b~ I ch~) + 2(c5~ I V e,0,) = Gh,F~; i + G~hP~;i. Analogously, ~r(G~i) = Ghir~;r + G~hF~;

6~(Gi~) = Gh~I'/h;, + GihI');~. Now, by (1.3.3) and (1.3.4) condition (iv) is equivalent to rh.

t;3 =

h

~j;i;

(1.4.11)

hence we get t;3 =

2V

(1.4.12)

where This is not enough to determine uniquely the connection, because the coefficients F~ are still unknown. To recover them, we contract with t, that is we compute the

28 following:

,,/~u j

l ~ h k [Gk,;ju j + Gk;i - Gi;k]

and

-~,~,J

= a h~ [ a ~ ; y

-

a;k],

where we used (1.4.1)-(1.4.5). Therefore Lemma 1.2.2 now yields -- l jkt

(1.4.13)

i U ,

and so Thus we get

F~ = 89 hk [Gkj;iu' + Gk;j - Gj;k] _ FjkGh k, [Gz;,u'

- G.t], .

(1.4.14)

Hence we have determined the coefficients F~ and so, by (1.4.12), the connection

forms w~. For the existence, a long but straightforward computation shows that, under change of coordinates, the coefficients F~r given by (1.4.10) behave as in (1.2.3), the coefficients F~ given by (1.4.14) behave as in (1.1.19), and the coefficients F b;i ~ given by (1.4.12) behave as in (1.2.3). Therefore the forms

~

= FL d ~ ' + rt~r ~ = rt;~ d~' + rtc du ~,

are the connection forms of a good vertical connection satisfying (i)-(iv), and we are done. Q DEFINITION 1.4.4: The good vertical connection whose existence we have just proved is the Cartan connection associated to the Finsler metric F. This is the Cartan connection introduced (in local coordinates) by Cartan and usually studied in classical Finsler geometry. Indeed, an easy computation shows that = so

--2FkrTijU

u

,

setting Gh ---- ~%ju , h ~u ---- ~l C-~k [ak;iu ' - G;k] , _

from (1.4.13) we get

r~ = 6,(ch); this is the way the Caftan connection is usually presented. It should be remarked that the Cartan connection is not the only linear connection on T.~/canonically associated to a Finsler metric; we just recall the Berwald connection (see [Rdl], [B]) and the connection recently defined by Sao and Chern (see [BC]). It would be interesting to know whether it is possible to give a global characterization for these connections similar to the one we just presented for the Cartan connection.

29 Coming back to our main concern, it is worthwhile to remark that the non-linear connection associated to the Cartan connection is homogeneous. Indeed, (1.4.5) and (1.4.10) show that F~c(#x(u)) = A-1F~c(u), and hence the assertion follows from (1.4.14), (1.4.1), (1.4.3) and (1.1.17). In defining the Cartan connection, we asked only for the vanishing of part of the torsion. We cannot ask much more; for instance, we cannot ask for the vanishing of the whole horizontal part of the torsion without trivializing the theory. Indeed, PROPOSITION 1.4.3: Let D: X ( V ) --* X(T*i~I | 1)) be the Cartan connection associated to a Finsler metric F: T M -* •+ on a manifold M . Then 8 ( X , Y ) E "12 for all X , Y E TJ~I iff F comes from a Riemannian metr/c on M. In this case, the non-linear connection associated to the Cartan connection coincides with the Levi-Civita connection induced by the Riemannian metric. Proof: Indeed/9(X, Y) E ~ for all X, Y E T-~/iff P~c - 0 for all a, b, c = 1 , . . . , m, thanks to (1.3.3). This happens iff Gabc -- 0 iff G~,b(p; u) depends only on the point p and not on the vector u. But this is the case iff

G(p; u) = Cob(p)u ,, is a Riemannian metric. Finally, if F comes from a Riemannian metric then (1.4.12) yields Fb. and the final assertion follows from Proposition 1.2.3.

i

So let D be the Caftan connection associated to a Finsler metric F. Since D is good, we have a horizontal bundle 7"( and a horizontal m a p O: Y --~ 7"/. Using O we m a y transfer the Riemannian structure from ~ to 7"/, by setting VII, K E 7-(

(H[K) = (O-I(H) [ o-l(g)).

We can then define a Riemannian metric on the whole ThT/, just by stating that 7"l is orthogonal to ~, that is X/H e ?/

WE1;

(HIV)=O.

We have already seen how to extend a good connection to a linear connection on T.AT/. It is then easy to check that the definitions imply

X(YIZ) = (VxYlZ)+(YlVxZ) for all X, Y, Z E X(T-~7/). Note that D is not the Levi-Civita connection associated to this Pdemannian structure on .~/, because its torsion is not identically zero. But, on the other hand, the existence and the definition of this Riemannian structure depends on the Caftan connection; so it is sensible to say that the Caftan connection is a good generalization to the Finsler situation of the Levi-Civita connection.

30

1.4.3. T h e h o r i z o n t a l flag c u r v a t u r e As we shall see more clearly in the next section, the aim of our approach to Finsler geometry is to provide a setting where, broadly speaking, the arguments used in Riemannian geometry can be used in Finsler geometry just rephrasing them in terms of the horizontal bundle 7-/and of the radial horizontal field X. As a consequence, we shall not be interested in the whole curvature tensor, but only in a particular contraction of its horizontal part, which we now define. DEFINITION 1.4.5: Take u 9 .~/; then the horizontal flag curvature at u is the bilinear form R~: 7"/~ x ~ u ~ R given by

VH, K 9 TI~

R~(H, K ) = <~(X(u), H ) K I x(u))~.

To work with R= we need at least to know that it is symmetric. This is the content of the following

PROPOSITION 1.4.4: Let F: T M ~ •+ be a Finsler metric on a manifold M , and D the Cartan connection associated to F. Then: (i) for all X , Y E T2VI we have

(1.4.15)

(O(X, Y) It) = 0; (ii) for any u E M the horizontal flag curvature R~ is symmetric, that is

v g , g e 7-l

(s

, g)g

[ x ) = (~2(X, K)H [x ).

Proof: (i) By (1.3.4) and (1.4.4) we see that (1.4.15) is equivalent to

=

and

a 0b(r ) = a a r L .

Now, (1.4.4) also implies GaG ab = ub; hence (1.4.6), (1.4.10) and (1.4.14) show that G,F~r = 0

and

G~F~' = G;i.

(1.4.16)

Therefore Go6,(U)

:

~,(G~U)-5,(G~)F; a

a

:

5,(a;j) a

- - " ~~ jGa;i

"4- r ' j.r i G abb

b

= G;ji - F i G~;j - FjG~;i + FjFiGab = Gagj(F~). Fhrthermore, Ga0b(F~) = 0b(G~rT) -- G~bF~ = Gb;i

-

-

l f~

G~bF~

c = l[Gb;i + Gi;b -- Gbi;lu l] + ~ i b c t l~ U

and (1.4.15) foUows.

I

= GaFF;i,

31 (ii) First of all,

w~(g) = F ab;s H ~

and

w~(X)

=

a u ~ = ~s;b ~ us ~b;s

F~,

=

where we used (1.4.11) and L e m m a 1.2.2. This yields dw~(X, H ) = [6d(r~;,)u' - 6s(r~) - r~r~;o - r~c(~d(r~) - ~ , ( r ~ ) ) u d] H ' ,

a n d so

(D(X, g)z, X) = G~ [dw~ - w~ A w~] (X, H ) g ~ = G. [6d(F~;s)ud -- 6~(F~) - F~F~;r - F~F~;. + F~F~;~] K ~ H ~ =

Go

-

-

r

r:;o

-

+ r~r~;~]K'S 9

= (~2(X, g ) g , X), where we used (1.4.11) and part (i), and the assertion follows.

[]

DEFINITION 1.4.6: So we may safely say that a Finsler metric F has positive (or negative) curvature if the horizontal flag curvature Ru is positive (resp., negative) definite for any u 9 hS/. By the way, it is easy to check that

VII, g 9 7"lu Ru~(u)(d(t~a)~(H), d(#a)~(K)) = A2 R , ( H , It'). As we shall see in the next section, the horizontal flag curvature is the correct contraction of the curvature operator to consider, because it is the one appearing in the second variation formula. This is a tipical instance of the phenomenon we described in the preface: it is easier to define and manipulate formally objects on the tangent-tangent level, but they are geometrically meaningful only when brought by contractions at least half-way down toward the tangent level. We end this section by proving that the curvature operator is antisymmetric: PROPOSITION 1.4.5: Let D: X(T2~I) ~ X(T*I~I | TI~I) be the linear connection on M induced by a good vertical connection. Then for any X , Y, Z, W 6 T ~ f we have

< n ( x , v ) z l w) = - ( Z l n ( X , Y ) W ) = - < e ( x , y ) w In particular, (~(X, Y ) Z [ Z) = 0 for all X , Y, Z 6 TI(/I.

l z>.

32

Proof: Indeed, Proposition 1.3.2 yields (ft(X,Y)Z l W) = (VxVYZ I W) - ( v y V x z I w) - (Vtx,YIZ I W) = x ( v y z I w ) - (v~.z I VxW) - Y ( V x Z I W) + ( V x Z I VyW)

- [ X , YI(Z I W) + (Z l V[x,ylW) = X ( Y ( Z l W ) ) - X ( Z I V y W ) - (VyZ I V x W ) - Y ( X ( Z l W)) + Y(Z I VxW) + (VxZ I VyW) - [X, Y l ( Z l W ) + (ZlV[x,ylW) = -(ZlVxVyW)

+ (ZlVyVxW)

+ (ZlVtx,ylw)

= -(ZI~(X,Y)W). []

1.5.

First

and

second

variations

1.5.1. The setting So far we have introduced a lot of forrnM objects more or less associated to a Finsler metric, claiming that we shall need them to study the geometry of a Finsler manifold; in this section we start doing exactly this. As mentioned before, a Finsler metric can be used to measure the length of curves.

DEFINITION 1.5.1: A regular curve a: [a, b] --* M is a C L c u r v e such that

v t e [a, b]

~(t) = d~,

~

# 0.

The length, with respect to the Finsler metric F : T M --~ R +, of the regular curve a is then given by 1" b

L(a) = ] F(&(t)) dt. Ja It is clear that, thanks to condition (c), the length of a curve does not depend on the parametrization; moreover, if a is a regular curve we can always assume that, up to a reparametrization, F(&) is constant. Needless to say, we can also measure the length of piecewise regular curves, just by adding the lengths of the regular pieces.

33 A geodesic for the Finsler metric F is a curve which is a critical point of the length functional. To be more precise: DEFINITION 1.5.2: Let a0:[a,b] ~ M be a regular curve with F(h0) = co. A regular variation of a0 is a C < m a p E : ( - e , e ) • [a,b] ---* M such that (i) ao(t) = E(0,t) for all t e [a,b]; (ii) for every s e ( - e , e) the curve as(t) = E(s, t) is a regular curve in M; (iii) r ( a , ) -= c, > 0 for every s 9 ( - e , e). A regular variation E is fixed if it moreover satisfies (iv) a,(a) = ao(a) and a~(b) = ao(b) for all s 9 ( - e , e). If E is a regular variation of a0, we define the function ~ : ( - e , e ) ~ R + by

is(s)

=

n(as).

DEFINITION 1.5.3: We shall say that a regular curve a0 is a geodesic for F iff dez

(0) = 0

for all fixed regular variations E of a0. Our first aim is to write the first variation of the length functional; we strall then find the differential equation satisfied by the geodesics, and we shall show that every geodesic for F is a geodesic for the non-hnear connection associated to the Cartan connection, and conversely. To write the first variation formula, we shall need to pull-back the Cartan connection along a curve. Unfortunately, the Cartan connection does not live on the tangent bundle, but on T_/17/; for this reason the pull-back procedure we shall presently describe is a bit more involved than the usual one in Pdemannian geometry. Let E: ( - e , e ) x [a, b] ---* M be a regular variation of a curve a0: [a, b] --~ M. Let

p: E*(TM) ~ (-e, e) x [a, b] be the pull-back bundle, and let 7: E*(TM) ---* T M be the fiber map such that the diagram

E*(TM)

~

TM

[a,b]

M commutes; 7 is the map identifying each E*(TM)(s,t) with T~(s,t)M. In particular, a local frame for E*(TM) is given by the local fields x

0

0

Oz ~ (s,,)='r-l(-b~z~r~(s,,)) '

34 for i = 1 , . . . ,m. So an element ( e X ( E * ( T M ) )

can

be written locally as

(,,o"

~(~,t) = u~(~,t) o

Accordingly, a local frame on T ( E * ( T M ) ) is then given by {O,,Ot, Oi}, where O, = O/Os, Ot = O/Ot and Oi = O/Ou i. Two particularly important sections of E*(TM) are T = 7 -1 and U = 7 -1

dE

-

((0)) dE

-~s

Ot Ox i'

0

,o

- - O s Ox i"

DEFINITION 1.5.4: The section U is the transversal vector of E.

We remark explicitly that, setting E*JI~/ = 7-1(.~r), by assumption we have T E X(E*37/); furthermore,

T(s,t)

= 7 - ' (#,(t)).

(1.5.1)

Now we may pull-back T.~ r over E*hT/by using % obtaining the commutative diagram 7,(T~r) :~ TM -----,

( - e , e) • [a, b]

--~

M

In other words, for any u E E*.Qr(s,0 = 7-a(h~/z(s,t)) the map "~ identifies 7*(T_~r)~ with TT(~)./~r. In the previous section we showed how to define a Riemannian structure on Tgtir. This induces a Riemannian structure on 7*(T)~r) by VX, Y E 7*(Th:/)u (X I Y ) , = (5(X) 15(Y))~(,). Analogously, on T . ~ / w e have the Cartan connection D; this induces a linear connection D*: X(,y*(T.~-I)) ---+X(T*(E*.Qr) | 7*(T)tT/)) by setting

v ~ y = ~-1 (vd~(x)~(Y)) for all X E T(E*A]r) and Y E X(7*(ThT/)). In particular,

x(Y[z)

= (v;,Y I z) + (Y I v;~z)

35 for all X e T(~*.A~/) and Y, Z 9 X(r*(T2~/)). Now take u 9 (~*]~/)(s,,); then d%(T(~*_~/)) C T.f(,>~/

and

~(7*(T~/)~) = T~<~).h:/.

Therefore we get a fiber map --: T(~*/Y/) -~ 7*(Th:/) such that the diagram

T ( E *_~I)

~=-

7 *( T-~I )

TM

commutes, that is ~ o E = dr. Since

OEi 0

d.r(O,) = - - ~

O~'oi

,,

dr(OA = Os

and

d r ( b , ) = 3~,

it follows that dT(T(E*AT/)) D V. Moreover, setting V* = kerdp, where p is the canonical projection of E*]~/onto ( - e , ~) x [a, b], it is clear that d 7 is an isomorphism between V* and 1). Therefore setting =

-1

n

we have decomposed T(E*]IT/) as T(E*hT/) = U* @ Y*. A local frame for 7"/* is given by

OY~i "

5, = a, - (r~ o r ) - ~ - a a ,

so that, setting T H = dr(6t) and

U H --~

" OEi b

~ = as - (r~ o r)-b- 7

a,

d7(68), we find

TH(u) = - ~ - ( s , t ) S i l ~ ( , ) = X-r(~)(r(T(s,t))) = X~(=)(&~(t)) e 7-/~(,), and

UH(u) = O:Ci "s, for all u E 2*~/(s,t). So T H and U H are the horizontal lifts of 7(T) and 7(U) respectively. Moreover, they are tangent horizontal vector fields along 7, and

T H (T(s, t)) = X(&~(t)).

(1.5.2)

We end this preparatory subsection with two final formulas: ~(V~cE(Y) - V~.E(X)) = Vd-y(x)dr(Y) - Vd~(y)dr(X) = [dr(X), dr(Y)] + O(dr(X), d r ( Y ) )

= d r [ X , Y ] + O(dT(X),dr(Y)),

(1.5.3)

36 and "~ o (V~-V~

-- V~,V*X

-- V[*X,y] ) ---- (VdT(X)~d~.(y)

-- ~ d T ( y ) ~ T d T ( X )

-- VdT[X,y])

o 'T

= ~(d~f(X), dT(Y)) o ;/,

(1.5.4) for all X , Y e X ( T ( 2 * , V I ) ) .

1.5.2. T h e first variation of the l e n g t h integral

We are now able to prove the first variation formula for Finsler metrics: THEOREM 1.5.1: Let F : T M -+ ~(+ be a Finsler metric on a manifold M . Take a regular curve a0: [a, b] ~ M , with F(#o) - co > O, and let ~: ( - e , e) x [a, b] ---* M be a regular variation of Cro. Then

den(O) =

I

~o

{ (U. iT,~>oolb z' _


}

In particular, if the variation T1,is fixed we have ds (0) = Proof: By definition,

( u H I V T , T~)~odt.

en(8)=/b(a(~,)),/2 dr;

therefore de~ -ds -

_

d8

C(#,))

1/2

dt =

__1 [ b A [ v ( & , ) ] d ~ = = 2c, J~ Os

C(~,

1

1/2

dt

b a _

~ / "~3(~(~t)lE(6t))Tdg'

where cs - F(6s), which is constant by definition of regular variation, and we used V(~,) = (X(#,) l x(~,))~, = ( ~ - l ( x ( ~ , ) ) I ~ - I ( x ( ~ , ) ) > T = (--(6t) I E(6t))T, which holds by (1.5.1) and (1.5.2). Now, lifting ( E ( 6 t ) ] E ( 6 t ) ) T to E*.~/in the obvious way, we get a function which does not depend on the vector variables; therefore 10 6 ~(-=( ,)l~(~,))r= ~,(~(a,)l~(~,)>r= r = ( V ; E ( 6 , ) I E ( 6 , ) ) T - (E([6t,6s])l--(6t))T -- (8(T H, u H ) ITH)~o,

37 by (1.5.3). Now, 8(T H, V H) 9];~, because we are working with the Cartan connection, and hence it is orthogonal to T H. Analogously, [Tt, 6s] 9 V* and so E([T,, 68]) is orthogonal to E(Tt). Thus

= 6t(E(Ts)[E(Tt))T - -

<E(Ts)]V; E(~t))T

(1.5.5)

= ~ ( U H [TH>~o --
up we have found

d2~.(s)=

1


< u H I V T , TH)~ dt

and the assertion follows.

[]

As a corollary we get the equation of geodesics: COROLLARY 1.5.2: Let F: T M --* R + be a Finsler metric on a manifold M , and a: [a, b] ---, M a regu/ar curve. Then a is a geodesic for F if[

VT.

T g -

O,

where TH(u) = Xu(&(t)) 6 7Q for a/l u 9 ~/~(,). Proof: T h e curve a is a geodesic if[(dg~/ds)(O) = 0 for any fixed regular variation ~,

of a. Since V, is fixed, we have that a is a geodesic iff V T n T

uH(o, a) = uH(o, b) = 0. By Theorem 1.5.1, it follows H =

O.

[]

In local coordinates, V T , T ~ = ~J [6~(~' o ~) + ( r L o ~ ) ~ ] 6 , = [~' + (r~ o ~)~J]6,,

(1.5.6)

and so we have proved

COROLLARY 1.5.3: Let F: T M ~ R + be a Finsler metric on a manifold M . Then a regular curve a: [a, b] ---, M is a geodesic for F if[it is a geodesic for the non-linear connection associated to the Carman connection if[ =

Proof: It follows from (1.5.6), (1.1.21) and (1.1.22).

[]

In particular, we have shown that a curve a is a geodesic for F iff & is an integral curve of the radial horizontal vector field X.

38

1.5.3. The second variation of the length integral Our final aim for this section is the second variation formula. It is contained in THEOREM 1.5.4: Let F: T M ~ R + be a Finsler metric on a manifold M. Take a geodesic a0: [a, b] ~ M , with F(&0) - 1, and let ~: ( - E , r x [a, b] ~ M be a regular variation of ao. Then

d2s (0)=" ds 2 A-~

[. TH)~,o] . ~ ][VTHUHll2o--(~(TH, u H ) u H I T H ) ~ o -

~(UH]TH)o o

dr,

where [[g[[ 2 = ( g [g ) u for all u e ]~I and g e TI,. In particular, if the variation is fixed we have

ds 2 (0) =

]IVT. UHI[2 o - ( a ( T H, u H ) U H [ T H ) a o --

(UR[TH)a o

dr.

Proof: During the proof of the first variation formula we saw t h a t dg~

fab

T h e integrand is a function defined over ( - ~ , c) x [a, b]; we m a y lift it over ~'21~/ and then compute

o [ (v; ~(_~,_.)[~(,~,)):r ] ,~,(V;,:%~,)I=(,~,))T = ((-=(,~,) I :=(,~,)):r)'/~

[ (~';,:=(,~,) I ~(,~,)):r ] 1 (V;--=(,~.)I--=(,~,))T - 2 (('=(,~,) I s(,~,))~rp/~ '~'(-=('~') I:=('~')):r"

Now we already saw in (1.5.5) t h a t ls,('~(St) l'Z(St))T =

I (v;,-=(,~.) I -=(,~d)T 2

(v~ _=(z,) I-=(6,))T; hence

I (V$ :=(,~.) I --=(,~,))T[ ~

((,~(,~,) I S(,~,))T)~/~ '~A=('~') I--=('~'))T = ((_=(,~,) I S(,~,))T)~/~ ((S(,~,) I :=(,~,)) T)3/~ 0 ~ I T')~,. [-~(V

_

(UHI VT.T~)~.[~

( (T H [ TH)~,,) 3/2 Furthermore

,~.(v;,~(,~.) I -=(,~,)):r = (%.%,-(,~.) I --(,~,)):r + (v;,--(,~.) I V;.-=('~,))T * = ( ~V,*v ; . ~ ( , ~ , ) I:=('~,))T-- (VI~,.~.]S(,~.) I--=(,~,))T (a(T~,U~)U" I T~)~," + (V;,S(,~.) I V;,=-(,~.)):r - (v; _=(~.) I _=([~,,~.]))T ( V T , , U H ] o ( T H , uH))~,., -

-

-

39 by (1.5.3) and (1.5.4). Now: V~ S(6,) and VTHU H are horizontal; O(T H, V H) and E([6t,/~]) = [T H, U H] are vertical; furthermore for any V e 12" one has (V~,E($,) I E(~,))T -- (Vd~,(v)UHiTH)~.

= v , b ( ~ , ) v ~ oo \ o~ ~ p + r~r

\--~; o p

~,

o \{- or,~ -~- o p/=o,

r by (1.4.16). Therefore

~,(v;, =.(6,) I -=(6,)) T =

( V ; , V ; E(6,) [ E(6t))T -- (f~(TH, uH)uHITH)a" + (V; E(,~:) I V ; E(~i,))T

= ~,(v;.-=(~,) I z(~,))r - ( v ; . z ( ~ , ) I v ; , z ( ~ , ) ) ~ - (~(T~,U~)U'iT'I~,

- 6 ,)lit2 + [PV *~,-'(

---- .-~(Vu~UH [ TH)b, -- (VuHU H I VT~TH)~,. _ (~'I(TH, uH)uH[TH)~, + IIVT, U g 2.

O" 8

9

Recalling the hypotheses on a, that is V T , T H = 0 and ]ITHtl = 1 for s = 0, we finally get the assertion. [] We remark that, by (1.5.2),

(~(T H, u H ) u H [ TH)~o = (~'~(X(Oo),UH)U H [ X(O'0)/&0 = R~o (U H, uH). So, as announced at the end of the previous section, from the point of view of variation formulas the only geometrically meaningful curvature is the horizontal flag curvature.

1.6. T h e e x p o n e n t i a l m a p 1.6.1. Parallel transport Let a: [0, a] ~ M be a regular curve, and ~ a vector field along a. In subsection 1.1.4 we said that ~ is parallel if V~,~ -- 0, where V is the non-linear connection associated to the Cartan connection. Recalling Proposition 1.2.3 we find that ~ is parallel iff

VT,~H -- o, that is, by Lemma 1.1.5, iff

= ~(~).

~

(1.6.1)

40 In local coordinates, (1.6.1) is expressed by ~a = _F~(~)&b

for a = 1 , . . . , m ;

therefore the Cauchy problem VT"(H

0, -

(1.6.2)

~(0) = ~0 9 M~(0), always a d m i t s a unique solution ~: [O,c) --- .M. A priori, e could depend on ~0; however, m o m e n t a r i l y we shall see that this is not the case. Indeed, we know t h a t G(~) = (~g, ( g ) e ; so if ~ is parallel one has d

d_~(G(~) ) = T H ( ( ( H , ~H)e) = 2 ( V T x ( H , ~H)e = 0.

(1.6.3)

So the length of a parallel vector field is constant. Being the Christoffel symbols of the C a f t a n connection homogeneous, it follows t h a t e could be chosen independent of ~0 - - a n d depending continuously on a(0). Since [O, a] is a compact interval, the uniqueness of the solution of the Cauchy problem (1.6.2) implies that we can actually choose ~ = a. Summing up, we have proved most of the following PROPOSITION 1.6.1: Let a: [0, a] --- M be a r e g ~ a r curve on a Finsler manifold M , with a(O) = p and or(a) = q. Define a m a p A~: Mp --* .~JIq by setting A,,(u) = ~(a), where ~ is the unique parallel vector field along a such that ~(0) = u. Then A# is a well-deigned bijective homogeneous isometry between f/lp and f/lq.

Proof: We are only left to show that A~ is bijective. Let a _ : [0, a] ~ M be the curve defined by a _ ( t ) = a(a - t), so that a_(O) = q, a _ ( a ) = p and &_(t) = - & ( a - t) for all t 9 [0, a]. Take u 9 _h~/p, and let f be the parallel vector field along a such t h a t ~(0) = u. If we set ~_(t) = ( ( a - t), then ~_ is clearly parallel along a _ ; hence A~,_ (Ac,(u)) = u. Analogously one proves t h a t A~ o A~_ is the identity. []

DEFINITION 1.6.1: curve

The m a p A# so defined is the parallel transport along the

or.

1.6.2. Geodesics

Having a way to measure the length of curves, we can define a distance on any Finsler manifold. DEFINITION 1.6.2: Let F : T M ~ R be a Finsler metric on a manifold M . Then the distance dR: M • M ~ R + induced by F on M is given by

Vp, q 9 M

dF(p,q) = inf{L(a)},

where the infimum is taken with respect to the piecewise regular curves a: [a, b] ~ M with a(a) = p and a(b) = q.

41 The proof that dF is a true distance inducing the manifold topology on M is identical to the corresponding one in Riemannian geometry. Now let us look to geodesics. In local coordinates, the geodesic equation is #a = _F~(&)&b,

for a = 1 , . . . , m ;

hence given p E M and u E /l~/p there exists a unique geodesic a: [0, ~] ~ M with a(0) = p and ~(0) = u. DEFINITION 1.6.3: Fix p E M; then the unique geodesic a , : [0,a] ---* M with a , ( 0 ) = p and #,(0) = u will be called the radial geodesic tangent to u E /17/p. If q = au(a), we shall say that a, connects p to q. A couple of properties of geodesics are trivial. First of all, we have already remarked in (1.6.3) that the length of the tangent vector to a geodesic is constant, that is F ( # . ) --- F ( # . ( 0 ) ) = F(u). (1.6.4) This readily implies that if or. is defined on [0, a] then

L(a,) = a F(u).

(1.6.5)

Furthermore, the non-linear connection associated to the Cartan connection is homogeneous; then, exactly as in the Riemannian case, one proves that

a,,(s) = a,(ts)

(1.6.6)

for all s, t E R such that one side (and hence the other too) is defined. Fix p E M , and denote by Sp the set of vector in h:/p of unitary length, i.e.,

= {u e Mp i F ( u ) = 1}. We know that there is r > 0 so that a,(t) is defined for all t e [0,e] and u E Sp; by (1.6.6), this implies that a~,(1) is defined. Setting aop(1) = p, we see that a , ( 1 ) is defined in a neighborhood of the origin in TpM; furthermore, it depends continuously on u, because

dF(a,(1),p) <_n(a,[[0,1]) = F(u),

(1.6.7)

by (1.6.5). DEFINITION 1.6.4: The exponential m a p expp from TpM into M is defined in a neighborhood of the origin by setting expp(u) = au(1). It is continuous where defined, and smooth outside the origin. As usual, we can patch together all these exponential maps getting the exponential map exp defined in a neighborhood of the zero section in TM.

42 The properties of expp are much the same as the ones enjoyed by its Riemannian analogue. To prove them, however, we need the notion of double of a differential manifold. Let M be a differentiable manifold of dimension m, and take p 6 M any point. If B is a small ball centered at p and M ' = M \ B, the double of M at p is the orientable manifold obtained gluing two copies of M ' along the boundaries; roughly speaking, we are separating the directions at p. More formally: DEFINITION 1.6.5: The double of M at p is a pair (JW/, rr) such that M is a differentiable manifold of dimension m, and rr: M + M is a proper smooth surjective map such that (a) 7r-l(p) = S is a compact submanifold of M diffeomorphic to the sphere Sin-l; (b) at any point ;5 6 S, and with respect to any system of coordinates near /7 on x~/and near p on M, the Jacobian determinant det(Orrh/cOxk) vanishes of order m - 1 at/$; (c) _AT/\S has exactly two connected components -~/1, -g3/2, and moreover the restrictions 7r: Mj --+ M \ {p} axe diffeomorphisms for j = 1, 2. These properties determine the double at a point up to diffeomorphisms commuting with the projection rr. Furthermore, the double may be described near p as follows. Let U be a coordinate neighborhood on M centered at p, B(e) C U the ball of radius e centered at p, and let V = zr-1 (B(~)). Let rr~ ( - ~ , ~) x S m-1 --~ B(~) be the map given by r~ b) = tb. Moreover, define rra:: B(r

\ {0} ---* ( - e , r

x S m-1

by

clearly, rr~

+=+id.

Then for e > 0 small enough there exists a diffeomorphism f : V + ( - e , e) x S 'n-1 such that = 0(six)), The map S is called a triviaJizathon of the double (.~/, re). For proofs and further details, see [St]. We need another remark. Take p 6 M; if jp: TpM --+ T M is the inclusion, then the differential d(jp)u: T,(TpM) --+ T~(TM) is a canonical isomorphism between Tu(TpM) and Yu for any u 6 TpM. So we can (and we shall) identify T(TpM) with 131TpM. In particular, in this way the maps ~=: TpM ~ l)u play the same role as the usual identification maps k~: TpM --+ T~(TpM) - - compare (1.1.5). Using the double we are now able to describe the properties of expp:

43 THEOREM 1.6.2: Let M be a Finsler manifold, and take p E M. Then the exponential map expp at p has the following properties: (i) the restriction of expp to any line through the origin in TpM is a (necessaxily smooth) geodesic; (ii) expp is continuous where defined, and smooth outside the origin; (iii) expp is Lipschitz in a neighborhood of the origin; (iv) expp is differentiable at the origin, and d(expv)o p = ~-1; (v) there is a neighborhood Uo of the origin in TpM such t~mt expp: Uo ~ expp(Uo) is a homeomorphism, and a diffeomorphism restricted to Uo \ {op} ; (vi) for any uo, u e I~lp the inapt ~ d(expp)tu (ttu(uo)) is smooth wherever defined.

Proof: (i) is true by definition and (1.6.6), and we have already remarked (ii). From now on, we shall work in a coordinate neighborhood U centered at p; in particular, we shall directly identify V~ -- T~(TpM) and TpM, disregarding all references to the identification maps tu. By (1.6.7), we can find r0 > 0 so that expp(tU) C U as soon as t e [0, r0] and u E Sp. Now the m a p t ~ expp(tu) = a~(t) is the solution of a ODE system with Lipschitz coefficients; therefore there is co > 0 such that

for Ul, u2 Z .~p and t E [0, r0], where I1" II is any fixed norm. In particular,

N o w t a k e Vl = t i l l ,

v2 = 1~2u2 9

TpM~w i t h

0 < t 1 __< t 2 __~ r 0 a n d Ul, u 2 ~ Sp.

Then II exp (v )

- exp (v )ll

_<

- o

-< coltllll

-

_< V (Co § w h e r e c 1 > 0 is independent of

uj and

tj

(t )ll

+

-

+ c ltl - t21

for j = 1, 2, and (iii) is proved.

Next, by definition, expp(tu) = tu + o(It[) for all u E Sp; since the remainder term is uniform in u, (iv) follows. To prove (v) and (vi), let (L, 7rl ) be the double of TpM at the origin, and (i~/, ~r2) the double of M at p. Let f: V ~ ( - e , e) x S m-1 be a trivialization of (j~r, 1r2) with respect to our chart U. Again, there is p > 0 such that expp(tU) 9 7r~ for all Pl < P and u E Sp. Define the map F: ( - p , p) X S m - 1 ~ ( - ~ , ~ ) x S m-1 b y setting

F(t, b) =

{

~r+[expv(~rl~

i f t > 0;

(0, b)

if t = 0;

~-[expp(~rl~

i f t < 0.

44 Now, F is s m o o t h everywhere, by (i) and (ii). Furthermore, dF(o,b) = id for every b 9 S ' n - l , by (iv); therefore F is a diffeomorphism in a neighborhood of {0} x S "*-1. Moreover, expp = r 2 o f -1 o F o g o T r ' l l, where g: V' --* ( - p , p ) x S n-1 is a trivialization of (L, Trl), and so (v) follows. Finally, expp O~rl o g-1 = 7r2 o f - 1 o F is smooth, and :rl o g-1 = 7r0; so d(expp)t~(u0) = d(exp v o~)(t,=tl,=/l=tD(ll~011, 0 u0) = d(~2 o f-~ o F)
[]

DEFINITION 1.6.6: A continuous m a p f : M ---* M I between two manifolds which is s m o o t h outside a point p 9 M , a homeomorhism in a neighborhood U of p and a diffeomorphism in U \ {p} will be called a local E-diffeomorphism at p; if moreover U = M we shall briefly say that f is an E-diffeomorphism at p. So expp is a local E-diffeomorphism at the origin. For r > 0 set

Bp(r) = {u 9 TpM I F(u ) < r},

Sp(r) = {u 9 2~Ip I F ( u ) = r};

in p a r t i c u l a r , / } p ( 1 ) is the f - i n d i c a t r i x at p, and Sp(1) = Sp. We shall moreover set

Bp(r) = expp(/~p(r)),

Sp(r) = e x p p ( S p ( r ) ) .

We shall see later on that for r > 0 small enough we have

Bp(r) = {q 9 M I d f ( p , q ) < r},

Sp(r) = {q 9 M I dF(p,q) : r}.

Using this terminology, Theorem 1.6.2.(v) says that there is a r > 0 so t h a t expp is a local E-diffeomorphism at op b e t w e e n / } p ( r ) and Bp(r). Then the following definition comes as no surprise: DEFINITION 1.6.7: The injectivity radius ir(p) of M at p is defined by ir(p) = sup{r > 0 I expp is a local E-diffeomorphism at % b e t w e e n / ~ p ( r ) and Bp(r)}.

45 1.6.3. T h e G a u s s l e m m a

Let M be a Finsler manifold, and fix p 9 M. For r < ir(p) we can define a vector field T over Bp(r) by 1 T(expp u) = F - - ~ d(expp)u(t(u)) 9 Texpp(u)M;

(1.6.8)

since we have identified T~(TpM) with V=, the vector field T is well-defined. Moreover, as we shall momentarily see, T(expp u) = ~

1

&,(1);

(1.6.9)

so T yields the unit tangent vector to the geodesics issuing from p, and thus it is sensible to denote it using the same letter used in section 1.5 for the tangent vector to a curve. Let us prove (1.6.9). LEMMA 1.6.3: Let F: T M -+ R + be a Finder metric, and t~x p 9 M. Then T(expp u) = &~(1) F(u) for all u 9 Y/Ip with F(u) < ir(p). In particular,

F(T) = 1.

(1.6.10)

Proof: A curve cr in .]l~/p with a(O) = u and a(O) = L(u) is given by ~(s) = u + su. Hence

T(expp u) = ~(u) 1 d(expp)u(t(u)) - F(u) 1 d {expp((1 + s)u) } ,=0 ds _

1

d

F(u) dt {expp(tu)

} ,=1

-

1

F(u)

d (o~(,)) ,=1

dt

e.(1)

= F(u)"

Since, by (1.6.4),

F(e~(1)) = r ( u ) , (1.6.10) follows.

[]

So t ~-, T(cr~(t)) is the unit tangent vector to the geodesic a~; in particular,

~7THTH = 0. Now take r < ir(p), u0 9 Sp(r), set q = expp(u0) 9 Sp(r) and take

u 9 Tq(Sp(~)) C TqoM.

(1.6.11)

46 Choose a curve 7 = expp -~: ( - c , ~) --* M in Sp(r) with 7(0) = q and q(0) = u, where "~ is a curve in Sp(r) with q(0) = u0. Now we define a variation of au0 by setting ~(s, t) = a~(s)(t ) = expp (t'~(s)), for t 9 [0, 1]. Since s Theorem 1.5.1 yields

= L(a:ds)) = F ( ~ ( s ) ) - r, the first variation formula

(UH [TH)T(q) = O, where U is the transversal vector, because a , 0 is a geodesic and

uH(o) =

o. But

U H (T(q)) = XT(q)(U(1)) = )lT(q)(U) = u H (T(q)), and we have proved a Finsler version of the classical Gauss lemma: PROPOSITION 1.6.4: Let F: T M ---+ R + be a Finsler metric, fix p E M and take r < Jr(p) and q e Sp(r). Then u 9 T q M belongs to Tq(Sp(r)) if[
1.6.4. Geodesics are locally minimizing Our next aim is to show that geodesics are locally minimizing. This can be proved by calculus of variations' arguments (see for instance [Rdl]); here instead we shall give a differential geometric proof, due to Bao and Chern (see [BC]). We begin with the following LEMMA 1.6.5: Let F: T M --* R + be a Finsler metric. Take p E M and uo E 2~lp. Then: (i) Fab(Uo)UaU b >_ 0 for all U E P~o, with equality iff U and ~(Uo) are linearly dependent; (ii) Fa(uo)u" <_ F ( u ) for all u E 1Vlp, with equality iff u and uo are linearly dependent; (iii) (t,,o(u) [ ~(u0)),o _< F ( u ) F ( u o ) for all u E IVlp, with equality if[ u and uo are linearly dependent. Proof: By assumption,

89176

b >_ 0

47 with equality iff U = O. Now, 1 G ~ = FFab + F~Fb; therefore

F,(uo)Fb(~o)V'Ub= (Fo(~0)U~ ~ =

G~176

1

- G~o) l
1

< c--(~0)IIuIli0a(u0)=

IIuIIi

= [r(uo)F=b(uo) + Fa(uo)Fb(uo)]U=U b, with equality iff U and e(u0) are linearly dependent, and (i) follows. Now, expanding F ( u ) at the second order near u0 (p is fixed) we get F ( u ) = F(uo) + F,,(uo)(U - uo) ~ + Fab(uo + r

- uo))(u - uo)=(u - uo) b,

for a suitable r 6 [0,1]. Therefore (i) yields F(uo) + F=(uo)(U - Uo) ~ < F(u), with equality iff u and u0 are linearly dependent. Since F~(uo)(u

-

uo) ~ = F~(uo)u ~

-

F(uo),

this yields (ii). Finally, (iii) follows from 1 1 F~(uo)u ~ - 2F(uo) G~(uo)ua = F(uo) (~,,o(u) [ t(uo)),,o 9 []

We can now prove the usual statement about the local minimizing property of geodesics: THEOREM 1.6.6: Let M be a Finsler manifold, and fix p 6 M and r < ir(p). Take a point q 6 Sp(r); let 7: [0, 1] ~ M be any piecewise smooth regular curve in M joining p to q, and let ao be the radial geodesic in Bp(r) connecting p to q. Then L(ao) <_ L(7), and the equality holds if[ 7 is a monotone reparametrization of ao. Proof: Take uo 6 ]lT/p such that expp(uo) = q; then ao(t) = expp(tuo). Assume first that the image of 7 is contained in Bp(r). Then we may write 7(s)

= expp

(t(s)u(s)),

48 where s 9 [0, 1], F(u(s)) =- F(uo) = r, t(O) = O, t(1) = 1, t(s) 9 [0, 1] and u(1) = u0. We now define a variation ~: [0, 1] x [0, 1] --+ M of a0 by setting = expp

so that ~(s) = ~ (s, t(s)); in particular

~(s) = U(s,t(s)) + T(s,t(s))~s,

(1.6.12)

where U is the transversal vector of E, and here T = 0 E / 0 t . Now, L e m m a 1.6.5.(iii) applied with u = +(s) and u0 = T(s,t(s)) yields L(7) -- ~01 F ( 7 ( s ) ; ~(s)) ds >_~01F(T(lt(s))) ( ~/(s)s ' TH(s,t(s)))T<s,t(8))ds. (1.6.13) The curves t ~-~ expp (tu(s)) are geodesics of constant length; so first of all

F(T(s,t(s))) = F(T(s, 0)) = F(u(s)) =_r. Furthermore, for each s 9 [0, 1] the vector V(so,t(so)) is tangent to the curve

s ~-* expp(t(so)U(S)) 9 Sp(t(so)r); by Proposition 1.6.4 (and the remark immediately after)

(uH(s,t(s)) l TH(s,t(s)))T(8,t(~)) = O. So (1.6.12) and (1.6.13) yield L(7) >_ lr Jofl(TS(s't(s)) dt l

= rl Joflr2dtds ds = r = L(ao),

with equality iff U(s,t(s)) - O, that is iff u(s) - u0, and so the equality holds iff 7 is a monotone repaxametrization of a0. Finally, if the image of 7 is not contained in Bp(r), let so 9 [0, 1] be the first s such that 7(s) 9 Sp(r). Then the previous argument yields 5(7I[0,~0]) -> r = L(a0); since L(7) > L(Tl[O,so]), we are done. [] As a corollary we have COROLLARY 1.6.7: Let M be a Finsler manifold, and fix p E M. Let r0 = Jr(p);

then Vu E Bp(ro)

dy(p, expp(u)) = F(u).

In particular, Bp(r)= {q E M [ dF(p,q) < r} for a/l r < ir(p).

and Sp(r) = {q E g [ dF(p,q) = r} =OBp(r)

49

Proof: Indeed, for every u

9

we have

dF(p, expp(u)) = L(a~l[0,1]) = F(u). []

Another consequence of Theorem 1.6.6 is: LEMMA 1.6.8: Let M be a Finsler manifold, fix p 6 M and take r < ir(p). q ~ Bp(r), then there exists q' 9 Sp(r) such that dF(p, q) = r + dF(q', q).

/t"

Proof: Let 7: [0, 1] --* M be a curve from p to q. Since q ~ Bp(r), there is a first So 9 (0,1] such that 7(s0) 9 Sp(r). Since, by Corollary 1.6.7, dF(P,7(So)) = r, we have L(7) > r + dF(7(so),q) >_ r + dF(Sp(r),q), where as usual

dF(S,(r),q)=

inf {dv(q',q)}. q' 6Sp(r)

Since 7 is generic, it follows that

dF(p, q) > r + dF(Sp(r), q). But by the triangle inequality the reverse inequality is true too, and so

F(p, q) =

+

F(sp(r), q). []

Since Sp(r) is compact, we are done.

We are now able to state and prove the classical Hopf-Rinow theorem for Finsler metrics: THEOREM 1.6.9: Let F: T M ~ R + be a Finsler metric on a manifold M .

Then

the following assertions are equivalent: (i) the distance dF is complete; (ii) dosed bounded subsets of M are compact; (iii) there is p 6 M such that expp is defned on all of TpM; (iv) exp is defined on all of T M . In addition, any of the previous statements implies that (v) for any p, q 6 M there is a geodesic a connecting p to q such that :

dF(p, q).

Proof: Thanks to Theorem 1.6.6 and Corollary 1.6.8 we can repeat word by word the proof of the Hopf-Rinow theorem for Riemannian manifolds described in ICE]. [-'l

5O DEFINITION 1.6.8: A Finsler metric F and the underlying Finsler manifold M are complete iff the induced distance dF is. T h e r e is a final consequence of Theorem 1.6.6 and of our version of Gauss' l e m m a t h a t shall be useful in the third chapter. Let F : T M ~ R + a Finsler metric, and fix p E M . Define p = pp: M --* R + by

pp(q) = dF(p, q). By Corollary 1.6.7, p = F o (expp)-i in B p ( r ) for any r < ir(p); in particular, it is smooth there (but possibly at p, of course). Now take r < ir(p) and q E Sp(r). Then a vector u E TqM belongs to Tq (Sp(r)) iff

0 = u(p) = dp(u); SO Proposition 1.6.4 says t h a t

dp(u) = 0

r

(uH I TH)T = O.

We can do b e t t e r t h a n this: PROPOSITION 1.6.10: Let F : T M --~ R + be a Finsler metric, fix p E M and take r < Jr(p) and q E Sp(r). Then for all u C TqM we have


(1.6.14)

Proof: If we write u = ua(O/Oxa), then u H = u~a; but 6~(po~r) = Op/c3x% a n d so uH(p o 7r) = u(p) = dp(u). We have already proved the first equality if u H is orthogonal to T H. On the other hand, if we take Uofi TpM so t h a t q = expp u0, we have 1

dp(T(q)) = F(uo)l dp o d(expp)~ 0 (~(u0)) = F(uo) dF~~ (e(u0)) ~ ( ~ - r(uo) u~)Fa(uo) = F(uo-----)u~)

) Ga(uo)

G(~o) = G(uo)

= 1 = G(T(q)) = (TH[TH}T(q), and (1.6.14) is verified for u = T(q). Finally, let u 6 TqM be generic, and write u = u0 + AT(q), with u H orthogonal to T H a n d A 6 R. T h e n

dp(u) = dp(uo) + Adp(T(q)) = A = (u H + )~TH [ TH)T(q) = (u H ] TH)T(q). []

51

COROLLARY 1.6.11: Let F : T M --~ R + be a Finsler metric on a manifold M , f i x v 9 i and take r < ir(p), u 9 Sp(r), and let q = expp(u). Then for d l

U 9Vu we have u = (d(expp)u(u)Hld(expp)~(t)H)T(q).

Proof: Indeed by (1.6.14) and Corollary 1.6.7, together with the definition of T(q), we have

(d(expp)~(U) H d(expp)u(,)H)v(q) = F(u)(d(expp)u(u)H l TH)T(q) = F(u)dp(d(expp)~(U)) = F(u) d(p o expp)u(V) 1 dG,~(V) = F ( u ) d F , ( U ) = -~ =

~Co(u)U ~ =

(u I~>~.

[]

1.7. Jacobi fields and A u s l a n d e r ' s t h e o r e m s 1.7.1. J a c o b i fields In this section we go on with our recasting of classical Riemannian geometry in a Finsler setting. After discussing variation formulas and geodesics, we start dealing with Jacobi fields. DEFINITION 1.7.1: A geodesic variation E : ( - e , e ) x [0, a] --* M of a geodesic a0: [0, a] ---+ M is a regular variation of a0 such that a~ = E(s,-) is a geodesic for any s 6 ( - e , e). This means that if we put

Yu 6 2t:/~,(t)

TH(u) : Xu(&,(t))

then we have W ~ (-~,

~)

V T . THI~, -- O.

Setting, as in section 1.5,

UH(u)=Xu (00~----~

52 we may then compute

0 = ~TuR~TT. TH ----~TTXVutxTH -I- V[UH,Tn]T H + ~ ( U H , T H ) T H = V T . V T . U g + V T . (IV H, T H] + 8(U H, TH)) + V[u-,Tn]T H - ft(T H, UH)T H.

(1.7.1) Now,

[UH'TH]= [Os 6 ~ , ~ 6 b -

os

~

[6o,6b] +

~

~o

-- -ST 6o

6b.

Since OEa/Os and oEa/ot do not depend on the vector variables, we get

6~

~

07=

- -

-b--~-~-~

= ~ 6~

and so

[U H,T H]= O~a 59~b Os Ot [5~,6b]

-

O~ a O~ b Os - - ~ {6b(P~)

-

6~(F~)}cw = - 6 ( U H , T " ) ,

because we are working with the Cartan connection. In particular, then, [U H, T H] is vertical; but for any V 6 V~,, we have

(VvTH)(~s) = [ \ Ot ) + r~J(i~')vJ(d~)b 6~ = O, because Y(OE~/Ot) = 0 and F~j(&~)(&~) b : 0. Summing up, we have proved PROPOSITION 1.7.1: Let E: ( - r r • [0, a] --* M be a geodesic variation of a geodesic a0: [0, a] ~ M in a Finsler manifold M. Set

J(t) = OEa(O,t) ~x ~ Os

6 Tao(t)M ~r0 (t)

and

JH(t) : XOo
~TTHVTn J H - ~ ( T H, JH)TH = O.

(1.7.2)

Note that, since TH(bo(t)) = X(b0(t)), the equation (1.7.2) can be expressed as

VxV• JH - ~(X, JH)x =--0 along &0.

53 DEFINITION 1.7.2: Let a: [0, a] ~ M be a geodesic. A vector field J along a is called a Jacobi field if it satisfies the Jacobi equation (1.7.2) for all t C [0, hi, where dH(t) = Xe(O(J(t)). Note that & and t& are Jacobi fields; the first one never vanishing, the second one vanishing only at t -- 0. The set of all Jacobi fields along a will be denoted by fl(a). In local coordinates, the Jacobi equation (1.7.2) is a second order linear differential system; hence, given the initial conditions J(0) and (VTH J/-/)(0), there exists a unique solution of the system, defined on [0, hi. Our first aim is to express it in a compact form, at least for Jacobi fields vanishing at zero. PROPOSITION 1.7.2: Let a: [0, a] ~ M be a geodesic in a Finsler manifold M , with p = a(O) and u = &(O). Take H e TI=, and set

Wt = it,, o , ~ I ( o - I ( H ) )

e Vt,,,

(1.7.3)

where O: V --* 7-[ is the horizontal m a p induced by the Cartan connection. Then the unique Jacobi field J along a such that J(O) = 0 and VT n JH(O) = H is given by Vt C [0, a]

J(t) = d(expp)t~(tWt),

where we are again identifying T t , ( T p M ) with 1;t~. Proof: P u t u(s) = u + s ~ i ( W 1 ) for s E ( - e , e ) , where e > 0 is so small that u(s) is never vanishing. Define E: ( - e , e) x [0, a] ~ M by E(s, t) = expp (tu(s)). E is a geodesic variation, and so OEa 0 t 0 J = - - 5 7 ( , )0xa is a Jacobi field. Clearly, J(0) = 0; moreover, OE ( 0 ,t ) = d(expp)t~(tWt) = t d(expp)t.(W,). J(t) = ~-~8 Therefore

VT~r j H = V T , [t (d(expp)tu(Wt))H] = (d(expp)tu(W,)) H + t V T , (d(exp,)tu(Wt))H, and, when t = O, V T I t J H ( o ) = ( t u l ( W l ) ) H = O ( W l ) = H,

because, by Theorem 1.6.2, t ~ d(expp)tu(Wt) is smooth and d(expp)o, = t--1 o,-

[]

In particular, then, zeroes of Jacobi fields and critical points of expp are one and the same thing, exactly as in Riemannian geometry. To express it better, let us recall a customary definition:

54 DEFINITION 1.7.3: Let a: [0, a] ~ M be a geodesic. The point a(to) is said to be conjugate to (r(0) along a, where to E (0, a], if there exists a non-identically zero Jacobi field J along a such that J(0) = 0 = J(to). Then: PROPOSITION 1.7.3: Let a: [0, a] --+ M be a geodesic in a Finder manifold M, and set p = a(0). The point q = a(to), with to E (0, a], is conjugate to p along a iff u0 = t0&(0) is a critical point ofexpp.

Proof: The point q is a conjugate point of p along a iff there is a non-identically zero Jacobi field J along a with J(0) = J(to) = 0. Let u = &(0) and H = VTgJH(o); by Proposition 1.7.2, J(t) = d(exp,),~ (tWt), where W, E Y,~ is given by (1.7.3); note that J is non-identically zero iff H # 0. Therefore q = a(t0) is conjugate to p along a iff

0 = J(to) = d(expp)u0 (toWto), that is iff u0 is a critical point of expp.

[]

In particular, it is important to notice that the zeroes of a non-trivial Jacobi field J axe discrete: indeed, if they were not, we would have J(to) =- 0 and VTR JH(to) = 0 for some to E [0, a], and the uniqueness theorem for solutions of a Cauchy problem would imply J _-- 0. We end this section with two results concerning the behavior of (jH ]TH)# along a geodesic a. PROPOSITION 1.7.4: Let J C J((~) be a Jacobi field along a geodesic a: [0, a] --+ M in a Finder manifold M. Then

(JH I TH)~(t) = t(VT,,JH I TH)~(O) + (JH I TH)~(O). Proof: We have

d (VT,,JH ITH)~ = TH(vT,~J H I TH)# = (VTn~TTnJHITH)a = (n(T H, J H ) T H I T H ) h = O, by Proposition 1.4.5. In particular,

(VT, JH I TH)~ =- (VT,,JH I TH)~(O).

(1.7.4)

Moreover~

d(j"

TH)~ = TH(JH I TH)a = (VT,~ JH I TH)~ = (VT,, J H I TH>a(0),

and we are done.

[]

55 COROLLARY 1.7.5: Let J 6 J ( a ) be a Jacobi t~eld along a geodesic a: [0, a] ---* M in a Finder manifold M . Assume that (jH [ TH)~(O) = (jH [ TH}•(a). Then

(jH

I TH)~

--

<jHi TH)~,(0), and WT,,J H I TH)~ -- O.

Proof: It follows from Proposition 1.7.4 and (1.7.4).

[]

DEFINITION 1.7.4: Let a: [0, a] ~ M be a geodesic in a Finsler manifold M. A proper Jacobi field along ~r is a Jacobi field J 6 J ( a ) such that

(jH ITH>~_O. For instance, by Corollary 1.7.5 any Jacobi field J with J(O) = J(a) = o is proper. The space of all proper Jacobi fields along a will be denoted by fr0(a).

1.7.2. T h e M o r s e i n d e x form

DEFINITION 1.7.5: Let a: [a, b] ~ M be a geodesic in a Finsler manifold M; we say that a is a normal geodesic if it is parametrized by arc-length, i.e., if F ( # ) =- 1. In particular, T( a ) = &. Let a: [a, b] ~ M be a normal geodesic in a Finsler manifold M; we shall denote by 2([a, b] the space of all piecewise smooth vector fields ~ along a such that

(~H [ TH)T ~ O. Moreover, we shall denote by 2(o[a,b] the subspace of all ~ G 2([a,b] such that ~(a) = ~(b) = o. DEFINITION 1.7.6: The Morse index form I = I~: 2([a, b] x X[a, b] ~ R of the normal geodesic a: [a, b] ~ M is the symmetric (by Proposition 1.4A) bilinear form

I(~,n) =

[(VT.~H i VT~H)T - (~(TH,sH),TH I TH)T ] dt

for all ~, ,7 6 X[a, b]. Note that, by Theorem 1.5.4, if U 6 Xo[a, b] is the transversal vector of a fixed variation ~ of a then

d2gn (0~

/(u, u) = - ~ / ,

,.

There is another formula expressing the Morse index form:

(1.7.5)

56 LEMMA 1.7.6: Let a: [a, b] ~ M be a normal geodesic in a Finsler manifold M , and take ( E X[a, b] smooth. Then

I(~,~?) = (VT,,~ H I

~H)T: fab --

(VT~VTH~ H - f~(TH,~H)T H I rlH)Tdt,

(1.7.6)

for edl rl 9 X[a, b].

Proof: Clearly we can assume r1 smooth (up to breaking a up in a finite number of pieces). Then d ( V T , ( H I r/H)r

TH(VT~( H I rln)r

VTn( H I~H)T + (VTH~ H JVT.rIH)T,


[]

In particular, then, the kernel of the Morse index form is composed by proper Jacobi fields: COROLLARY 1.7.7: Let a: [a, b] ~ M be a normaJ geodesic in a Finsler manifold M , and take ~ E X[a,b]. Then I(~,Xo[a,b]) = {0} iff ~ is a proper gacobi field. In particular, ker IIx0t~,b] = X0[a, b]n 3"0(a).

Proof: Assume that I(~,Xo[a,b]) = {0}. Then by (1.7.6) for every r/ E X0[a,b] smooth we have

0 = I((,rl) = -

s


and so ~ 6 rio(a). The converse statement is obvious.

[]

As we have already remarked in the previous subsection, there is a strong connection between Jacobi fields and conjugate points. Our next aim is to exploit this connection to describe a relation between the Morse index form and the minimizing properties of geodesics. DEFINITION 1.7.7: Let a: [a, b] --* M be a normal geodesic in a Finsler manifold M. We shall say that a contains no conjugate points if a(t) is not conjugate to a(a) along a for all t E (a, b]. We shall also say that a(b) is the first conjugate point to a(a) along a if a(b) is conjugate to a(a) and all previous points of the image of a are not.

57 PROPOSITION 1.7.8: Let a: [a, b] --* M be a normal geodesic in a Finsler manifold M containing no conjugate points. Then the Morse index form I~ is positive definite on Xo[a, bl, Proof: The assumption is that expa(a ) is a local E-diffeomorphism in a neighborhood of the image of a; so we can argue as in the proof of Theorem 1.6.6 to show that a is length-minimizing among nearby curves. By (1.7.5), then, I is positive semidefinite on X0 [a, b]. Now assume that ~ E Xo[a,b] is such that I(~,~) = 0; we claim that ~ E k e r I . Indeed, take y E No[a, b]; then for any e E R*

0 < I(~ + ~ , ~ + ~ ) = ~ [2I(~, ~) + ~I(~,

~)].

Dividing by e and letting e --+ 0 + (respectively, e --+ 0 - ) we get I(~, r/) > 0 (respectively, I({, r/) _< 0), and so I({, 77) = 0, as claimed. It follows, by Corollary 1.7.7, that { is a Jacobi field with J(a) = J(b) = o; being a(b) not conjugated to a(a) along a this implies ~ - 0, and we are done. [] As a consequence, the proper Jacobi fields realize the minimum of the Morse index form among vector fields with given boundary values: COROLLARY 1.7.9: Let a: [a, b] ~ M be a normal geodesic in a Finsler manifold M containing no conjugate points. Let ~ E X[a, b] and J E flo(a) be such that ~(a) = J(a) and ~(b) = Y(b). Then I(J, j ) < I({, ~), with equality iff ~ - J. Proof: By (1.7.6) H

I(Y,~) = (VT,,JHI~

b

H

)T a = ( V T " J H [ J

b

)T a = I ( J , J ) .

Then if ~ ~ J we have 0 < I ( ~ - J, ~ - J ) = I ( ~ , ~) - 2 I ( ~ ,

J) + I(J, J) = I(~, ~) -

I(J, J). []

Of course, the Morse index form becomes positive semidefinite at the first conjugate point: PROPOSITION 1.7.10: Let a: [a,b] -+ M be a normal geodesic in a Finsler manifold M such that a(b) is the first conjugate point to a(a) Mong a. Then I~ is positive semidefinite on A'o[a, b] and b

kerLIxoto,bj = Xo[a, b] n ao(o) # {0}.

58

Proof: By Corollary 1.7.7, it is enough to show that I b > 0. Take b' E (a, b) and define a m a p Tb,: X'o[a, b] -* X0[a, b'] by setting Tb,(~)(t) = r

The m a p Tb, is clearly an isomorphism; so we can define a symmetric bilinear form Ib,:Xo[a,b] x Xo[a,b] --+ I{ by Ib,(~,~) = I~ s (Tb,(~),Tb,(~)). Then

-C(r ~)

~i~, _r,,,(~,r

=

o

for all r E Xo[a, b], by Proposition 1.7.8.

[]

To complete the picture, we prove the following: PROPOSITION 1.7.11: Let a: [a,b] ~ M be a normM geodesic in a Finsler manifold M. Then there is to E (a, b) such that a(to) is conjugate to a(a) along a if[ there exists ~ e Xo[a,b] such that I~(~,~) < O. Proof: If there exists such a ~, by Propositions 1.7.8 and 1.7.10 there must be a

to E (a,b) such that or(to) is conjugate to a(a) along a. Conversely, let to E (a, b) such that a(to) is conjugated to a(a) along a. Then there is a non-identically zero Jacobi field J E Xo[a, to]. Take t' E (a, to) and t" e (to,b) such that J(t') 7s 0 and dF(a(t'),a(t")) < i r ( a ( t " ) ) . In particular, ai[e,e,] contains no conjugate points to a(t"). Let 7: ( - r --* M be a curve in M such that 7(0) = a(t') and ~(0) = J(t'). If we write 7 = exp~(v,)('~), then we may consider the geodesic variation E given by E(s, t) = expa(t,, ) (t~(s)), as usual. The transversal vector U of E then is a proper Jacobi field (by Coroll a y 1.7.5) belonging to X[t',t"] such that U(t') = J(t') and U(t") = O. We now define ~ E X'0 [a, b] by J(t)

~(t) =

iftE[a,t']; if t E [t',t"]; if t E [t", b].

U(t) 0

We also denote by J ' E X[a, t"] the extension of J obtained by setting J'(t) = 0 if t E [to, t"]. Clearly, J ' is not smooth at to, and so it is not a Jacobi field on [t', t"]. Therefore I,b(~, () = I~ts (~, ~) + Itst " (4, ~) = I ,t ' (J, J ) + Itst t~ (U, U) tt

t st

t

tt

< zo (J, J) + • ( j , j') = Io (J, J) + d,o(J, J) = I~~

J) = O,

by Corollary 1.7.9 and Corollary 1.7.7.

[]

59 In particular, a geodesic containing a conjugate point cannot be length minimizing: COROLLARY 1.7.12: Let a:[a,b] --* M be a normal geodesic in a Finsler manifold M . A s s u m e that there exists to 6 (a, b) such that ~r(to) is conjugated to a(a) along ~. Then r; is not length-minimizing, that is dF(a(a),a(b)) < L(a). Proof: If a were length-minimizing, then by (1.7.5) the Morse index form Iab along a would have been positive semidefinite, against Proposition 1.7.11. []

A final corollary, which is a major step in the proof of the Finsler version of the Cartan-Hadamard theorem: COROLLARY 1.7.13: Let a:[a,b] --* M be a normal geodesic in a Finsler manifold M . Assume that the Morse index form I b is positive detlnite on X0[a, b]. Then a contains no conjugate points. Proof: By Propositions 1.7.10 and 1.7.11.

[]

1.7.3. T h e C a r t a n - H a d a m a r d and B o n n e t t h e o r e m s

In this subsection we complete our presentation of real Finsler metrics providing a proof for the Finsler versions of the classical Cartan-Hadamard and Bonnet theorems. The theorems were originally proved by Auslander in [Aul, 2]; our proofs, however, are devised so to stress the similarities with the Riemannian case. We start with a consequence of Corollary 1.7.13: LEMMA 1.7.14: Let M be a complete Finsler manifold, and fix p 6 M . Assume that the Morse index form I~ is positive definite on X0[0, a] for all a > 0 and along every radial normal geodesic issuing from p. Then the exponential m a p expp? T p M --* M is a local diffeomorphism on ff/lp - - and an E-diffeomorphism at the origin. Proof: Indeed, by Corollary 1.7.13 no radial normal geodesic contains conjugate points, and thus, by Proposition 1.7.3, expp has no critical points. []

The assumptions in this Lemma are slightly weaker than the usual hypotheses of the Cartan-Hadamard theorem, as shown in LEMMA 1.7.15: Let M be a complete Finsler manifold of nonpositive (horizontal flag) curvature. Then the Morse index form I~ is positive definite on X0[0, a] for all a > 0 and along every radial normal geodesic issuing from p for all p 6 M .

60 Proof: Let a: [O, a] ~ E X0 [0, a]. Then

M be a radial normal geodesic issuing from p, and take

I({,{) =

[IIVT,~HII 2 - RT({H,{H)] dt >_ O,

because the horizontal flag curvature is nonpositive. Furthermore, I({, ~) = 0 implies V T , { H -- 0, that is that ~ is parallel along a. But ~(0) = %; hence, by (1.6.3), { is identically zero, and we are done. [] We do not directly assume here that the horizontal flag curvature is nonposirive because in chapter 3 we shall encounter a situation where we shall be able to control the sign of the Morse index form without controlling directly the sign of the horizontal flag curvature. Anyway, from now on we can proceed with the standard proof of the CartanH a d a m a r d theorem. First of all we need the following result: LEMMA 1.7.16: Let F j : T M j --* R + be a Finsler metric on a manifold M j , for j = 1, 2, and assume that F1 is complete. Fix a point Po E M1, and let f : M ~ N be a local ditfeomorphism of M1 onto M2 (and a 1ocM E-diffeomorphism at Po, but still differentiable there) such that F2(dfp(u)) > F,(u) for MI p E M l and u E TpM1. Then f is a covering map. Proof: We can repeat word by word the proof of Lemma 3.3 in [DoC].

[]

And finally: THEOREM 1.7.17: Let M be a complete Finsler manifold, and Fix p E M. A s s u m e that the Morse index form I~ is positive definite on X0[0, a] for all a > 0 and along every radiM normal geodesic issuing from p (e.g., assume that the horizontal flag curvature is nonpositive). Then expp: TpM ~ M is a covering map, smooth outside the origin. In particular, if M is simply connected then expp is a E-diffeomorphism at the origin. Proof: By Lemma 1.7.14, f = expp: TpM ~ M is a local diffeomorphism on _~/p, an E-diffeomorphism at the origin, where it is still differentiable, and it is onto, because M is complete. We can then use it to pull-back a Finsler metric onto TpM, which coincides with the original F at the origin. This metric is complete, because the geodesics through the origin are straight lines. Hence we can apply the Lemma 1.7.16, and we are done. [] We end this chapter by proving the Bonnet theorem for Finsler manifolds by using our second variation formula:

61

TItEOREM 1.7.18: Let M be a complete Finsler manifold of dimension at least 2 such that R=(H, H) > c(H I H>u (1.7.7) for some c > 0 and all u 6 M and H 6 ~ . Then the diameter diam(M) is bounded above, namely 7r diam(M) _< - ~ .

In particular, M is compact with l~nite fundamental group. Proof: Let S C R 3 be the euclidean 2-sphere of radius 1/v~; then it has constant sectional curvature c and (intrinsic) diameter ~r/v~. Then for any a > Zr/v~ there is a non-minimizing geodesic ra: [0, a] --+ S, and so the usual proof of the Bonnet theorem for the 2-sphere yields a vector field Ua = faEa - - where Ea is a parallel unit vector field along r~ orthogonal to/-a - - such that Ua(0) = U~(a) = 0 and

0 >

/0~[llv~og~ll ~ -

cllg~ll ~] dt =

/0 (IAI ~

-

cf2~) dr.

Now, let cr: [0, a] --* M be a length-minimizing normal geodesic; we must prove that a _< 7r/v/'~. Assume, by contradiction, that a > 7r/v~. Let ~ be a parallel vector field along a such that ~H is orthogonai to T H and H~H[[/, - 1 along b; since d H ITH)~ = TH((H [TH>~ : (VT.~H [TH)~ + ((S [VT.TH)a = 0 ~(( and

d

d-7<( ~ I ( ~ ) ~ = T " < ( H I(H>~ = 2 ( v ~ (

~ I ~'{>, = 0,

such a ~ exists. Put U = fa~, where fa is the function defined above. Being a a globally minimizing geodesic, we should have d2&. ds 2 (0) > 0

for any fixed variation ~ of a. But on the other hand the second variation formula applied to a variation ~ with U as transversal vector yields , j -- f0 ~ [IIVT, uHII~ -- R~(U H, uH)] dt d2~z (0~

Ill:a]2 - c ( g H I U g ) o ] at

~ =

~0 a

[I]~1 ~ - c f ~ ] dt < O,

(because (U H [ TH)~ =_ O, V T , ( H -- 0 and II~H[10 - 1 along o), contradiction. Finally, let 7r: M --* M be the universal covering space of M. Since 7r is a local diffeomorphism, we can pull-back the Finsler mettle on M to a Finsler metric on .~/'. Clearly the horizontal flag curvature on M satisfies again (1.7.7), and so M is still compact. Hence for any p 6 M the set 7r-l(p) has finite cardinality, and so rl(M) must be finite. ['-I

62 In this way it is possible to recover even the Rauch comparison theorem (see,. e.g., [BC]); but for our aims this is enough, and in the next chapter we shall start dealing with the complex case.

CHAPTER

2

Complex Finsler geometry

2.0. I n t r o d u c t i o n In this chapter we s t a r t dealing with complex Finsler metrics. T h e main idea is exactly the same as in the real case: replacing the Finsler metric on the tangent bundle by a H e r m i t i a n metric on a suitable bundle it becomes possible to get geometrical results on the Finsler manifold using Hermitian techniques. Contrarily to the real case, we did not find in the literature much references on complex Finsler geometry. We anyway would like to recall here the pioneering work of R u n d [Rd2], Fukui's [Fu] e x a m i n a t i o n of the C a r t a n connection in the complex case, F a r a n ' s [F] work on the equivalence p r o b l e m for complex Finsler metrics, and above all Kobayashi's paper [K], which inspired our approach. So let M be a complex manifold, and F : TI'~ ~ •+ a complex strongly pseudoconvex Finsler metric on M (see section 2.3 for the precise definitions). Then, as in the real case, using the Levi form of G = F 2 it is possible to define a Hermitian metric on the vertical bundle 13 in such a way that the canonical section t: JP/---* ]2 turns (where this time AT/stands for the complement of the zero section in TI'~ out again to be an isometric embedding of M into ]2. Now to any H e r m i t i a n metric on a vector bundle it is associated the Chern connection, which is the unique complex linear connection (or connection of t y p e (1, 0), as it is also called) for which the given Hermitian metric is parallel. It turns out t h a t the Chern connection is automatically good, and so we get a complex horizontal bundle 7-/such t h a t T 1,~ H O P , and a bundle isomorphism O: Y ~ H. Using O we can define an Hermitian metric on 7-/, the associated Chern connection and a canonical section )C = O o L; and this is the setting we need to work. T h e torsion of the Chern connection is, in general, not zero. Generalizing the usual definition in Hermitian geometry, we shall say that a complex Finsler metric is Kghler if (a suitable contraction of) the horizontal part of the torsion of the Chern connection vanishes. As we shall see in sections 2.3 and 2.4, K/~hler Finsler metrics enjoy properties similar to the ones of Hermitian Ks meti-ics. In particular, we can compute the first and second variation of the length integral in complex terms, and prove the local existence and uniqueness of geodesics for Kghler Finsler metrics even when the associated real Finsler metric is not convex - - t h a t is in cases where the existence of geodesics is not a consequence of the real theory.

64 We shall describe geometrical (and function theoretical) applications of this theory in the next chapter; the finals sections of this one are instead devoted to the study of the holomorphic curvature of a complex Finsler metric, and to the comparison of the Chern-Finsler connection with the Cartan connection. In particular, we shall give a variational characterization of the holomorphic curvature. More precisely, the content of this chapter is the following. In the first two sections we review the theory of horizontal bundles and maps, and of non-linear and vertical connections, adapting the presentation to the complex case. In the third section we introduce the vertical Chern connection associated to a complex Finsler metric, and we discuss the notion of K/ihler Finsler metric. In the fourth section we compute the first and second variation of the length integral in this context, proving the existence of geodesics for strongly pseudoconvex weakly Kghler Finsler metrics. In the fifth section we examine the notion of holomorphic curvature of a strongly pseudoconvex Finsler metric. Finally, in the sixth section we compare the ChernFinsler connection with the Cartan connection under various Kghler hypotheses.

2.1.

Complex

non-linear

connections

2.1.1 Preliminaries

Let M be a complex manifold of complex dimension n. Let {zl,... ,Z n} be a set of local complex coordinates, with z ~ = x~+ix n+~, so that { x l , . . . , x n, x n + l , . . . , x 2n } are local real coordinates. Lowercase greek indices will run from 1 to n, whereas lowercase roman indices will run from 1 to 2n. Let T~M denote the tea/ tangent bundle of M; it is a real bundle of rank 2n equipped with a complex structure J. With respect to the local frame {O/Oxl,..., O/Ox2n}, the complex structure J acts in the following way:

j(O/Oz ~) = { O/Ox a+n --O/Ox a-n

if 1 < a < n, ifn+l
In other words, writing J(O/Ox ~) = J~ O/Ox b we have J:=

{~b+~ --~-n

ifl
(where ~ is the Kronecker delta), that is J= Let ToM = T~M | bundle of rank 2n. Set

o (o

- -

Oz ~

-

2

0-~

(o In

0

'

C be the complexi~ed tangent bundle; it is a complex

-i

0

~

)

and

- -0-

02 ~

I

2

(

0

i

0--~--g+ ~

0

)

;

65

then

{a/az',..., a/a=",

is a local

frame for TcM.

The

complex

structure J acts complex linearly on T c M , and j2 = - I ; so T c M splits as the sum of the two eigenbundles TcM= TI'~ | T~ where T ' , ~ = {v 9 T e M I J v =- iv} and T ~ = {v 9 T e M I J , = - i v } . The bundle T I , ~ (which we shall sometimes simply denote by T M ) is the holomorphic (or c o m p l e x ) tangent bundle of M. It is easy to check that { O / O z l , . . . , O/Oz n} is a local frame for T : , ~ and { 0 / 0 2 ~ , . . . , 0 / 0 2 n} one for T ~ The real splitting ToM = T~M @ i(T~M)

induces a conjugation on T o M commuting with J; since O/Ozi = 0 / 0 2 j, we have TI,O M = TO,I M .

As well known, the bundles T I , ~ and T~cM are isomorphic, and both axe sensible representations of the tangent bundle of M. In this book we choose as explicit isomorphism the bundle map o: Tl,O M ___+T~cM given by Vv 9 T I ' ~

v ~ = v + ~.

Note that (O/Oz~) ~ = O/Ox ~, (iO/Oz~) ~ = O/Ox ~+'~ and that J v ~ = J ( v + o) = J v + J v = iv + iv = (iv) ~ = i ( v -

O),

that is o is a real isomorphism preserving J. The inverse o: T ~ M ~ T l ' ~ by 1

Vu 9 T ~ M

Uo = -~ (u - i J u ) .

Furthermore, locally an element of T I , ~ V .~ V

is given

(2.1.1)

could be written as

O OZCr

(2.1.2)

Then, setting v ~' = u ~ + iu c~+'~, it is easy to check that 0 v ~ = u a Ox a .

(2.1.3)

Conversely, if u = u a O/Ox a 9 T R M , then uo = (u ~ + iu"+n) 0-0z".

(2.1.4)

Let T ~ M denote the real cotangent bundle of M; a local frame for it is {dxa}. Also T ~ M is endowed with a complex structure J* given by V7 9 T ~ M

Vu 9 T ~ M

J*7(u) = 7 ( J u ) .

It follows that, writing J * ( d x ~) = (J*)~ dx b, we have

{-*~

+~

(J*)~ =

5; - n

if 1 < a < n, ifn+l
66 in other words, j . ( d z a) = ~ - d z ~+~ i f l < a < n , ifn+l
and

d5 ~ = dx ~ - i dxa+~;

the set {dz ~, d5 ~} is a local coframe for T ~ M . The map J* induces the complex splitting T ~ M = AI'~ 9 A~ (2.1.5) where Aa'~ = {7 E T ~ M [ J*7 = i~,} and A~ = {3' e T ~ M I J*7 = - i 7 } . and A~ = (TI,~ so Aa'~ is naturally Note that A~'~ = ( T ~ 1 7 7 isomorphic to (TX,~ * and {dz ~} is the dual frame of {O/Oz~}. Again, the real splitting T ~ M = T ~ M @ i T e M induces a conjugation commuting with J*; since dz ~ = d5 ~, clearly Aa'~ o: A1,0M ~ T ~ M is given by

Vw ~A~'~

= A~

The real isomorphism

wo = 5(`0 1 + co),

so that (dz~) ~ = dx% (i dz~) ~ = dx ~+~ and i S*`0 o = (i`0) o = 5(,0 - co). The inverse isomorphism o: T ~ M ~ A a'~ V7 E T ~ M

is given by

% = ,,/- i J.;,.

Finally, the splitting (2.1.5) induces a splitting K(TcM) =

~

A"aM

la+q=r

where AP'qM =/~P(A~'~176 As before, AP'qM = Aq'PM, a n d / ~ ( T ~ M ) is both a subbundle o f / ~ ( T ~ M ) and isomorphic (via o) to A~'~ We shall denote by AP,q(M) the space of smooth sections of AP'aM, that is the space of smooth (p, q)-forms on M, and we shall set A(M) = ~

AP,q(M).

p,qEN

Let us now look at the tangent-tangent level. Since T ~ M ~- TI,~ is a complex manifold, we have a complex structure J on it. Thus we have at least six spaces to consider: TR(T~M), Tc(TRM), TI,~ Ta(TI,~ Tc(TI,~ TI,~176 and so on. Luckily, the situation is not so complicated as it seems at first glance. Indeed, (2.1.3) and (2.1.4) show that the isomorphisrns o and o preserve the real coordinates of T 1,~ and T~M. This means that, as fax as only the tangent-tangent spaces axe concerned, we can (and we shall) safely identify TX,~ and TRM. In

67 particular, M will denote either T I , ~ or T•M minus the zero section, depending on the actual situation, without any further warnings. A local frame over R for T~flT/is given by {0f,... ' 0 2~n ' 0"~ 1 ' ... ' 0~n }, where 00=

0

OX a

-o

artd

analogously, a local frame over C for Tl,~ where O0 z~

0,~ =

0.

0~, = Oua,

is given by {01,..., 0n, b l , . - . , 0 n } ,

and

0o,

0

Ova,,

and, as always, z a

= x a + ix a+n and v ~' = u a + iu a+n. There is a natural isomorphism, still denoted by o: T1,0A?/__. TRfl)I, given by VX

9 T 1,~]~/l

X ~ = X + -X ,

with inverse o: TR/t7/--* T l'~ (O~) ~ . 0 %

given again by (2.1.1). It is easy to check that

(iO)o .

0o+,,.

There also are T~/17/and A1'~

@~)o .

0o,

and

(i0o) ~

Oo+,.'~

with dual coframes and

{dx ~. . . . , d z 2n, du l , . . . , du2n}

{ d z l , . . . , dz n, d v ~ , . . . , dv n}

respectively. Note that, as in the real case, dz~

is different from dz~'lp.

2.1.2. C h a n g e of c o o r d i n a t e s Let us now summarize what happens under holomorphic change of coordinates. Notations, proofs and arguments needed are exactly as in the real case (cf. subsection 1.1.1). This time the relevant matrices are

Oz~ (,.TBA)~ = OZfA

02z~ and

Therefore the change of coordinates on T I , ~ = (YBA)

and

and, conjugating everything, = (YBA)

and

( H B A ) ~ . y - OZflAOZ~A.

are given by

Oz~

A

.

68 Recalling that the change of coordinates are holomorphic, we have Oz~/O 5~A = O, that is

0~

0 ~ +"

OX ~A 0 ~ +" OX flA

-

-

Oz~

OX flA+n = Re Oz---~A= R e [( J B A ) '~],

Oz~

Oz~

OX ~A+~

-

-

Im OZ flA = Im [( ff B A ) '~],

and

(Re0~A) Oz~'~

d~

= 0~

d~ =

#

(imOZ~

dXA-- ~

za-n I m p /

dx#+ .

Oz~/

if 1 < a < n,

a-n

ozA /

dXflA+ (ReOZB:'*~ dx#+ n i f n < a - n < n ,

\

04 ]

=

that is

OxbA ] = Tl,~

Imj.BA

ReJ'BA ] "

Now we briefly examine TI,~ We know that {0~,, 0~} is a local frame for let {dz~',dv ~ } be its dual coframe. Then again we find

dz~ = Oz~ dz~ = (J.~)~ dz~, (2.1.6)

d v ~ -.=- OZz.__~Ad V ~AJr OZ flA 2ZO~

V ~Ad Z flA = ( J B A ) ~ d V ~A+ ( H B A ) ~ ~ v I d Z flA,

and

Oz~

04

022~ vl o d

- (J~A)o( = --

"A)~(J~A).

( J ~ A ) ~ (O~)A.

2.1.3. H o r i z o n t a l and vertical bundles

The next step is to define the vertical bundles. DEFINITION 2.1.1 : The real vertical bundle is ~

= ker d~r C T~/17/,

while the complexified vertical bundle is VC = ]3~ |162 C = ker dzr C TCA~/.

A (&)A,

(2.1.7)

69 Since the projection 7r is holomorphic, dTr commutes with J. So 12c splits as 12c = 121,0 ~ )20,1 and we may define the (complex) verticM bundle = ],~1,0 = k e r d r

C Tl'~

Clearly, #: 1) --+/V/is a holomorphic vector bundle of rank n: whose local frame is given by {c51,..., 0n}. Analogously, a local frame for ~ is {0~',..., c5~}. Take u E 2t~/. Then the natural isomorphism ~ : (TRM)~(,) --+ (12~)~ commutes with J; hence the (12e),-valued extension to (TcM)~(~) sends T ~(u)~a ~'~ " onto 12,, and commutes with J and the isomorphisms o and o. In particular, we can define the canonical isomorphisms ~v: T ~(~) 1'~ M --* ~ by setting

DEFINITION 2.1.2: The rea/radiM vertical vector t~eld t~ 2t7/~ Y~ and the (complex) radiM vertical vector/~e/d ~:/V/--* Y are defined as usual:

t~

= L~(u)

for all u E T R M and v E T I ' ~

and

~(v) = ~,(v),

They of course satisfy

Vv e T I ' ~

t~ ~ = (t(v)) ~

Now take ~ E X ( T ~ M ) . Then we know that d(: T ~ M ---* Ta(TRM) is given by a 0~ b ~o + u bSz~oh.

d~(u) = u~ Analogously, we can take ~ e X ( T I ' ~

v =

getting d~: T o M ---+T c ( T I , ~

if

v~ 0 v~ O Oz------g+ O-----gE T c M ,

then

e~(,) = ,~ 0o + 0 - - ~ 8 + ~

8j +

0~ + oz

o-7 8].

(2.1.s)

In particular,

Yu e T R M because u e T R M iff u s = ~

d~(u) = d(~ for a = 1 , . . . , n. Moreover,

(2.1.9)

7O LEMMA 2.1.1: Let M be a complex manifold, and take p E M and ~ E X ( T I ' ~ Then the following assertions are equivalent: (i) (ii) (iii) (iv)

~ is holomorphic at p, that is O~p = O; d((T~,~ C T~('~176 d~p o J = J o d~p on TcM; d~; o J = J o d~; on T~M.

Proof: (ii)==:v(i):==>(iii): it follows from (2.1.8). (iii)==~(iv). Take u E (T~M)p; then Ju E (TRM)p and (2.1.9) yields d~;( Ju) = d~p( du) = J(d~p(u)) = J(d~;(u)). (iv):==t-(ii). Take v E T :10 ' M, and set u

~p(v) = 89

~yo.

Then (2.1.9) yields

~d~,(J~)] = 89

~d~;(J~)].

Applying J we then get 1

J(d~,(v)) : ~ [Z(~;(~)) -~J(d~;(Zu))] : 89

-~;(J~)]

i = ~ [d(p(u) - i d(p(Ju)] = i d(p(v). []

We shall denote by Xo(l~/I) the space of holomorphic sections of _AT/, where when AT/is considered as a subset of T~M "holomorphic" means that the condition given by Lemma 2.1.1.(iv) is satisfied. More generally, if p: E --+ B is a complex vector bundle on a complex manifold, we shall denote by 2(o(E) the space of holomorphic sections of E. DEFINITION 2.1.3: Let pl,0: Tc(TI,OM) __+ TI,O(TI,OM) be the natural projection; note that pl,0 restricted to T~(TI,~ coincides with the isomorphism o. Then setting d1,0=pl,0od we have

dl'~ for all ~ e X ( T I ' ~

Vv e TI'~

= v~ a~ + 0-TZuq + ~ 0--~u~,

and all v E TcM. It is easy to check that

d1'~176 -- ( d O~ v ( ~ ))o,

and so d 1,~ is the correct analogue on TI,~ commutes with J iff ~ is holomorphic at p.

(2.1.10)

of d on T~M. Furthermore, dl,~

71

-

Now let D: X ( T a M ) ~ X ( T ~ M | be a (real) non-linear connection on M. ~ X ((Ta,~ | TI'~ The usual isomorphisms give rise to a m a p Do: X ( T l'~ where ( T I , ~ is the space of R-linear functionals on T ' , ~ - - by setting -

=

.

Clearly,/)o satisfies ~ t t Do~p bo~p=*~a o(d 1 0' ~p-d"~

for all p 9 M and ~, ~' 9 X ( T I , ~ need the following

such that ~(p) = ~'(p) = v. At this point we

LEMMA 2.1.2: Let V be a complex vector space. Then every R-//neax functional w 9 V~ is the sum of a C-linear functional w' 9 V* and a C-antilinear functional w" 9 V*; moreover, this decomposition is unique. In other words, V~ = V* ~ ~*.

Proof: Assume w = w' + w", with wl 9 V* and w" 9 V*. Then w(v) = w'(v) + J ' ( v ) ,

w(iv) = iw'(v) - iw'l(v),

and hence

w'(v) - w(v) - iw(iv) w"(v) - w(v) + iw(iv) 2 ' 2 On the other hand, it is easy to check that w' and w" so defined are as required, and we axe done. [] In our case, this implies that we may w r i t e / ) o =/~1 + / ~ , , with

D'~ 9 X ( ( T I ' ~

| TI'~

b"~ 9 X((T',~

and

* @ TI'~

Recalling the canonical isomorphisms, we can think o f / ) ' ( as belonging to the space , 0 ',0 X ( A ',0 M | M ) , a n d o f b tt ( a s b e l o n g i n g t o X ( / ~ ' 0 1 M | M). We are then ready for the following definition: DEFINITION 2.1.4: A complexified non-linear connection is a map

D: X ( T I ' ~

- . X ( T ~ M | Ta'~

such that Do--0 and - b(p

for all p 9 M and ~, ~' 9 X ( T 1,~

10

t

4 - ' o (d ' (p -

with ~(p) = {'(p) = v.

(2.1.11)

72 It is easy to check that if we define/~o: X(TRM) ~ X(T~M | TRM) by setting bo~(u) = (b~o(Uo)) ~ we get a (real) non-linear connection, thanks to (2.1.10). I f / ) is a complexified non-linear connection, we can uniquely write

b = b' + b", with

D':X(Ta'~

~ X(AI'~174176

and

D":X(TI'~

---*X(A~174176

DEFINITION 2.1.5: We shall say that a complexified non-linear connection/) is a

complex non-linear connection if V~ E Xo(TI'~

L)"( - O.

Since b ' ~ o J = JoD'~ and D"(oJ = - J o b " ( is equivalent to

V~ E Xo(TI'~

(2.1.12) for any ~ E X(TI'~

Z)~ o d = d o D~.

(2.1.12) (2.1.13)

Note that (2.1.13) is supposed to hold only for holomorphic vector fields. However, the locality o f / ) immediately implies that b~p o 3 = J o b~p

for all ~ E X ( T I ' ~ which are holomorphic in a neighborhood of p E M. As in the real ease, we can associate to complex non-linear connections both complex horizontal bundles and complex horizontal maps. DEFINITION 2.1.6: A complex horizontal bundle is a complex subbundle ~ c C Tc.AT/ which is J-invariant, conjugation invariant (that is ~ c = Hc) and such that

TcM = Nc r Vc. Since 7"/c is J-invariant, we can write 7-/c = ~ 1 , 0 ~ ~0,1, where 7./1,0 = "Hc N T 1'~ moreover, being 7"/c conjugation invariant, ~0,1 = 7.ll,0. This means that a complex horizontal bundle is completely determined by its (1,0)-part 7-/1,~ later on, we shall often simply write 7-I instead of ~1,0. DEFINITION 2.1.7: A complex horizontal map is a complex linear bundle map O: FC ~ TcI~I commuting with J and the conjugation mid such that

(dTr o Oh, lv~,o = ,~'lJv~,o for all v E -~/. Since Vc = 121'~ 2~ ~ 0 , 1 ~- ~)1,0 and O commutes with both J and the conjugation, it is clear that O(~ 1,~ C TI'~ O is completely determined by its behavior on 121,~

73 It is easy to recover in this case too the correspondences among complex nonlinear connections, horizontal bundles and horizontal maps. Let 7"/c be a complex horizontal bundle, and x: T c 2 ~ / ~ Pc the associated vertical projection. Since 7"/c is J-invariant and conjugation invariant, x commutes with J and the conjugation; in particular, x sends T1,~ onto yl,0. We define a complex non-linear connect i o n / ) n c on M by setting - I o ~(p) o dl'~ b,~c~p = t ~(p) for any p E M and ~ E X(Ta'~

Since dl,~

- dl,~

sends (TcM)p into ~ , 0 as

~

soon as ~(p) = ~'(p) = v, it is clear that D ~ c is a complexified non-linear connection. Fhrthermore, if ( is holomorphic then both x and dl,~ commute with J, and so D ~ c is complex. Now let b be a complex non-linear connection. Take v E -~/~ and ( E X ( T l'~ such that ~(p) = v. Then define Oh'. - v -vYa'~~ T~,I0Mby" O~b = dl,~ G o ~:~ - ~,, o b,~), o ~-1.

This definition does not depend on ( but only on v, by (2.1.11). If we extend O b to ~0,1 asking that O commutes with conjugation, and then to Fc by complex linearity, we obtain a complex horizontal map. Indeed, O D commutes with conjugation by definition. Next, since we may take ( to be holomorphic in a neighborhood of p, we have shown that it commutes with J. And so, since d r o dl,~ = id on TI,~ O b actually is a complex horizontal map. Finally, if O is a complex horizontal map, it is clear that 7"/~ = @(l,'c) is a complex horizontal bundle. The arguments used in the real case can be then repeated almost word by word to show that we have defined a one-to-one correspondence among complex horizontal bundles, complex non-linear connections and complex horizontal maps. We end this subsection describing what we mean by homogeneity in this context. DEFINITION 2.1.8: We shall say that a complex non-linear c o n n e c t i o n / ) is homo-

geneous iff b(~r

for all ( E C* and ~ E 2d(Tl'~

o ,~) = ,,.,~ o b~:

where #r TI,~

~ TI,~

is the multiplication

bye. DEFINITION 2.1.9: A complex horizontal bundle ~ c is homogeneous iff (~C)~,(~) = d ( ~ ) ~ ( ( ~ c ) o ) for all ( E C* and v E 2~/.

74 DEFINITION 2.1.10: A complex horizontal map O is homogeneous iff d(pr

o O~ o ~. = O,;(~) o %r

for all ( E C* and v E/17/. Again, it is easy to prove exactly as in the real case that the previously defined correspondences preserve homogeneity.

2.1.4. L o c a l c o o r d i n a t e s Our next aim is to find the expression in local coordinates of complex non-linear connections. Let b be a generic complexified non-linear connection, with associated p E M and set v = ~(p). Then covariant differentiation V. Take ( E X(TI'~ dl'~

= G + ~

= 0-F~,~.

By definition, ~,((7o/ozo~) - dl'~ and t ~ ( V a / 0 ~ ) - dl'~ depend only on v. Therefore there are F~(v), r~(v) E C such that

I.v(~ a/Oza ~)

-

-

dl'~

") must

OI OzCt) = -Oc~ q- r~(v)bfl,

,~(va/o~o~) - d"~

") = r~(v)b~.

Thus

for all w = w~(OlOz ~) + w~(O/O~ ~) ~ (TcM)p. In particular,/) is complex iff r~_--o. The F~'s are the Christoffel symbols of the complex non-linear connection/). Assuming now b complex, it is easy to show that in local coordinates | given by ~ = oh(G)

is

= 0~ - r ~ b z

Clearly, {6,,... ,6n} is a local frame for ~..~1,0 and { ~ 1 , . . . , ~ n , b l , . . . ,bn} a local frame for Tl,~ r. We shall denote by { d z l , . . . , dz n, r Cn} the dual coframe; clearly,

r '~ = dv ~ + r'~ dz ~.

75 DEFINITION 2.1.11: We also set

Xv

10

|

M~

1,o.

then the (complex) radial horizontal vector field X 6 ,'lf(~L~1'0 ) i8 given by x=Oot. Clearly, X(O/Oz ~') = 5,~. Finally, let 9/c be a subbundle of Tc/IT/such that TcAT/-- ~'/c ~i:c. Let {6a,6~} be a local frame for "Hc such that d~r(5,~) = O/(gz c' and dTr(6~,) = 0/02, a. Then we should have

for suitable functions I'~, I'~, I'~ and I'~. Therefore ~cc = 7"/c iff ~ = 6~, that is iff F~ = F"~

and

F~ = F'-~.

(2.1.14)

Analogously, "/'/c is J-invariant iff JS~ = i5,~, that is iff r~ -- 0 - r ~ . We end this subsection with two remarks proved exactly as in the real case. First of all, a complex horizontal bundle ~ c is homogeneous iff its Christoffel symbols are, that is iff r~(,~(v)) = cr~(v) for all ~ 6 C* and v 6 AT/. Secondly, and lastly, the behavior of the Christoffel symbols under change of coordinates is (r~) B

=

/5

3,

-1 3

--

,8

--I

9'

76

2.2. C o m p l e x v e r t i c a l c o n n e c t i o n s 2.2.1. Definitions DEFINITION 2.2.1: A complexit~ed vertical connection on a complex manifold M is a linear connection on the complexified vertical bundle Fc, that is a linear map D: 2((]2c) ~ 2 ( ( T ~ # I | Vc) such that D ( f V ) = f D V + df | V

for every f 6 C~176

and V 6 2((]2c).

This definition makes almost no mention of the complex structure of 2~/, and thus it is too general for our aims. So we make three assumptions. First of all, we assume that D commutes with the conjugation, that is that VV E 2((]2c)

D(V) = D V .

(2.2.1)

In particular, since ]2c = V 1'~ ~3 V ~ and ]20,1 = ]21,0 this implies that D is completely determined by its behavior on 2((V1,~ Second, we shall assume that D restricted to F 1,~ commutes with J, that is that V V C X ( F 1'~

(2.2.2)

D( J V ) = J o DV.

As a consequence, D V E X(T~/~/QV 1'~ for all V E X(FI'~

indeed, (2.2.2) implies

J o D V = D ( J V ) = D ( i V ) = iDV.

Finally, we shall assume that for holomorphic sections V of ]21,0 the 1-form D V commutes with J, that is VV 6 2(0(]21'0)

(DV) o J = J o (DV),

(2.2.3)

where, as before, 2(0 stands for holomorphic sections. Since T ~ . ~ / = ]~l'~176 we may decompose the connection D as a sum D = D' + D",

with D': X(]21,~ ~ 2((Ai'~ (2.2.3) is equivalent to V V E 2(0(]21'~

@ F 1,~ and D": 2((]2,,0) ~ 2((A0,13)/@ Vi,0). Then D " V =_ O.

DEFINITION 2.2.2: A complex vertical connection is a complexified vertical connection D satisfying (2.2.1), (2.2.2) and (2.2.3). In particular, we shall consider V as defined over 2((]21,~ and with values in 2((T~/'~/| 121,~ extended, when necessary, to the whole ]2c by (2.2.1) and complex linearity.

77 Let D: X(1;r ~ X(T~.f/i | 1;c) be a complexified vertical connection. In local coordinates, writing V = V~c~a + VacUa, we get D V = dV ~"| 0,~ + d Y ~ | Oe~+ V~DO,~ + V~'DOe,, and

D0~ = w~ | 0~ + w~) | (~

and

Dc~ = w~ | 0~ + w~) | 0~,

where w~, w~,) w~a and waa are locally defined 1-forms. Then (2.2.1) is equivalent to

~ =~

and

~ =~;

(2.2.2) on the other hand is equivalent to ~ - 0 = ~. Finally, (2.2.3) is equivalent to requiring that every w~ is a (1,0)-form, that is that

~ = ~ , ~z, + ~ v ~ , for suitable coefficients F~;~, and F ~ . Summing up, locally a complex vertical connection is given by

DV = [~v~ + v ~ ]

|

o~

for all V E X(1;l'~ where the w~'s are local (1,O)-forms. In particular, V x V = [X(V ~) + w~(X)V~]O~,, v ~ v =~(v~)a~,

V x V = V~-V, v ~ v = VxV,

(2.2.4)

for all X E Tl'~

2.2.2. G o o d vertical connections From now on, we shall write 1; in place of 1;1,~ The idea is that, exactly as in the real case, we can associate a complex non-linear connection to some complex vertical connections. Let D be a complex vertical connection, and denote again by A: T l ' ~ 1; the bundle map defined by h ( x ) = Vx~.

In local coordinates, A(X) = [ ; b + ~;(X)v~]0~. DEFINITION 2.2.3: We say that the complex vertical connection D is good if the bundle map AIr: Y ~ Y is a bundle isomorphism.

78 Exactly as in the real case, we see that D is good iff 7"/c = 7-/@ 7-I is a complex. horizontal bundle, where 7-I = ker A C TL0~/. In local coordinates, D is good iff the matrix L~ = 5~ + FT~v is invertible. If ((L-1)~) denotes the inverse matrix, a local frame for 7-I is given by --I # ~ u

~=as-(L

) ~ r ~ ; s ~7a" .

~

as

_ _

r~a~,

where I~

,u ~ v = (L - 1 )~r~;sv

are the Christoffel symbols of the complex non-linear connection associated to 7"/. So let D: X(F) ~ X ( T ~ I Q | Y) be a good complex vertical connection. We shall denote by {dz ~', Cs} the local coframe dual to {6~, cgs), where Cs = dv s + r'~ dz ~'. Correspondingly, we may decompose

~$ = r~;.

dz. +

U . r .,

where

F ~s = F~ s~

and

F~;, = F~; u - P ~ . f , .

In particular,we agsJn have

r~;.v~ = r~. Fhrthermore, writing X = X~'5~, + X'rO.~, we have

89 =

s) +

s) }

The behavior of w~, F~'r and F~;, under change of coordinates is exactly as expected:

= (J~A)s(~)A(JBA)~ -- (:~A)s(HB~)~. (r~;/z)B

dz ,

, 7 -1 v 5 /~ - 1 3' -1 u fl 9 = (,.7~A)a(fl[~A)~,(F.r;v)A(,.7t~A)~ -- (,.7[~A)e,(Jf3A)~,(HI3A).r~, ,

(r~.)B =

(2.2.5)

--1 p --1 a I" fl (:~A)s(Y~A)~(r~)A(:BA)~.

Let O: F ~ 7"/be the complex horizontal map associated to the complex horizontal bundle 7"/c. In local coordinates, O(0s) = 5s. We may then define a complex linear connection on 7"l by setting VxH

= {9 [Vx(O-1H)]

for any X 6 Tc-~/ and H 6 X(7-/). By linearity, this yields a complex linear N connection on T c M , still denoted by D.

79 2.2.3. Connections

on complex tensor bundles

DEFINITION 2.2.4: Let D: X(I;) --* X'(T~.~I | l;) be a good complex vertical connection, and let D also denote the associated complex linear connection on Tc2~/I. Then we can define a complex linear connection on T~/Q, which we shall denote by V: X(T~.~I) ---, X(T~_~I | T~2Q), by requiring (V~0)(X) + ~o(DX) = d(~o(X)), for all ~ e X(T~37/) and X e X(Tc-~/). It is easy to check that ~7 commutes with the conjugation, that is V ~ = V~, and with J* on A1'~

In particular, it preserves the type, that is

In local coordinates, if ~o = ~ot,dz ~' + #~r

then

Clearly, we can extend V to a complex linear connection V: X(T(c"'S)M) ~ X( T~2Q | T(c"")J~I) commuting with the conjugation and preserving the types (i.e., the behavior with respect to J and J*), where, of course, Tc(r's).~/= T(~,s)37/| Then, still following the real example, we introduce an exterior differential for Tc(~'~)/~/'-valued (p, q)forms. DEFINITION 2.2.5: If ~p = ~a | e~ is a TC(r's)fiT/-valued (p,q)-form, where {ca} is a local frame for Tc(r")JQ composed by tensor products of {6~,, 0~,}, {dz t', r176and their conjugates, we set D ~ = d~ ~ | e~ + ( - 1 ) P + q ~ ~ A Ve~.

It is easy to check, exactly as in the real case, that D~0 is well-defined. Furthermore, we can decompose D as D = D' + D", according to the type. In local coordinates, D'~o = O~ ~ | e~ + (-1)P+q~ ~ ^ Ve~, D " ~ = c5~0~ | e~, (2.2.6) where we used the fact that the connection forms of a complex linear connection are of type (1,0).

80 2.2.4.

The

torsions

DEFINITION 2.2.6: The tangent bundle T I , ~ with a (1,0)-form, the canonical form

(and hence .~/) is naturally equipped

rl = dz" @ O. + dv" | 0 , e X ( A I ' ~

| Tl'~

It is easy to check that ~ is a well-defined TL~

global (1,0)-form on 217/.

Now let D: X(V) ~ X(T~.h:/| 1/) be a good complex vertical connection, and let D also denote the associated complex linear connection on .hT/. Then it is easy to see that = dz" @ 6. + r | 0,,; in particular, r/o~=t

and

r/ox=

X.

We have introduced in the previous subsection an exterior differential D; so it is very natural to consider 077 = ( - d z ~ A w2) N 5, + (de ~ - r Now D q is a Tl'~ is the sum of a T~'~

A w~) @ 0~.

2-form obtained by differentiating a (1,0)-form; thus it (2,0)-form D'rl and a Tl,~ (1,1)-form D"rb

DEFINITION 2.2.7: We shall call 0 = D'r/ the (2, O)-torsion of the connection, and r = D " y the (1, 1)-torsion of the connection. Locally, we may write 0 = O" |

+0~ |

and

r = r a | C~,,

where 0# :

-dz

v A~,~

:

1 p, - r ~ , l d z ~ ^ d z ~ + r , . , r , ^ dz~ ; ~[r~;~

= ~1 [ e A r ,ct) - ~,(r~)] d z , A d z " + [b~(r~) - r~;,] r + ~ [ r ~ - r~z ]

Adz t'

(2.2.7)

A Cv;

r ~ = 0 r ~ = - 5 o ( r ~ ) dz" A de ~ - cSa(I'~) dz" ^ r We shall see in the next section that one of the prominent features of the ChernFinsler connection is that 0~ = 0, a fact without references in the real setting. Anyway, an immediate corollary is COROLLARY 2.2.1: The (1, 1)-torsion r vanishes iff the frame { ~ , c~c,} is holomorhic. Proof: Indeed the frame {5~, 0~) is holomorphic iff its dual coframe {dz ~, ~/,~'} is, which happens iff the forms r are holomorphic, that is if[ r ~ = cqr a = 0 for [] a= l,...,n.

81 The torsion(s) and the covariant derivative enjoy the usual relationship: PROPOSITION 2 . 2 . 2 : Let D: X(Tc](/I) ~ X(T~I(/I | Tr be the complex linear connection induced by a good complex vertical connection. Then for any X, Y 9 X(Tl'~ we have

V x Y - V y X = [X, Y] + 8(X, Y); V x Y - V v X = [X, Y] + r(X, Y) + "~(X, Y). Proof: Write X = X~6~, + J~"~O~ a n d Y = Y~6~ + ~'~0~. Then V x Y = [X(Y') + ~v;(X)V u] 6, + [X(]J~) + w~(X)]J~] c~,

v v x = [ v ( x ' ) + ~;(Y)X.] 6, + [Y(X~) + ~ ( y ) x * ] b~, 8(X, Y) = [w;(X)Y u - w~(Y)X"]6, + [w~(X)]~' - w~(Y)X'~]b~ + OOZ(X, Y)O~, [X,Y] = ( X ( Y ' ) - Y ( X ' ) ) 6 , + r or

YI)O~,

Y) = X(] ~z) - y ( _ ~ ) - r

Y]),

where we used the fact that [6u, 6~], [6,,, 0~] 9 yl,0, and the first formula is proved. For the second one, V x Y = X(Y-'~)6~ + X ( ? ~ ) O j ,

9) = _~(s

= Y(x")G + Y(R~)G,

~(x, P) = a ~ ( x , 9)b~,

~ ( x , V ) = Or Or

vvz

_r

g]), ar IX, Y--]= x(Y-w)& - 7 ( x " ) G + r

9) = x ( ? q - ~ ( [ x , 9]), + ~([x,Y])bz,

where we used the fact that [6~, 6~], [6u, 0~] 9 Yc, and we are done.

2.2.5.

The

[]

curvature

DEFINITION 2.2.8: The curvature tensor R: X(Ta,~ --* X(A2(T~M)|176 a good complex vertical connection D is given by R = D o D, that is VX E X(TI'~

of

R x = D(DX).

Analogously the curvature operator ~ 9 X(A~(T~2~I) @ Al'~174 Ta,~ defined by ~(x, v ) z = Rz(X, y). Locally, ~ is given by = f l l | [dz~ | 6~ + r

| 0o],

where =

--~fl

A o.'~,.

of D is

82 As in the real case, everything is globally well-defined. Decomposing ~2 into types, we get fl = ~Y + fl", where fl' is a (2,0)-form and fl" a (1,1)-form. Locally,

(n')~ =

a~; - ~

^

~?,

(n");

= a~;.

f~ has no (0,2)-components because the connection forms are (1,0)-forms. As we shall see in the next section, fl' vanishes identically when we consider the curvature operator of the Chern-Finsler connection, leaving ~ " as the only meaningful part of the curvature. The relation between curvature and covariant derivatives is the usual one:

PROPOSITION 2.2.3: Let D: X(TI,~ ---, X ( T ~ I |176 be the complex linear connection on ]~I induced by a good complex vertical connection. Then for any X, Y E X(TI'~ we have VxVy

- VvVx

= Vtx,vl +

a ' ( x , Y);

VxV-v - VyVx = Vtx,V1 + ~ " ( x , Y ) ; V ~ - V y - VyV~- = V[-~,-V]. Proof: Write X = X~'6~ + X ~ 0 ~ , Y = YV5 v + l;'~0~, and take a vertical vector field V = V~3,. Then VvV = [Y(V ") + w~(Y)Va]3~,

VxV = [X(V ~) + ~ ( X ) V ~ ] b ~ ,

V x V y V = [X(Y(V'~)) + X(o:~(Y))V ~ + w~(Y)X(V ~) +w~(X)Y(V ~) + w~( X)w~(Y)V a] b~,, V y V x V = [Y(X(V'~)) + Y(w~(X))V/~ + w~(X)Y(V ~) +w~(Y)X (V ~) + w~(Y)w~( X)V ~] a~, v t x , , , y = [x(r(v~

- Y(X(V~

+ ,,,;(IX, r])va] &,

fl'(X, Y)V = [Ow~(X, Y) - oa~(X)w.~(Y) + w~(Y)w~(X)] V~c~.,

aw;(X, Y) = X(w;(Y)) - Y(w;(X)) - ~;([X, Y]), mad so the first formula (applied to a vertical vector field) is verified. For the second one,

v T v = g(v~)&, VxV = [x(v") + o4(x)v~]bo, VxV~V = [x(Y(w)) + ~;(x)Y(v~)] b~, VvVxV = [Y(x(v~ + V(~;(x))v~ + ~;(x)Y(v~)] &,

Vtx,vlv = [x (Y(v~

-v(x(vo))

+ ,,,;(Ix, 71)v ,~] &,,

83 ~"(X,Y)V

= (&G(X,~))VeOo = [-V(w;(X)) - w;([X,~])] Ve0,,

and we are done. Finally,

v ~ v v v = ~(-f(vo))bo,

v~,v~v = [x(-f(P'))

v v v ~ v = V(-Z(v~

- v(X(v~)) + ~([~,v])w']

ao,

and, since [X,Y] 9 T~ that w ; ( [ X , ~ ] ) = 0, we are again done. The same computations prove the formulas for horizontal vector fields. [] We end this section recovering the Bianchi identities in this setting. PROPOSITION 2.2.4: Let D: X(TI,~ ~ X ( T ~ I |176 be the complex linear connection on ]VI induced by a good complex vertical connection. Then D'0 = r/A f~',

D " r = 0,

D"O + D' r = r1 A ~", D'fl' = O,

D"f~" = O,

D"~' + D'~" = O. Proof: It suffices to compute. First of all, o e . + e" ^ ~

= dz" ^ o ~

= r

^

- dz ~ ^,.,~ ^ ~

= dz ~ ^ (f~')~;

(~')~,

and so D'8 = rI A f~'. Next

(.O/'~u \-/L',

O0 ~' = d z ~' h Ow#. = d z ~' ^

~o.,

a~'~

= a o ~ ,~ _ & ~ ^ ~

+ r

^

= aor ~ _ or

+ r

^ (e")~,

a~-" + r~ ^ ~ ; = a0r ~ + ar

^~

^ ~ = -0ar

+ &~ ^ ~,

and so D " 8 + D ' r = 7/A ~2". Now

O(~')~

-

(~')~ ^ ,~ + (~')~ ^ w? = -~,~ ^,.,.,~ +,.,.,~ ^ o,.,.,~ -,.,.,~ ^ o,,.,., +,,.,~ ^,.,,.., +0~^~ - ~ ^ ~ '~A ~ ~ = 0 ,

and D'f~ ' = O. On the other hand, a(~')~ = aa~

- a~

^ ~

a(~")~ - ~ ^ (~ " )~" + (~,,)~ ^ ~ ~ = a a ~

+ ~ - ~

^ a~, - o + a~ A O~.,

a n d D"f~' + D'f~" = 0. Finally, D " r = 0 and D"f~" = 0 are trivial.

^ ~ ,~" []

84 2.3.

The Chern-Finsler

connection

2.3.1. Complex Finsler metrics DEFINITION 2.3.1: A complex Fhasler metric on a complex manifold M is a continuous function F : T I , ~ --* R + satisfying (i) G = F 2 is s m o o t h on 2~/; (if) F(v) > 0 for all v E h:/; (iii) F ( # r = I~lr(v) for all v E T I ' ~

and r E C.

We shall systematically denote by G the function G = F 2. Note t h a t it is i m p o r t a n t to ask for the smoothness of G only on Jr?/: as we shall show momentarily, iff F is the norm associated to a Hermitian G is s m o o t h on the whole of TI,~ metric. If this is the case, we shall say t h a t F comes from a Hermitian metric. Again, the easiest example of complex Finsler metrics not coming from a Herm i t i a n metric is given by a Minkowski metric. DEFINITION 2.3.2: A complex Minkowski space is C n endowed with a complex Finsler metric F : C" x C n -~ T 1,~ --+ R + given by

Vp E C n Vv E T~'~

'~ C n

F(p; v) -- Ilvll,

where I1 I1:c" --, R § is a complex norm (with strongly convex unit ball) on C". If II. II is not the norm associated to a Hermitian product, then F does not come from a H e r m i t i a n metric.

In the past twenty years, a number of (not necessarily smooth) complex Finsler metrics have become a very useful tool in geometric function theory of holomorphic mappings - - a n d indeed this is the origin of our interest in complex Finsler geometry. The two most i m p o r t a n t such metrics are i n d u b i t a b l y the Kobayashi and Caratht~odory metrics, which we presently define. DEFINITION 2.3.3: Let M be a complex manifold; then the Kobayashi metric ~ R + of M i s given by

FK: T I ' ~

Fg(p;v) ----inf{l~l [ 3~o E

H o l ( A , M ) : q0(0) = p,

W0(~) = v},

for all (p; v) E TI'~ where A is the unit disk in C and Hol(A, M ) is the space of holomorphic m a p s from A into M . DEFINITION 2.3.4: Let M be a complex manifold; then the Carath~odory metric ~ R + of M is given by

Fc: T I ' ~

Fc(p;v) = sup{IdL(v)l [ f for all (p; v) E T a'~

E Hol(M,A)

f(p) = 0},

85 In general, the Kobayashi metric is only upper semicontinuous on _~/, and the Carathdodory metric is only continuous. However, Lempert in his fundamental paper ILl proved that if D CC C n is a strongly convex bounded domain of C n then the Kobayashi metric (and the Carath6odory metric too, for they coincide there) is a smoo$h complex Finsler metric, furthermore enjoying a number of very interesting properties (existence of complex geodesics, of Monge-Amp~re potentials, and so on). Most of our chapter 3 is devoted to recover Lempert's results under only differential geometric assumptions on the (smooth) complex Finsler metric, instead of under a priori assumptions on the geometry of the manifold. Our assumptions will imply that the given complex Finsler metric necessarily is the Kobayashi metric of the manifold; therefore the unique part of Lempert's results actually depending on the strong convexity of the domain is the proof of the smoothness of the Kobayashi metric. For more informations on Kobayashi and Carath6odory metrics see, e.g., [A] and references therein. To start working on general smooth complex Finsler metrics, we need a few notations and general formulas. We shall denote by indexes like a, r and so on the derivatives with respect to the v-coordinates; for instance,

G~

=

O2G Ov~O~.

On the other hand, the derivatives with respect to the z-coordinates will be denoted by indexes after a semicolon; for instance,

02G G;~ = Oz,Oz~

02G or

G~;~ = O~Ov .

For our aims, we ought to focus on a smaller class of Finsler metrics. DEFINITION 2.3.5: A complex Finsler metric F will be said strongly pseudoconvex if the Levi matrix (G~t}) is positive definite on 2t7/. This is equivalent to requiring that all the F-indicatrices 10 IF(p) = {v e T~' M [ F ( v ) < 1}

are strongly pseudoconvexes. As we shall see in 2.3.2, this hypothesis will allow us to define a Hermitian metric on the vertical bundle. The main (actually, almost the unique) property of the function G is its (1,1)homogeneity: we have V(p; Cv) = r v) (2.3.1) for all (p; v) E T l ' ~ and ~ E C. We now collect a number of formulas we shall use later on which are consequences of (2.3.1). First of all, differentiating with respect to v '~ and v~ we get Go(p; Cv) = ~a~(p; v),

V~z(p; (v) = GaB(P; v),

(2.3.2)

86 Thus differentiating with respect to ~ or ~ and then setting ~ = 1 we get G~3 v---~= G~,

G ~ v ~ = 0,

(2.3.3)

and G a ~ v ~ = -G~t3 ,

G~t~7

= G~,

G~v

~ = 0,

(2.3.4)

where everything is evaluated at (p; v). On the other hand, differentiating directly (2.3.1) with respect to r or ~ and putting eventually r = 1 we get Go, v ~ = G,

Gat~ v~v ~ = O,

G~3 v~v--~ = G.

(2.3.5)

It is clear that we may get other formulas applying any differential operator acting only on the z-coordinates, or just by conjugation. For instance, we get Ga;~ ~ = G; u,

(2.3.6)

and so on. By the way, this is enough to prove the claim about the smoothness of G: LEMMA 2.3.1: Let F be a complex Finsler metric on a complex manifold M . Then G = F 2 is smooth on T I ' ~ i f f F comes from a Hermitian metric on M. Proof: One direction is clear. Conversely, assume G smooth (C 2 is enough) on TI,~ Take (p; v) 6 it:/; then (2.3.5) yields

G(p; v) = Go~(p; v)v%~. Take t > 0; by (2.3.2) Ga~(p; tv) = G~3(p; v). Hence Vt > 0

G(p; v) = GaB(p; t v ) v % ~ .

Letting t go to zero, by smoothness we obtain

a(p; v) = a,z(p; o,),%~. It is easy to check that setting g~[~(p) = G~3(p; %) one gets a Hermitian metric on M , whose associated norm is exactly F. [] Assuming from now on (unless explicitly noted otherwise) F strongly pseudoconvex, we get another bunch of formulas which we shall need later. As usual in Hermitian geometry, we shall denote by (G ~a) the inverse matrix of (G~$). First of all, applying G Sa to the first equation in (2.3.3) we get G ~ G ~ = v--~,

(2.3.7)

G ~ ; ~ G ~ G ~ = G;t,.

(2.3.8)

and thus, applying (2.3.6),

87 Recalling that (G ~a) is the inverse matrix of (G~3), we may also compute derivatives of G ~ :

DG ~ = - G ~ G ~ (

DG ~ ),

(2.3.9)

where D denotes any first order linear differential operator. As a consequence of (2.3.4) and (2.3.9) we get for instance

G~a'~-~"x = -G~'~G~UG,~,~-~ -x = O,

(2.3.10)

and recalling also (2.3.7) we obtain

G ~G ~,~ = _G ~G ~UG ~ G u~,.t = _c:,_f'r

~ u~'t- vU = O.

(2.3.11)

2.3.2. T h e C h e r n - F i n s l e r c o n n e c t i o n

Now we may start to work. To any Hermitian metric is associated a unique complex linear connection such that the metric tensor is parallel: the Chern connection. The main goal of this section is to define the analogue for strongly pseudoconvex Finsler metrics: The first observation is that F defines a Hermitian metric on the vertical bundle 1). Indeed, if v 6 i f / a n d W1, W2 6 1)v, with Wj = WTO~ , we set

(w1, W~)v = a~(v)WTW~. Being F strongly pseudoconvex, (,) is a Hermitian metric as soon as it is welldefined. But in fact = (J$A)~(JhA)AG~)A,

(W;)~ = ( J ~ a ) . ~( w ~ ) A , and everything works. Note that the third equation in (2.3.5) says that

a = (~, ~); so e is an isometric embedding of )tT/into 1). The main theorem of this subsection states that there is a unique good complex vertical connection D making this Herraitian structure parallel: THEOREM 2.3.2: Let F: TI,~ --* R + be a strongly pseudoeonvex Finsler metr/c on a complex manifold M, and let (,) denote the Herrnitian structure on 1) induced by F. Then there is a unique complex vertical connection D: X ( F ) --* X ( T ~ I | 1)) such that X (V, W) = (VxV, W) + (V, V-~W) (2.3.12) for a / / X 6 Ti'~

V, W 6 2((!)). Furthermore, this connection is good.

88 Proof: Assume such a connection exists; we shall recover the connection forms w'~, showing its uniqueness. The tensor = G~r

~| r

(where here {Ca} is just the dual cofrarne of {vh,,}) is clearly globally defined. Since the covariant derivative V commutes with contractions, (2.3.12) is equivalent to V~ = 0, that is to

dGo,~ - G-r3 w~ - G,~r w~ = O. Since the w~'s are (1,0)-forms, this is equivalent to OGo3 = G.13w~, that is to

w~ = G ' " OGr

(2.3.13)

and the uniqueness is proved. In particular,

so I'~.tv # = 0 by (2.3.4), and the connection is good. Conversely, take V E X 0 ) ), v E .~r and X E Tva'~ To define V x V E Y~ it suffices to know its product with any other element of V,; so we set

(vxv, w) = x(v, w) for any section W E X(V) which is holomorphic in a neighborhood of v, so that X(V, W)(v) depends only on the value of W at v, and not on the derivatives of W. Once defined V x V , we define V T V by (V~V, W) = X(V, W) - (V, V x W ) , where this time W is any section of I;. Then it is easy to check that the map D: A'(V) --, X(T~.,~/I | I;) so defined is a complex vertical connection satisfying (2.3.12), and we are done. [] DEFINITION 2.3.6: The unique good complex vertical connection D whose existence is asserted in the previous theorem is the Chern-Finsler connection associated to the strongly pseudoconvex Finsler metric F. The connection matrix is given by

w~ = GfC'OGa~ = G~'C'(Gaf.;~dz ~' + G~§ dv'l), that is

f"~'r = G~~ #e't

and

"~ = G rt~;u

In particular, r ~ = P~;.v a = Ge~

~Ot

Ga§

89 Denoting by 7-/ C TI'~ the complex horizontal bundle associated to the Chern-Finsler connection, the natural local frame {~1,..., Sn} for ~ is given by 6 r = 0 r - F~0~ = 0 r - ( G ~ G § From now on we shall work only with the frame {~r,0~} and its dual coframe { dz r, Ca}, where r

= dv ~ + P~ dz r = dv ~ + G~'G-~;r dzr"

Writing we get O~

Ot

(2.3.14)

Pa;~ " = G ~ S r ( G a f ) = G ~ (Ga,;r-G~r~). Note that P~; r = 0 , ( F : ) ;

(2.3.15)

for this reason from now on we shall write F~r instead of F~. Furthermore,

r ~ = r~, more or less by definition - - cf. (2.1.14). Being D a good complex vertical connection, it extends to a complex linear connection on 2~/. Using the complex horizontal m a p | l; ~ 7-/we can transfer the Hermitian structure ( , ) on 7"/just by setting VH, K 9 7 - l , ,

(H,K),, = (O-I(H),O-I(K))v,

and then we can define a Herrnitian structure on TI,~ by requiring ~ to be orthogonal to )2. It is easy to check that these definitions are compatible enough so to get X ( Y , Z) = ( V x Y , Z) + (Y, V T Z ) for any X E TI'~

Y, Z E A'(TI'~

2.3.3. Some computations The Chern-Finsler connection enjoys a number of interesting features, making computations much simpler than for a generic good complex vertical connection. For instance, we have LEMMA 2.3.3: Let D be the Chem-Finsler connection associated to a strongly pseudoconvex Finsler metric F, and let {61,..., 6n } be the corresponding locM horizontal frame. Then [~., ~ ] = O,

[~., ~]

: ~(r~'.)&

~ ~.(rr~)&,

[~, b,~] [~.,&]

r

"

~ 9 : r~;#Oa,

[oo, ,9~1 = o, [oo, b,] = o,

9O for a/1 1 __ a, ~, #, u < n, where F~-;, = 0a(F~). P r o o f : It suffices to compute. First of all,

[~., ~ 1 = ( r ; . . - r ; . . + r ~ ; ~ r ; . - r ~ ; . r ; . ) 0 ~ , where r;~ v = O~(r~) and so on. Now, r;,~ = a

(G.;,.

- G~.;.F[,).

r.;~r;,

= G

(a..;.r;,

- a..,r;~r;,).

and so [6~, ~ ] = 0. Note that we have actually proved that

~.(r;~) = ~.(r;~).

(2.3.16)

Next, [ ~ .,b ~ ] = [o.

r ;~. 0 ~ , 0~] "

0"~ ( r ; .~) 0 ~"

~

"

Now~ [~., ~1 = [~. - r ; ~. 0 ~", o~ - r;~0z] = r ~ o & ~(r;.)0~

- r~.0~

- r ; Z ~ r ~ , . & + r ; . r o ~~o 0 ~"

- ~ . ( r ~ ) 0 z,

where P;.~ = 0o(F~), and so on. Furthermore, [~.,0~1 = [0.

rT.b. ,&]

0"a ( r ; .~) 0 ~ "

~

" []

The remaining assertions are trivial.

LEMMA 2.3.4: L e t D be the Chern-Finsler connection associated to a s t r o n g l y pseud o c o n v e x Finsler metric F , and let { ~1, . . . , &~} be the corresponding local horizonted frame. T h e n

~.( a ) = ~,( a) = ~.( a~) = o for all l <<_a, l~, u < n. P r o o f : Indeed, ~.(a)

= a;.

- r~'.a~ = C;. -

C'~C~;.a~=

a;. - G~;.F

= 0.

Next ~g(G) = ~,(G), because G is real-valued. Finally, ~(a~)

= a o ; ~ - F;~G~e = G~;~ - G~;~ = 0. []

Another feature of the Chern-Finsler connection is that the radial horizontal vector field X induces an embedding of ~/" into ~ which respects the Lie algebra structure.

91 DEFINITION 2.3.7: A (never vanishing) vector field ~ 9 X(]I~/) may be lifted in two different ways to vector fields in TI,~ via the horizontal lift

and via the vertical lift

A consequence of Lemma 2.3.3 is that the horizontal lift is a Lie algebra homomorphism: PROPOSITION 2.3.5: Let D be the Chem-Finsler connection associated to a strongly pseudoconvex Finsler metric F on a complex manifold M. Then:

(i) [X(~), X(~)] C X ( ~ ) and [X(V), X(V)] C X(V); (ii) ff~l, ~2 E X(hS/) then [~H,~H] = [~l,~2]H, [~V,(V] = 0 and [~H,~V] 9 X(I)). Proof: (i) Take H1, H2 9 X(7-/). Locally, Hj = H~6~; hence

[H,,Hd = (Hr~.(H~) - H;~.(HD)~.

(2.3.17)

(where we used Lemma 2.3.3) is horizontal. Analogously, if V1, V2 9 X(P) with Vj = Vj~ba, we get

[vl, v2] = ( v f & ( v : )

- v~o~(v~'))&,

(2.3.18)

which is vertical. (ii) Locally, ~j = ~(O/Oz ~') and ~H = (~, o rr)6,; so (2.3.17) yields [C,C]

= ((~' o ~)~(r

o ~) - ( ~ o ~ ) ~ ( r

o ~))~..

Now 6~(~ o r) = (O~/Oz ~) o r; therefore

[r162162Lk.lOz~ r(-0r

,az.] ~

] ~#=[r

H"

On the other hand, ~y = (r o r)ch~ and cha(~ o r) = 0 yield [C,

r

= o.

Finally,

L\ again by Lemma 2.3.3.

'Oz.)

o~ + ((r162162

0o, []

Note that, as a consequence of (ii), the obvious map of X(F) into Pc'('/-/) induced by the complex horizontal map O: 12 ~ A' is not an isomorphism of Lie algebras; it suffices to remark that O(~ V) = ~H for all ~ E k'(.~/).

92 2.3.4. T o r s i o n s and c u r v a t u r e s Also the torsions and the curvatures of the Chern-Finsler connection enjoy a n u m b e r of nice properties. Some of t h e m are collected in the next couple of results.

PROPOSITION 2.3.6: Let D be the Chern-Finsler connection associated to a strongly pseudoconvex Finsler metric F on a complex manifold M. Then (i) 0~ = 0 for a = 1 , . . . , n ; (ii) a'= O.

Proof: (i) It follows from (2.3.16), (2.3.14), (2.3.15) and (2.2.7). (ii) By definition, w~ = G ~ O G ~ . So

Oto~ = OG ~/3 ^ OG,~§ = -G~'~'G~'ZOG ~,~ A O G ~ []

If we write 77 = r/~ + r/v, where

r/~ = dz ~ | 6~

and

r/v = ~b~ | 0~,

t h e n Proposition 2.3.6.(i) amounts to say 0 = Dr~ ~ = D ' ~ ~

and

T = Dr~ v = D " ~ v.

For future references, we record here the local expression of 0 = O# | r = r a @ 0~ for the Chern-Finsler connection: Oa

=

1

a

o" r r dz" A d z u + F~.

-

T a =--~o(F~),

A dzU;

dz ~*Ad5 v - r ~ ; ~ , d z ' A r

6# and

(2.3.19)

As a consequence of Proposition 2.3.6.(ii), ~ = D" for the Chern-Finsler connection; in particular, VXVy -- V y V X : V [ x , y ] , (2.3.20) for all X , Y 6 Tl'~ From now on we shall always write ~ instead o f ~ " . T h e Bianchi identities too assume a simpler form:

COROLLARY 2.3.7: Let D be the Chern-Finsler connection associated to a strongly pseudoconvex Finsler metric F on a complex manifold M. Then D ' O = O,

D"8 = r / n A D,

DI.r = ~ v A f~,

D"T

DID = 0,

D"f} = 0.

= O,

P r o o f : It follows from Propositions 2.2.4 and 2.3.6.

[]

93 In local coordinates, the curvature operator is given by a ~2~ = R~;,r, dz t' ^ d~ ~ + R~;r, r

where

5

a h d~ ~ + R~;~, dz t' A r

n~;.~ = - , ~ ( r ~ ; . )

+ R~

r

A -r

- rL,~(r;~).

(2.3.21) Ra~;~ = - 0 ~ ( r a ; v ) - Pa,P~; v,

R~ = -04r~) = R~. In particular, since

+ R~;, r

A d z ~' A r

= -R~,;.~ dz ~ ^ r

+ R~

r

Ar

^ d2 ~ - R ~ ; , d z ~ ^ r

A

~,~,

the vanishing of r implies the vanishing of most of the curvature; cf. Corollary 2.2.1. Another consequence of (2.3.21) is an unexpected relation between ~t and r: LEMMA 2.3.8: Let D be the Chern-Finsler connection associated to a strongly pseudoconvex Finsler metric F on a complex manifold M. Then

= ~ ( . , .)~. Proof: Recalling (2.3.21), (2.3.3), (2.3.4) and

r ~ ; . v ~ = r;~.,

r~v~

= O,

we have

R~;~v ~ = 0,

R~;vv~ = - r ~ ; ~ , R ~ v ~ = o, and the assertion follows from (2.3.19).

[]

2.3.5. K~ihler Finsler metrics

In the Hermitian case, the vanishing of the torsion is equivalent to the metric being K/~.ler. In our case the situation is a bit subtler, for instance because our torsion 0 has a horizontal part and a mixed part. DEFINITION 2.3.8: A form 7 E A(_~/) is horizontal if it vanishes contra~ted with any V 9 X(1;). The decomposition T 1 , 0 37/ = 7/@1; induces a projection p ~ of A(37/) onto the horizontal forms; the horizontal part of a form 7 is then P~/(7)-

94 There is a corresponding projection on the vertical forms, of course, but we shall not need it now because the vertical part of both torsions 0 and 7- is zero. For this reason, the form 0 - p*H(O) will be called the m/xed part of 0. In local coordinates,

p*H(O) = (F~;, dz" h dz ~) | ~

and

0 _ p*H(O) = (F~. r r

^ dz ~) | ~ .

The next proposition discusses the meaning of the vanishing of the (2,0)-torsion O or of one of its parts. PROPOSITION 2.3.9: Let F be a strongly pseudoconvex Finsler metric on a complex manifold M . Then: (i) the mixed part of the (2, O)-torsion 0 vanishes iff F comes from a Hermitian metric; (ii) 0 itself vanishes iff F comes from a Hermitian Ks metric.

Proof: (i) The mixed part of the torsion vanishes iff G~g7 = 0 for all 8, # and 7. Conjugating, this is equivalent to having vS-r(G~#) = 0 = O~r(GB#), that is G~#(v) depends only on lr(v) - - and this happens iff F comes from a Herrnitian metric. (ii) It follows from (i) and the fact that when F comes from a Hermitian metric g = (ga~) one has (~Z#

*

[]

The meaning of the next definition is then clear: DEFINITION 2.3.9: We say that a strongly pseudoconvex Finsler metric F is strongly Ki~ller if the horizontal part of the (2,0)-torsion vanishes, that is iff

VH, K 9 7-l

O(H, K ) = O.

It turns out that this definition is too stringent. The point is that, as we shall see in the next two sections, from a geometrical point of view only the vanishing of suitable contractions of 0 is relevant. DEFINITION 2.3.10: We shall say that F is Kifuhler if VH e 7-/

0 ( g , X) = 0,

and that F is weakly K ~ / e r if

V H 9 7"l


95 In local coordinates, F is strongly K~hler iff i~;~, = Fg;~) a . it is K~hler iff a

r~;vv

it is weakly Ks

/z

c~

/z.

= rv;~v ,

iff

that is iff 0 = [ a , ; , - av;~ + avvr~,]v" = [a,~;,

-

av,;, + award',lye7.

(2.3.22)

(and so this is the condition we called Ks in [AP2]). In particular, if F comes from a Hermitian metric then these three conditions are all equivalent to the usual K~'der condition, because Gvae - 0 for a Finsler metric coming from a Hermitian metric. There are other characterizations of strongly K/ihler Finsler metrics. DEFINITION 2.3.11: To a complex Finsler metric F we may associate the fundamental form = i G ~ dz ~ A dzZ, which is a well-defined real (1,1)-form on _h~/, thanks to (2.1.6) and (2.1.7). Then the strong K ~ d e r condition is equivalent to the vanishing of the horizontal part of d~2. To express it more clearly, set dH

= P*H o

d,

OH = P*H o 0

and

~H = P*H o O,

so that again dH = OH q- OH. THEOREM 2.3.10: Let F be a strongly pseudoconvex Fins]er metric on a complex manifold M . Then the following assertions are equivalent: (i) (ii) (iii) (iv) (v)

F is a strongly K~dller Finsler metric; V H K -- V K H = [H, K] for all H, K E X(7-l); d H r = O; 0HCI) = 0;

for any Po E M there is a neighborhood U of po in M and a real-valued smooth

function r e

such that 9 =

0 0Hr on

Proof: (i) ~=~ (ii): Proposition 2.2.2. (iii) ,v==~ (iv) holds simply because r is a real (1,1)-form. (iv) ~ (i). Indeed, (2.3.14) yields h e ( x , Y, Z) = i{O(X, Y ) , Z) for all X , Y, Z E Tl'~ strongly K ~ l e r .

hence OHr vanishes iff p ~ o 0 vanishes, that is iff F is

96 (v) r (iii) One direction is clear. For the converse, let 7 be any horizontal form. In local coordinates, defined on a coordinate neighborhood of the form 7 r - l ( U ) , one has 7[(p;~) = TAB(P; v) dz A A d"~-g, for suitable multi-indeces A a n d / ~ . On U we m a y then consider the family of forms

%,lp = 7A[~(P; v ) d z A A d'~ff E A(U), where here {dz j } is the dual frame of {O/OzJ}; in other words, we are considering the v-coordinates just as parameters. The gist is t h a t the following formula holds:

( d . 7 ) ~ = d(~v). T h e n we m a y now a p p l y the Dolbeault and Serre theorems (with p a r a m e t e r s to ~ in a possibly smaller neighborhood of P0 - - still denoted by U - - to get a function r E C ~ ( U , R) depending smoothly on v such t h a t ~,, = i00r Then setting

r

v) = r

we get 9 = iOH~Hr as required.

[]

From this point of view, a strongly pseudoconvex Finsler metric is K/ihler iff

dHr and it is weakly Ks

X, ") = 0,

iff

dHq~(., X, X) -- O.

2.4. V a r i a t i o n s of t h e l e n g t h integral 2.4.1. The setting Let F : T ] ' ~ ---+R + be a strongly pseudoconvex Finsler metric on a complex manifold M . To F we m a y associate a function F~ TRM ~ R + just by setting

Vu E T ~ M

F~

= F(uo).

F ~ satisfies all the properties defining a real Finsler metric but perhaps the last one: the indicatrices are not necessarily strongly convex. Nevertheless, we m a y use it to measure the length of curves, and so to define geodesics; and one of the m a i n results of this section is a theorem ensuring the local existence and uniqueness of geodesics for weakly K~hler Finsler metrics - - a striking by-product of the complex structure. Of course, it is conceivable (cf. [P2]) t h a t the indicatrices of F ~ could be necessarily strongly convex if F is weakly K/~hler; nevertheless, the m a i n point

97 here is that it is possible to develop a theory of geodesics in weakly Ks Finsler manifolds without assuming the real convexity. Let us fix the notations needed to study variations of the length integral in this setting. Let a: [a, b] --~ M be a regular curve; we define &: [a, b] ~ M by setting

&(t)= -j-C~ da~" t)" ~ 0

~(t) 9

Then the length of a is given by

L(a) =

F(&(t)) dt,

exactly as in the real case. Let E : ( - r • [a,b] --* M be a regular variation of a given regular curve -~ ( - e , e) • [a, b] a0: [a, b] --* M. We shall again set g~(s) = L(as). Let p: E * ( T I , ~ be the pull-back bundle, and 7: E * ( T I ' ~ ~ TI'~ be the bundle m a p such that the diagram ~_~ Tl,O M E*(T1,0M)

(-~,~)

• [a,b]

commutes. An element ~ 9 X(E*(TI,~

~(~,t)

= va(~,t)

----*

M

can be written locally as

0

=

(s,,) o,

ua(~,t) ~

where we used the notations introduced in subsection 2.1.1. TR(E*(TI'~ is given by {Os, Or, Oa}, "o where

Os = O/Os,

Ot = O/Ot

A local frame on

Oo = O/Ou a,

and

or by {O~,Ot,ba,iba}, where 0a -- O/Ov a. Two particularly important sections of E * ( T I ' ~

are

Ot Oz ~' and U=7-1

(dl'~ (o)) ~

=

o

Os Oz a'

the section U is still called the transversal vector of the variation E. Again, setting E * ~ / = 7-1(.~/), we have T 9 X(E*_~/) and

T(8,t) =

'~--l(~s(t)).

98 Now we pull-back Tl'~

E*ff/by using 7, obtaining the commutative diagram

7,(T,,Off/)

_.__++ T I .off/

1

l

~*ff/

~

if/

-~,

M

1

(--r e) X [a, b]

.,

l

note that 7*(Tl'~ is a complex vector bundle over a real manifold. Tim bundle map "~ induces a Hermitian structure on 7*(Tl'~ by VX, Y

e 7*(T l'~

(X, Y)v = (~(X), ~(Y)),(v).

Analogously, the Chern connection D gives rise to a complex linear connection --+ X(T~(E*ffI) | 7*(T 1,~

D*: X('y*(Tl'~

where T~(~*ff/) = T~(~*ff/) | C, by setting

V } Y = ~-a (Vd,,O.~(x);/(Y)) ,

for all X + T~(~*)~/) and Y E ,~(7"(T1'~

We remark here that if we write

"a'o x = x'o~ + x'o, + XoO~ ~ T~(~*ff/)

then

d',~ where X'r

= [Ix ~oz~ as + x ' _~_] o~, + 2 " & = (d.r(X))o,

X'oa + zX o 9

"

a + n

. Furthermore we have

X ( Y , Z) = X ((~(Y), ~(Z))~) = d'r(X)((~(Y), ~(Z))) + da,~

= (dl'~

q(Z)))

(2.4.1)

= (v}Y, z ) + (Y, V x Z ) + (VxY, Z ) + (Y, v } z ) ,

for all X E Ta(E*2(/I) mad Y, Z E X(7*(TI'~ We may also decompose T~(~*]~7/) = 7"/* G 12", where as usual a local frame for I;* is given by {0~, i0~}, and a local frame for 7"/* is given by 6, = 0, - (r;~ o v ) -0E~o6 g/-,, -

Therefore, setting T H = da,~ OE~'

6. = 0 . - ( r ; . ~ o

and U H = d1,~

~176..

7)--~--

we have

T"(v) = ---~-(~,t),5.1~(~) = x.,<~)(a.(t)) e n.,(~>

99

and

OE~' UH(v) -- ---~-s (s,t)5,l.r(,) -- xT(v)(7(U(s,t))) E 7-17(~), ~

for all v E E*M(~,t). In particular, TH(3'-I(dr,)) = X(&s). Again, we have a bundle map E: TR(E*2tT/) --, 7*(T 1,~ TR(E* ~/)

~-

dr'~ ~

(2.4.2) such that the diagram

7*(TI'~

i~ T1,0~/

commutes. Using E we may prove three final formulas: "y(V~(Y)

-- V~-~(X)) = Vdl,OT(X) dl,~

) - VdX,O~(v)dl'~ = [d,,oT(X),dLOT(y)] + 6(dl,oT(X),dLoT(y)),

(2.4.3)

for all X, Y 9 X(T~(E*/tT/)); ;'[ 0 ( V ~ V~. -- V ~ V ~ )

~-

(Vd,,O.,l(X)Vd,,O..,l(y) -- Vd,,oT(r)Vd,,OT(X) )

0

(2.4.4)

= V[dt,OT(X),d,,oT(y) ] 0 ;'[,

and o (V~r v -

V y V ~ ) = (Vtd,,O.r(X),d~,oT(y)] + ~(dl,~

~

o -~, (2.4.5)

for all X, Y 9 TR(E*21~/).

2.4.2. T h e first v a r i a t i o n f o r m u l a We are now able to prove the first variation formula for weakly K~hler Finsler metrics:

THEOREM 2.4.1: Let F : T I , ~ ~ R + be a weakly KgJz/er Finsler metric on a complex manifold M. Take a regu/ar curve ao: [a, b] --* M with F(&o) =- co > O, and a regular variation E : ( - ~ , e ) • [a,b] --* M ofao. Then

dt~ds ( 0 ) = c'-ol Re(UH,

- Re

(U H, VTU+-f-~TH)~,o dt

.

In particular, i r e is a fixed variation, that is E(.,a) -- ao(a) and P,(.,b) =_ ao(b), we have

d~r, (0) = ds

_ l__R e

co

[ b ( u H , VT~+~--~TH)~o dt. d~

(2.4.6)

100

Proof: By definition, e~,(s) =

~a b (G(&s)) 1/2 dr;

therefore ds ds

b o

1 j[ ~_.~s[G(&,)]dt= 1 tb O _ 6 2c~

where c~ = F(&~) and we used

a(a~) = <x(a.), ~(a~)>~. = {--(6,), =(6,)>~, by (2.4.2). Now, using (2.4.1) and (2.4.3), we get

1 0--~(:~(6,), ~(6t)> T = ~16,(_=(6,),_=(6,)>~ = ~1 {< V*~,_=(6,),--(6,)>~ + <_=(~,),vk_=(6,)>~ + ~ + {--(6,), v* _=(6,)>~}

= Re {(V~-(6t), E(6t)}T + (V~-E(6t), E(6t)>T} = Re{(V;'Z(6,),E(6t)>T + ([UH, T H] + V-O-~-TH, TH>a.

-}-(~(UH,TH),TH)b"}. Since F is weakly Ks

(2.4.2) yields

(~(U ~, T~), Tn)~. = O. Furthermore,

[UH'TH]=[ Os b,k---~-- ) - ~ = 0--~-6~

\ 02 )

6~,

6.;

since

OtOs

= ot

(2.4.7)

kO2)+--~-6~ka2)'

we get

[UH,THI+v_o__mTH = 0-~ Ot 6~\(OEz~ Os ] 6~ = V~-~-UH.

(2.4.8)

101 Then

10 2 0 s (E(6t), E(6t))T -----Re { (V; E(6~), E(6t))T -t- (V~'E(6a),

E(~t))T}

V;,+T' =(

Re

(2.4.9)

= Re { ~ < U H , T H ) o , -- (uH, vTu+'~-~TH)o. } , []

and the assertion follows. As a corollary we get the equation of geodesics:

~ R + be a weakly Kgdder Finsler metric on a COROLLARY 2.4.2: Let F : T I , ~ complex manifold M , and let a: [a, b] --* M be a regu/ar curve with F( &) =- co > O. Then a is a geodesic for F iff VTU +~--~TH = O,

(2.4.10)

where T H ( , ) = Xv(#(t)) E 7-/, for all v E flip(t). Proof: It follows immediately from (2.4.6).

[]

COROLLARY 2.4.3: Let F: T I , ~ ~ R + be a weakly Ki~hler Finsler metric on a complex manifold M . Then for any p E M and v E flip there exists a unique geodesic a: ( - e , e ) --* M such that a(O) = p and &(O) = v. Proof: In local coordinates we have VTn+~--~TH = [(&~'6, + ~-~-6~)(h~) + F~;~(h)&~'0 "] ~= = [ ~ + F~(&)&~']6~. So (2.4.10) is a quasi-linear ODE system, and the assertion follows.

[]

So the standard ODE arguments apply in this case too, and we may recover for weakly K~hler Finsler metrics the usual theory of geodesics, as developed for instance in subsection 1.6.2. In general, looking at the proof of Theorem 2.4.1 we see that the equation of geodesics for a strongly pseudoconvex Finsler metric F not necessarily weakly K~ihler is VTU +~--~TH -- O*(T H, T H) = 0, (2.4.11) where 0* is defined by VH, K, L E 7~

(O(H, K), L)~ = (H, O*(L,-K)),.

In local coordinates, O* = G ~ G~(F~;~ o - F ~~;,~)dz ~ h dU' |

102

2.4.3. T h e s e c o n d variation formula Our next goal is the second variation formula, which holds for K ~ f l e r Finsler metrics. To express it correctly, we need two further ingredients. DEFINITION 2.4.1: The horizontal (1,1)-torsion r ~ is defined simply by

rn(X,g) = O(r(X,7)) = a(X,7)X. DEFINITION 2.4.2: The symmetric product ({,)): 7-/x 7 - / ~ C is locally given by

VII, K 6 7-(v

((H, g)) v = Gc~(v) H ~ K ~.

It is clearly globally well-defined, and it satisfies VH e 7"/

((H, X))

=

0.

Needless to say, one can define a symmetric product on 1; in the same way.

Then the second variation formula for K~hler Finsler metrics is: THEOREM 2.4.4: Let F: TI,~ --* R + be a K//h/er Finsler metric on a complex manifold M. Take a geodesic ao: [a, b] --+ M with F( &o) - 1, and a regular variation P~:( - e , e) x [a, b] ~ M of ao. Then

d2s (0~ = Re(Vun+-~-ffU H, TH)oo b

TH) ,o 2 _

Re[(~(TH,-u-g)UH, o r TH)a

_

(~(uH,'~)UH,TH)ao

In particular, if ~ is a t~xed variation such that I~(U H, TH)ho -- 0 we have d2~

~,

=/[{ ilv .+ m'll .o - Re[(fI(TH,-O-ff)UH, TH)a ~ -- (fI(UH,T--ff)UH, TH)a ~

+ ((.~(v.,yw), u"))~o - <<.~(T', ~-~), v"))~o] } dt P r o o f : During t h e proof of the first variation formula - - in (2.4.9) - - we saw t h a t

tier

[b (v; z(~.),-=(~,))r + (v~z(~,), --(~,))r

ds (s) = Re Ja

dt.

103 So we need to compute

o

(v~ z(6~), ~(~,)>~. + < v~ z ( ~ ) , z(~)>~ 1

(~)l/:

((z(~,), Z(~,))T)~/:

2

(2.4.12) Since, when s = 0, the denominator of the first term on the right-hand side is equal to 1, and the denominator of the second term is equal to 2, we do not need to consider them any further. Let us call (I) the numerator of the first term, and (lI) the numerator of the second term. First of all, (2.4.9) yields 21Re(lI) = Re [[&' O-~lUt't' T H\.'~' - (uH, VT~+-y-gTH>~. 2 . in particular, for s = 0 we get 1Re(li)(0)

=

Re(UH, TH># ~ ~,

0

(2.4.13)

because a0 is a geodesic. The computation of (I) is quite longer. First of all, using (2.4.3), (2.4.4) and (2.4.5) we get (I) = (V;oV;,E(6~), E(St))T +
+ (v~ _(~.), v6 z(~,))r + ~ *

*

~

~'

*

*

~

= (V;,V~.E(6.),E(St))T- (V[TH,u.]UH, TH>~o + #. -
+ (VT. UH, O(UH, TH))*, + (V~-V; S(b~), E(6t))T- ( V [ y ~ y . ] U H, TH)a. + (f~(uH,'T-g)UH,TH)#. + (V~V~-=(6~), E(6~))T- (V[-~--b--ff]U H, TH)ao + i v y = ( 6 . ) , V~,--(6.))T +
-k (V'~'U H, O(UH, TH))#..

104 Recalling (2.4.8), (2.4.2) and that F is KEhler we get (I) = (V; V; E($s), E(6,))T + (V;, V~E(6s), E(6,))T

+ (v~v; . s(,.), ~(*,))T + ( v ~ v ~ s ( , , ) , s(**))~ _ (f~(TH,-O'Y)UH, TH)~. + (f~(uH,'T-Y)UH, TH)~.

UH' TH)a"

-- (V[T/I ,UBI+[T H ,U~r]+[T ~' ,USl+[T",O"]

+ (V*, -(~,),v; ~(~,))T + (v~-=(~,),v;,E(~))r + (vL _=(,.), v~_=(,.))~ + (v~s(,,), v~:=(,.))~. Now Lemma 2.3.3 yields

[T", U"] =

"k, Os / - ~

[T,,-~-~] _ O2 ~' ON"

6~,

6,,

~, 9

~ [6.(r:~)oz _ ,~(r?.)bo] [TH, UH ] _ 0N____~0E__~ Ot Os

~'k Os ,] - ~

[T", UH] =

~'

6,,

and so (2.4.7) and (2.3.19) yield

[TH, U H] + [Tn ,U H] + ITH, U H] + [TH, U H] = r(UH,T H) - r(T H, U H) + r(UH,'T-'~) _ r(T H, UH). Fhrthermore, if V 6 Y we have

(VvUH'TH)8"= Gaj(#s)V'~ [r,~"/~,(o~<,,~ as ,] + = a o ( ~. . ) r ,~, ( ~. D -O~N r v = ((o(v),

.~

_<,

. . .

or:~-

l'~7{'asJ-~-s ] &f

= a~,(o.)

V. ~

u"))~.,

and

(V-VUH, TH)~~ = Gaj(#,)V--~ [0- ( O~c"~] #"-~= O. [ "t Os )} Therefore V H H ( [T.U.I+[T.,U.I+[T.,U.]+[T.,v.]U ,T )~~

= ((~'(u",~--~),u")>~. _

((r~(TH,U--H),

UH))~.,

105 and thus

(I) = *,(V;.+X---=(*,),--=(*,))T -- (V,*.+~--=(*,), V~+W--=(*,))T -

-

(~(TH,-~--~)UH,TH)~. + (n(uH,'T--H)UH,TH)~o

- ((vX(uH,'T--H), uH))a. + ((rn(TH,U--H), uH))a. +

(2.4.14) Recalling that for s = 0 we have V ~ + ~ ( * * ) -- 0 because a0 is a geodesic, (2.4.12), (2.4.13) and (2.4.14) evaluated at s = 0 yield the assertion. [] In particular, as we shall discuss in chapter 3, Theorem 2.4.4 will allow us to express the Morse index form in complex terms.

2.5.

Holomorphic

curvature

2.5.1. Symmetry properties of the horizontal curvature The aim of this section is to study in some details the curvature operator of a strongly pseudoconvex Finsler metric. So let F: T I , ~ ~ R + be a strongly pseudoconvex Finsler metric on a complex manifold M, and let D denote the associated ChernFinsler connection. As we have seen in the previous section, from a geometrical point of view the most interesting part of the curvature is the horizontal one, that is (fl(H, K ) L , M) where H, K , L, M E 7-/~ and v E AT/. In local coordinates,

(f~( H,'-K )L, M ) ,, = G ~,)R~;~,f,H *'K v L '~M a. Of course, it is important to know its symmetries: PROPOSITION 2.5.1: Take v E 2VI and H, K , L, M E 7"l~. Then

(fl(-K, H)L, M ) , = - ( f l ( H , K ) L , M)~;

(2.5.1)

(~( K,-H)M, L ), = (n( H,-K)L, M ) , .

(2.5.2)

Furthermore, if OH8 = 0 we also have (n(L,-K)H,M)~ = (n(H,-K)L,M)~ = (~(H,-M)L,K),. Proof: (2.5.1) follows immediately from the observation ~2~(K, H ) = To prove (2.5.2), we start from

~'j = Ow'~ = O(G"'~OG~f ) = -G~'G~'~'OG~,~ A O G ~ + G~'OOG~;

(2.5.3)

106 in particular, G ,~ft'j = - G e U OG ~,:~A OG ;~e + OOG t~#.

On the other hand, ~2~ = GfUGe'"OGu~ h OG~§ - GC, r ~ - ~ ,

.

hence In our case, this means that ( ~ ( K , g ) M , L ) ,, = G ~,.r~ 'j ( g , -ff ) M Z-ffq = - G ~ ~,f t c~( I f , "-H) M ~-ffq = G,~h~,~(H,-I-()L'~MZ

= (~(H, K)L, M),, and (2.5.2) is proved. Now, (2.5.3). First of all, OHO = p*H(D"O), by (2.2.6). Corollary 2.3.7 says that D"~ = y u A f/; in local coordinates, (~u

A ~)'~ =

dz ~

A~

--R~,;i,~dz ~ ~

'~ A dz~'A d2 ~ + R~,~;~,dz Ar

+ R ~ ; , dz ~ A d z ~' A r

in particular, then, OH~ = 0 iff R~;uo

= R ~ ; a ~ ,.

(n(L, K)H, M)~ = G~§

+ R~6~, dz ~" A r

A dE" A :~~;

(2.5.4)

Hence ~

= G,~R'~;~,~L~'Ii~H~M " = (f~(H, K ) L , M ) v .

Finally, ( ~ ( H , - M ) L , g ) , = ( ~ ( M , H ) K , L)~, = ( ~ ( K , " H ) M , L)~ = (gl(H.-K)L, M ) , .

O We remark that (2.5.2) is equivalent to ( ~ ( H , K ) L , M ) v = (L, ~ ( K , Y ) M ) v

(2.5.5)

for all H, K, L, M E T/. and v E /~. In Hermitian geometry, quite a relevant role is played by the holomorphic sectional curvature; so it is natural to look for an analogue in our situation. A first attempt could be: DEFINITION 2.5.1: The (horizontal) holomorphic flag curvature K F ( H ) of a strongly pseudoconvex Finsler metric F along a horizontal vector H E ?-(v is given by 2 K F ( H ) = (H,H)---~, (12(H,'-H)H,H),,.

107

Exactly as in the Hermitian case, if ~H o = 0 then the holomorphic flag curvature completely determines the horizontal part of the curvature tensor: PROPOSITION 2.5.2: Let TI be a complex vector space, and let R, S:7"/x ~ x 7"/x ~ ~ C

be two quach'ilinear maps satisfying (2.5.2) and (2.5.3), that is R( K,-H, M,-L ) = R( H,-K, L,-M)

(2.5.6)

R(L, -K, H,-M) = R( H,-K, L, -M) = R( H,-M, L,-K)

(2.5.7)

and for all H, K, L, M 9 7-l. Asstune that

VH 9 7"l

R ( H , H , H , H ) = S(H,H,H,H).

Then R =- S. Proof: Up to replacing R and S by R - S and 0, without loss of generality we can assume R(H, H, H, H) = 0 for all H 9 7-/, and we should prove R - 0. Let G be the group of all permutations on the set of four letters {H, K, L, M}. The permutations (H, K, L, M) ~ ( g , H, M, L) and (H, K, L, M) ~ (L, K, H, M) generate a subgroup G1 of order 8; in particular, G/G1 consists of 3 lateral classes. As representatives of these classes we can take: the identity; the transposition ( H , K , L , M ) ~ (H,L,K,M); and ( H , K , L , M ) ~ (L,H,K,M). Then (2.5.6) and (2.5.7) imply that the function ( H , K , L , M ) ~ Re{R(H,'-K,L,-M) + R(H,-L,K,-M) + R(L,'H,K,-M)} is symmetric. Since it vanishes on the diagonal, it is identically zero:

R e { R ( H , K , L , M ) + R ( H , T , K , M ) + R(L,-H,K,-M)} =- O. PutL=HandM=K;weget

Re{2R(H,Y,K,'-K) + R( H,-K,H,-K) } = O. Replacing H by iH we obtain

Re{ 2R( H,-H, K,-K) - R( H,-K, H,-K) } = O, and hence

VH, K 9 7%

R(H, H, K, K) = 0,

because R(H, H, K, K) is real, by (2.5.6). Now replacing K by K + L in (2.5.8) yields

0 = R ( H , H , K + L , K + L) = R ( H , H , L , K ) + R ( H , H , K , L ) = 2 ae{R(H,g,

(2.5.8)

108 by (2.5.6). Replacing H by H + M we get 0 -- R e { R ( H + M , H + M , L , K ) } = Re{R(H,-M,L,-K) + R(M,-H,L,-K)} = R e { R ( H , K , L,-M) + R(H,-M, K , L ) } ; hence

Re{R(H,-K, L , M ) } = - Re{R(H,-M, K , L ) } .

(2.5.9)

Replacing H by iH we get - Im{R(H,K, L,M)} = Im{R(H,M, K,~)}.

Replacing here K by iK we obtain Re{R(H, K, L, M)} = Re{R(H, M, K, L)}; so, comparing with (2.5.9) we have

VH, K , L , M 6 ~

Re{R(H,-K,L,-M)} = O.

Replacing again K by iK we also get

VH, K , L , M e Tl

Im{R(H,-K,L,-M)} = O, []

and we are done.

Unfortunately, the holomorphic flag curvature is not the right object to study. In fact, as we have seen in the previous section, the geometrical meaningful objects (e.g., the variation formulas) involve contractions with X. The geometry of the manifold still lives in the tangent bundle, and the radial vector fields X and e are just the means we use to read what we need in T2t:/, where the algebra is easier. In a certain sense, we might even say that the holomorphic flag curvature contains too many informations - - and, conversely, that the condition ~H~ = 0 is tOO restricting. So in the next subsection we shall concentrate on a notion of holomorphic curvature on the tangent bundle (and not on the tangent-tangent bundle) associated to a strongly pseudoconvex Finsler metric, which is more suitable to our purposes.

2.5.2.

The

holomorphic

curvature

DEFINITION 2.5.2: Let F: TI,~ ~ •+ be a strongly pseudoconvex Finsler metric on a complex manifold M, and take v 6 -iV/. Then the holomorphic curvature KF(V) of F along v is given by 2

gF(v) = gF(X(V)) -- a(v) 2 (fl(X, ~2)X,X)o.

109 Clearly, gf(l~(v))

(2.5.10)

= gf(v)

for all ~ E C*. Note that, by Proposition 2.5.1, the holomorphic curvature is necessarily real-valued. In local coordinates we get 2

g f = -~-G~(F~)vI'~

"w.

(2.5.11)

This definition is compatible with the usual notion of holomorphic sectional curvature in Hermitian geometry, as it is easily checked. But to better show its geometric content - - as well as a variational aspect - - we need to recall a couple of concepts. DEFINITION 2.5.3: Let # = g d~ | d~ be a Hermitian metric defined in a neighborhood of the origin in C. Then the Gaussian curvature K ( # ) ( 0 ) of # at the origin is

given by g(#)(0) =

1 2g(0)(/kl~

w h e r e / k denotes the usual Laplacian O2u A u = 4 0~ 0~.

Now take a holomorphic m a p ~0: A ~ M (where A is the unit disk in C) such that ~o'(0) ~t 0. Then # = ~0*G is a Hermitian metric in a neighborhood of the origin, and so g(~o*G)(0) makes sense. Wu (in [Wu]) has proved that if r comes from a Hermitian metric on M then the holomorphic curvature of F along v E M is the m a x i m u m value that K(~0*G)(0) can attain when ~o varies among the holomorphic maps with ~0'(0) = Av, A E C*. This is true in our case too, as we shall see as a corollary of the following: ---* R + be a strongly pseudoconvex F/nsler m e t ~ c on a complex manifold M . Let ~: A ---, M be a holomorphic map, and set p = ~(0) and v = ~l(O), with v ~ op. Then

THEOREM 2.5.3: Let F: T I , ~

K(~o*G)(0)

=

KF(v)

2 G(v) 2 v(~,),,(~') ~ -

(v(~')"(~")~'x)" x i" (X,X)v

Proo[: Writing ~o*G -- g d~ | d~, we have

g(r

= G(~(r

~'(0),

g(0) = G(v),

and K(v*G)(0 )=

2 0)

(Atogg)(0)=

... ,

~,[ v )

02(logg)(o] O~0(

k--Z"

(2.5.12)

110 The computation of the Laplacian yields O~(log g)

a(a(

1 G;.(~; ~')(~'p + G~(~; ~')(~")~ G(~; ~,)2 1

ago; ~')

{G;,~(~; ~')(qo')"(~ii;' + aa$(~; ~')(~o")a(~") # + 2 Re [Ga;~,(~; ~')(~')u(~")~'] }.

Hence writing w = ~"(0) we get

K(~*a)(0) = - ~

a;~o

2 [G~ ~2 G2 Re

G~.] G~;u

w~w~ G

(2.5.13)

vUw~ '

where everything is evaluated at v.

Now,

v(~.).(~')~(~) = [w~ + rT.v,]eo e ~.;

hence

(x,x).

=[r;%c,;,

= I1%")"(~')"11~- I(V(~')"(~')u' x)"t2G'

G;-~;o],,~+[G~ z G~GZ]w~w~

+ 2Re { [G~;~

G;GG~] vU'w-g}. (2.5.14)

Finally, by Lemma 2.3.4, gF(v) = - ~

GJ~(FT.)v"F =

2 6~(G~F~)v,.b- ~ G2

(2.5a5)

G2 The assertion then follows from (2.5.13), (2.5.14) and (2.5.15).

[]

COROLLARY 2.5.4: Let F: TI,~

--~ R + be a strongly pseudoconvex finsler metr/c on a complex manifold M, and take p E M and v E l~lp. Then Kv(v) = sup{K(~*a)(0)},

"~i~F.16)

where the supremum is taken with respect to the family of all holomorphic maps ~:A -, M with ~(0) = p and ~'(O) = Av for some A E C*. The supremum is achieved by the maps ~ such that V(~,),(~')'(~'(0)) = az(~'(0)),

(2.5.17).

111 for some a E C. Proof: It suffices to show that a holomorphic map ~: A ~ M ~d(0) = Av and satisfying (2.5.17) exists. Now, we can certainly Ar = {~ 9 C I ]r < r} and a holomorphic map r ~ M r = v and V ( , , ) H ( r = 0. Then ~: iX ~ M given by ~(r ~(0) = p, ~'(0) = rv, V(~,)H (~')H(rv) = 0 and K(~*G)(O) = g ( r

with ~(0) = p, find a small disk with r = p, = r satisfies

= gF(v),

by (2.5.10) and (2.5.12).

[]

In the third chapter we shall study holomorphic maps ~: A ~ M satisfying =

for suitable functions A; we shall see that their existence implies that the holomorphic curvature is constant.

2.6. The Cartan connection vs. the Chern-Finsler connection 2.6.1 The radial vector fields Let F : T I , ~ --* R + be a complex Finsler metric on a complex manifold M. As we have already remarked at the beginning of subsection 2.4.1, using F we may define a function F~ T R M --* R + by setting Vu e T R M

F~

= F(uo).

(2.6.1)

Since o is a R-isomorphism preserving the complex structure J, setting ~ = ~ + ~/J with ~, ~ E R we have Vu 9 T R M

F~

= Hr~

where I$l = (~2 + T/2)1/2. In particular, F ~ satisfies all the properties of real Finsler metrics but possibly the last one: the indicatrices might be not strongly convex. DEFINITION 2.6.1: We shall say that the complex Finsler metric F is strongly convex if F ~ is a real Finsler metric.

In this section we shall always assume that F: TI,~ M ~ R + is a strongly convex F i n , let metric. Thanks to (2.6.1), the F-length of any curve in M is the same as its F~ in particular, F and F ~ have the same geodesics and induce the same distance function on M. On the other hand, the Cartan connection induced by F ~

and the Chern-Finsler connection induced by F might (and actually shall) differ:

112 this section is devoted to compare them. As we shall see, this is not a trivial task; so we shall pursue the comparison only up to obtaining the results we shall need in the next chapter. However, it will be enough to get an idea about what is going on. We start recalling that, as we already remarked in subsection 2.1.3, the canonical isomorphism o: T1,0)~/_.. T~/tT/induces an isomorphism o: y _+ 1~ such that (Oa) ~ = 3 ~

and

(iOn) ~ = Oa+,.'~

(2.6.2)

In particular, if V E 13 is expressed in local coordinates as V = Va0a, then VO = U ~0a, "0 where V a= ~l(va+v-if)=ReV

[

a _

~

= Im Y

ill
(2.6.3)

if n + 1 < a < 2n,

where we recall that greek indeces run from 1 to n = d i m c M, latin indeces run from 1 to 2n = dima M , and we used (and shall thoroughly use) the following convention: if in an equality there is a free latin index on one side and a free greek index on the other side, the greek index is equal to the corresponding latin index taken mod n. For example, in (2.6.3), the a on the right-hand side and the a on the left-hand side are related by ~={a a-n

ifl
Conversely, if U E Ya with U = U a 0a, 'o then Uo = V~c~ with V a = U a + iU ~'+n.

We also recall that both the real vertical radial vector field co: T a M --* ])R and the (complex) vertical radial vector field t: T 1,~ ~ 12 are well-defined and satisfy e~ ~ = (t(v))o.

Vv e T X ' ~

Let G: -~/--+ R + be the square of the Finsler metric F (or F~ by (2.6.1), it is the same). We have seen how using G we can define both a Hermitian metric ( , ) on ~ and a Riemannian metric/]) on ~ by setting

V Vl, V2 E ]]v

(Vl, V2)v : C o ~ ( v ) V ~ V / ,

and

VUx,Uz Our first aim is to compare these two metrics. Of course, it is tantamount to comparing the Hessian and the Levi form of G. Using (2.6.2) we easily find that Ga=

{ G a + G~ i(Ga-Gc~)

if 1 < a < n, ifn+l
(2.6.4)

113

and GaB + Ga# + GaB + GaB

i(G,~/3 + Ge~O - Ga B - G~$) Gab = { i(aao - ac~ + G,~9 - ae, B) - ( G , ~ - G ~ - GaB + GoB )

if if if if

1 1 n n

_< a,b < n, < a < n and n + 1 < b < 2n, + 1 < a < 2n and i < b-< n, + 1 _< a, b _< 2n.

(2.6.5)

Now take Va, V2 9 1)~, and set Uj = V / f o r j = 1, 2. Then (2.6.3) and (2.6.5) yield

= 2 P, [aoBvT

+ ao#Tv/].

So we have proved the PROPOSITION 2.6.1: Let F: TI,~ M --* •+ be a strongly convex Finsler metric on a complex manifold M. Then

VV, W e lr

(V~ I W~ = Re[(V, W> + ((V, W))].

Note that if F comes from a Hermitian metric then G~# - 0 and so

( V~ [ W~ = Re(V, W). In general, however, the symmetric product is not identically zero; as a consequence, the C a f t a n and the Chern-Finsler connection are less related than in the Hermitian case. Nevertheless, there is an important case where this relation still holds: since ((', t)) = O, we have

V V 9 1)

( V~ I ,o> = Re
(2.6.6)

To compare the Cartan and the Chern-Finsler connection we need to compare the corresponding horizontal bundles; we start by considering the horizontal radial vector fields. We recall that the complex horizontal radial vector field X 9 X(7"/) is given by where F.O,~,= G~/3Gf,;o. Using X we can define a section X~ of TR-~/just by setting

Vu 9 TRM

X~

= (X(Uo)) ~

On the other hand, the real Finsler structure F ~ determines a real horizontal radial vector field, that we shall denote here by ;~ 9 2((7-/R), given by

Vu 9 T ~ M

2(u) = ua$, = ua(O ~ - F~bg),

where the I'~ are the Christoffel symbols of the non-linear Cartan connection, and so are given by equation (1.4.14). We anticipate that, as we shall see in the next

114 subsection, in general the elements 6a ~--- .~o v a - - ~fib t l v bho of the standard local frame of 7-/R are not of the form (6,~) ~ or (i6a) ~ So we must compare )~ and X~ Now, if u = v ~ C -~/we have = (X(V)) ~ = ( , 4 0 ~ 1 7 6- (v~r~o0~) ~

X~

So to compare )~ and X ~ we ought to compare a^,

{ 89

with

u F~

+ b-'~F;3a)

.~(v~162

ill
Since (Gb~) is an invertible matrix, we can multiply everything by Gbc and then compare the outcomes. Using (1.4.14), (2.6.3), (2.6.4) and (2.6.5) we first get ~^b Gbc u ~a = Gc;d ud -- G;c

12Re[G~;~v~ +G~;j~'g-G;-~]

I. - 2 Im[GT;~v ~ + G ~ ; $ ~ - - G;7 ]

=

if1 < c < n , if n + 1 < c < 2n. (2.6.7)

On the other hand, we have 1

e~

/9

+

,

+

1

a

v~

{ 2Re[Gz~v~F~ + G.r,~- ] =

-2Im[G~.rv6F~+GT;~v"~ ]

if 1 < c < n, ifn+l
It turns out that we have proved the following PROPOSITION 2.6.2: Let F: T I ' ~

~ R + be a strongly convex Finsler metric on a

complex manifold M . Then vv e v

(x ~ - 2 I V

~

= Re(0(e(V), x), x).

(2.6.9)

In particular, X ~ -- 2 iff F b weakly K~dder. Proof: First of all, note that X~ - :~ is a section of FR, and so the left-hand side of

(2.6.9) makes sense. Then (2.6.9) itself follows subtracting (2.6.7) from (2.6.8) and recalhng (2.3.22). Finally, we get the last assertion writing (2.6.9) both for V and for i V . [] So we have obtained a new characterization of weakly K~_ler metrics. Note that this is coherent with the fact that F and F ~ have the same geodesics: indeed, the geodesic equation depends only on T H = X(&). Finally, we explicitly remark that, as a consequence of Proposition 2.6.2, F is weakly Ks iff U o~b l a a _~o b =

(vor~oo~)o.

(2.6.10)

115 2.6.2. T h e h o r i z o n t a l b u n d l e s The next step consists in comparing the horizontal bundle 7-I C Tl'~ to the Chern-Finsler connection, and the horizontal bundle 7"/~ C T~_h:/associated to the C a f t a n connection. Let O: 13 ~ 7-/denote the complex horizontal map associated to 7-/, and let (~: 1~ ~ 7"/R denote the real horizontal map associated to ?-/~. Since ~ Y ~ 1~ is a R-isomorphism, we get a R-isomorphism ^: 7-I --* T/~ given by VH e 7-/

/:/= O((O-I(H))~

Note that 6"~ = 6~ and i6"'~ = 6~+n. Now, the canonical isomorphism o sends 7-I into TR/I~/; if the image would be 7"/~, then for all practical purposes we could identify 7-/ and ~ . So the comparison between 7-/ and T/~ boils down to comparing H ~ and /~ for any H E 7"( (note that both have the same coordinates with respect to the corresponding usual local frames). The first step is: PROPOSITION 2.6.3: Let F: Ti'~ M --+ R + be a strongly convex Finsler metric on a complex manifold M . Then: (i) O c o m m u t e s with the complex structure J iff i'H = J[-I for all H E ~ iffTl~ is J-invariant iff

~+.

= -f'~+"

and

-~+-"e+"= ~

(2.6.11)

for all 1 < ~,/3 <_ n; (ii) H = H ~ for all H E 7"l iff T"l~ is J-invariant and

r~ = ~ + ~ + "

(2.6.12)

for all 1 < a, ~ < n. Proof: (i) Since 0 -1 commutes with i, and ~ transforms multiplication by i into

the complex structure J, it is clear that O commutes with J iff i'H = J / t for all H E 7-/. In turn, this happens iff ^b

"0

.

.

.

.

r~0c)

= 0~+. -

^r ~ r~JOc

for all 1 < a _< n, that is iff (2.6.11) holds - - and this clearly implies that 7-/R is J-invariant. Finally, if 7-/R is J-invariant we should have J(O; - r^~ ~ 0 ~"~ ) = 0 ~ + . - r^~ ~ ( J 0 ~"~ ) ~ ~/~,

and then (2.6.11) again holds, because ~ + n is the unique element of T/~ whose image under &r is O/Ox ~+'~, and d r commutes with J. (ii) We have /~ = H ~ for all H e 7-/ iff (5~)o = 6, and (i~'~)o = i 5 , for all 1
(i~".)o = i(o~ _ ,-.+.r#*'+"_ i f ' ~ + . ) & ) , and so [-I = H ~ for all H E 7-I iff (2.6.11) and (2.6.12) hold.

[]

116 Set { Re F;~a ImP~ - Im F;~a Re F~a

~ =

ifl~a,b~_n, if 1 < a < n and n + 1 < b < 2n, ifn+l
Then Proposition 2.6.3.(ii) says that .fI = H ~ for all H ~ 7"( iff

for all 1 < a, b _< 2n. Arguing as before, we must compare GbCF~ with GbcF~ - and the computations begin to become long and tedious. First of all, 1 d abc'~ba = ~{ [Gab;dU "-I-Gb;a - aa;b] --

^d e )}. aobd(r~u

Using (2.6.3), (2.6.4), (2.6.5) and their equivalents for third order derivatives, and assuming that F is weakly Kghler, that is using (2.6.10), we get:

Gab;dU d

=

2Re[(G,~;6+G~,~;6)v6+(G~,I3;~+G~,~;~)-~ -g] ifl
1

Gb;a=

Ga;b=

2 Re[a~;~ + a~;~] -2Im[Gz;~+Gz;~] --2 Im[G~;a - G#;~] - 2 Re[Gz;a G~;~]

if 1 < a,b < n, ifl
2 Re [G~;z + G~;~] -2Im[G~;~+G~;~] -2Im[G,,;~-G~;~] - 2 Re[G~;#-G~;~]

if 1 < a, b < n, ifl
Gabd(rdu e)

[ 2Re[(G,~ + G~,z,)r~v' + (G,~ + G~,~t)r~[~] = -2Im[(Go,~ - G ~ , ) r ~ , 9+ (Go,~ - G ~ ) F ~ ] -2Re[(a.z~

-

+

-

if l <_a,b < n, and n + 1 < b < 2n, if n + 1 < a < 2n and 1 < b < n, if n + 1 < a,b < 2n.

117 Furthermore,

GbcPCa =

if 1 _< a,b___ n, ifl
- 2 I m [ G ~ r ; ~ + G~;~] --2 Im[G~FT~ - G~;~] - 2 Re [G~F;~ G~;~]

Now set K~ = [G~;~ - G~;~ +

a~or~.]v~.

Since F is weakly K~hler, K~ - 0 by (2.3.22). So we get 7 $ 0 = ( g ~ ) ~ = G~;Z - G~;~ + G#TF;~ + [-G~z;~ + G~.rF;~ + G~Tr~;~]v , (2.6.13) 0 = (K~)~ = aa;~ + [ - G ~ ; ~ + G . ~ r ~ , +

G

~ r . ;7~ ] v .

Moreover,

Summing up we have obtained -

tb[V~(r~;~

r ~~;~~v~ + v ~ ( r ~ ; ~ --

~ ~ if 1 < a,b < n,

-

-

-F~;~)v

- u ~

~;6

-

ifl
im[a~(r~;~

-Re[a~(r~;~

-

r ~ ~v~

-

- W~;~j~v~ - G ~ ( r ~ ; ~ -


~ ~

v ~ r ~ 6 v ~ - a ~ r ~ ;-g ~-7 ]

if n + l

< a < 2n and 1 < b < n , +

-

if n + 1 _< a,b < 2n. (2.6.14) So we have proved most of the following THEOREM 2.6.4: Let F: T l ' ~ ~ R + be a strongly convex weakly K ~ l / e r Finsler metric on a compJex manifoJd M . Then:

1[

(ill- H ~ lV~ = ~ Re (O(H, X), O(V)) + <(O(H,X), O(V))) I

(2.6.15)

+ + (
for all H E Tl and V E V. In particular, [t = H ~ for all H E ~ iff O(H, X) = 0

and

r(X, W) = 0

(2.6.16)

118 for all H E Tl and W E V.

Proof: First of all, / I - H ~ belongs to Y~, and so the left-hand side of (2.6.15) makes sense. Moreover, if K = H ~ and U = V ~ we have

( [ t _ H o l V o ) = 1~Vbc(r.-~

~ . U; b r.)K

so (2.6.15) follows from (2.6.14) and (2.3.19). We are left with the final assertion. If (2.6.16) holds, then (2.6.15) immediately yields [-I = H ~ for all H E 7~. Conversely, assume that [-I = H ~ for all H E 7"(. Then Proposition 2.6.1 and (2.6.15) imply that ( O - I (O( H, x) ) ~ + r ( x , e ) - ' ( g ) ) ~ l V ~ = 0

for all H E 7-/and V C V, and so e - I (0(H,x)) 4- r ( Z , O - ' ( H ) )

= 0.

Replacing H by i H we get the assertion.

[]

In particular, if F is K/ihler we find V H E T-I

1T(X,O-I(H)~ ~

[-I = H ~ +

that is

O ( v o) = ( e ( v ) ) ~ + { , ( x , v ) o .

VVEV

If F comes from a Hermitian metric, then r(X, V) - 0, and so we do not see this difference between the Levi-Civita and the Chern connections; otherwise, the mixed part of the (1,1)-torsion measures how far apart are the Cartan and the ChernFinsler connections. Finally, we should remark that there is one instance where the behaviors of and H ~ agree: COROLLARY 2.6.5: Let F: T I , ~ --~ R + be a strongly convex weakly Kii~hler Finsler metric on a c o m p l e x manifold M . T h e n

VH e ~

(H I Z~ = ( H ~ IX"~ = R e ( H , x ) ,

(2.6.17)

(where we recall that :~ = X ~ since F is weakly Kii~ler).

Proof: Indeed, since F is weakly Ks

we have

<e(H, X),

X> =

O.

Furthermore, we know that ((., t)) - 0. Finally,

<~(x, V), ~> = - G ~ F ~ ; ~~ V ~ v -

and (2.6.17) follows from (2.6.15).

-

~ v~ --

= -(G~P~;~)V-Cv ~ = 0,

[]

119 2.6.3.

The

covariant

derivatives

The next step in our comparison program concerns the covariant derivative. Let denote the covariant derivative associated to the Cartan connection, and V denote the eovariant derivative associated to the Chern-Finsler connection. We are not interested here in comparing V and V in their full generality; for our aims, it is enough to compare the covariant derivatives appearing in the second variation formula, namely ~7~ and Vx+ ~. So we shall compare ~7~V~ and (Vx+~V) ~ for any V 6 X(V). As usual, this is equivalent to comparing

(V? I V~V~) for all V1, 1/2 6 X(V). First of all, by (2.3.14)

and

(V? I (V,:+~V2) ~

Go;rF~;~v~' = ~(Ga~), and so

(vi~l(v~+~v2) ~ = R~[(V1, V~V2) + ((V1, V~V~)) + (V~, V~V~) + ((V~, V~V~))]

+ ao~v;~,(V~) + a~V-f~(V~) + a~V;~(V~) +

G~(V~)}.

On the other hand, writing as usual (1.4.4) and (1.4.6), we have

=

[x(u2)rb;j'~ Ui]U 1

1 __ ~GacVlafi(U~ ..[_ I

=

i{2

Uj = Vj~ for j = 1, 2 and recalling (1.4.12),

6b(Gaj) --

88[~(aob)+ Go;, a~v:~(v~)+ aozV~(v~)+ a~.~(v~) + a~z~ ~(v~)} I

+ 88[~(G.~) + Go;, - a,;o]v ~ v ~ + ~ [~(Go~) + ao;~ - G~;o] V ~ V ~ I

+ 88[,(G~) + aa;~ - a~;~]VT"V~ + ~ [X(G~) + Ga;~ - a~;~]VT~l"V~. We now assume that F is weakly K ~ l e r , that is that ;~ -


X~ = X +

X. Then

~v~)

[2a.,rl;,~, + V~;~

-

ao;~

+ [(~ - x)(a~ z) + aZ;o

-

a~

= 1 Re{

-

(x + ~)(ao~)]

v;~v~

120 Now since F is weakly Kghler, recalling (2.6.13) and (2.3.22) we get 0

=

G~;~

-

G~;~+

G~(r.;~ ~

+

r~;~)v -r

- x(Go~),

(2.6.18)

0 = GB;~ + G~,FB; o - x ( G ~ z ),

Therefore " " ( W I (v~+~v~) ~ - *~v~ ~ = g1 Re{ [ a ~ , ( r ~ ; , - r.;~)~

+ [ao4r~;.

a r3;.N Vv?

- r~;z)~-~- a~,r~;,v']V;~y?}, (2.6.19)

and we have proved the THEOREM 2.6.6: Let F: T 1,~ ~ R + be a strongly convex weakly K~daler Finsler metric on a complex manifold M . Then 1 o % V ~ = (Vx+~V) ~ - ~ [O -1 (O(O(V), X)) + r ( x , V)] ,

for all V E X ( F ) . In particular, F is Kghler iff ~v

o=

(VxoV)

o -

~(x,V) ~

for a t / V ~ X ( ~ ) . Proof: Indeed (2.6.19) exactly says ( V ( l (Vx+~V~) ~ -

fT~Vr

=

1 0 g(V, I[e-l(o(e(v2),x))+ r(x,V)]~ []

a n d w e are done.

So once again it is the mixed part of the (1,1)-torsion which distinguishes between the Caftan and the Chern-Finsler connections, even for Kfihler metrics. So it could be useful to register a couple of facts about it: PROPOSITION 2.6.7: Let F: T I , ~ ric on a complex manifold. Then

~ R + be a strongly pseudoconvex Finsler met-

(i) for any V, W E ~ we have

(v, r(x,W)) = (w, r ( x , V ) ) ; in particular,

(r(x, v), ~) =_o. (ii) i f F is weakly KgLhler, /'or any V, W E Y we have

((r(x,Y), W)) = (O(O(W), X), O(V));

121

in particular, if F is K~hler we have

((r(x, v), w)) = o.

Proof: (i) Indeed

(v, ~(x,W)/~ = - a ~ r ~ ; Z v ~ w ~

= - ~ ( G ~ ) V ~ W ~ = (W, ~(x,~)),.

The second assertion follows from ~( Gao)v ~ = ~( Ga#v ~) = O. (ii) Indeed

a ~ r ~ ; v " = x ( a ~ ) - a~;~ = a ~ ( r ~ ; . by (2.6.18) because F is weakly Ks

- r~;~)~.,

(2.6.20)

and we are done.

[]

In particular, we have

<%,V~ l ~) = ((v,,~ V)~ l x ~ = ~e (V~~ V, xl. Proceeding with our task of comparing the two second variation formulas - the real one Theorem 1.5.4 and the complex one Theorem 2.4.4 - - we now want to compare

( % , V ~ ,~

and

((V~,~176 ,~

for any H E 7"[ and V E X(I)). As usual, let us start with some computations in local coordinates.

((v,-,~176 ~ = P~[(v,,v, 0 + ( v g v , , ) ] = R e [ G a H ( V a) + a~r~;~VeH ~ + a~g(V~)]

= Re[a~g~

~) +

(Ga;~ -

a~.r".~)Yeg~].

On the other hand, setting U = V ~ and H = Ka6,, we have

<%v~

,~ = 8 9

~) + f,~b;~Kcrrbl ~

+ $~(G.~) - $.(Gb~)]u"gr ~

= 89

~) + 88

= 89

1 ~) + ~[ab;~ + ao;b -- 2 ( G b e ) ] g c v b.

Now,

G,[-t(U") = G~[-I(V ~) + G a H ( V ~) = 2 Re[G~/~(V~)], [ab;c + ac;b -- X( Gbc)]KcU b = 2 Re{ [G#;-~ + G-t; # - ~( G #-~)]V# H "t

+ [G~;~ + e~;~ - 2(G~,)]V~B-~}.

122 So subtracting we get


-

(VHo V)~ l t ~ = Re[G~([1

-

H~

_ l Re{[G~.~ - G ; ~ - 2Ge~r[~ + ~ ( C ~ ) ] V ~ H ~ 2

+ [2(Gzq) - GZ;~ - Gq;~]VZH--~}. If we assume that F is weakly Ks

then )~ = X + 2 and, by (2.6.18), we get

(fVfl V~ - (VHoV) ~ [t ~ = Re[G~([1 - H~ -

89Re{[G~(F~;. - r;;,)v. + Ge~F~;ov-~VZH ~" .

~

-

r~;~)~v~--~.

)

(2.6.21) As a consequence, we have proved the following: THEOREM 2.6.8: Let F: T I , ~ --+ R + be a strongly convex weakly Ks Finsler metric on a complex manifold M , and take H 6 ~ and V 6 2((12). Then: (i) if V does not depend on the vector variables, i.e., if V = ( v for some vector field ~ 6 X ( T I ' ~ then ( V H V ~ -- ( V H o V ) ~ It ~ = <[t - H ~ I V~

(ii) if V 6 2((12) is any vertical vector t]eld then


o _ (VHoV)~ I d) = ([1 - HO)( (VO l d> ).

Proof: Recalling Theorem 2.6.4, and setting W ~ = [1 - H ~ 6 12~, we see that (2.6.21) can be written as


-

(VHoV) ~ It~ = Re [G~W~

+ <W ~ I V~

and (i) is proved. In general, we have aoW~

~) = W ~

~ - W~

and (ii) follows from Proposition 2.6.1.

~ = W~

~>) - ( + <>), []

So in general the Cartan and the Chern-Finsler connections actually differ, and the difference is not a tensor: it depends on first-order derivatives - - unless F is K~ihler, hypothesis that we are willing to accept, and T ( X , V ) -- O, hypothesis that we are not willing to accept, because as we shall see better in the next chapter we are able to study the geometry of Kghler Finsler manifold without assuming it. At this point, then, one might even wonder whether there exists another canonical connection on real Finsler manifold which agrees with the Chern-Finsler connection on K~hler Finsler manifolds. However, we stop our direct comparison of Cartan and Chern-Finsler connections here, devoting the last subsection to the curvatures.

123 2.6.4. The horizontal curvatures

We are left to compare the last piece in the s~cond variation formulas: the horizontal flag curvature. We should compare (~(~,/t)/t[:~} (where ~ is of course the curvatttre operator of the Cartan connection) with

{(f~(H,~)X)~ l H ~ - ((f~(X,'-ff)X)~ [ H~ where this expression is suggested by Theorem 2.4.4 together with Lemma 2.3.8 and (2.5.2). Let us start computing the expressions involved. First of all, ~TH~ -- 0 for any /:/ ~ 7-/~, because V is a good connection; hence, recalling Propositions 1:4.5 and 1.3.2, we have

Now ~'[,.B]t ~ is the vertical part of [:~,HI; therefore (1.1.20) yields

<~(~,1})/~ I ~> =

~O,b[ar ~ ^ F ^ d) - g~(~: )]K~u~ K b,

w h e r e / t = KbSb 9 "HR. From now on we assume F g~_ler, so that /~ = H ~ + 89r ( x , O - ~ ( g ) ) ~ In particular, Re(F~ + ~_~;._ ~

~=

if I < a, e < n; Im(r;~+~F~;~v ) ifn+l
A long but trivial computation then yields 1 ^ ^a e *b ~G~bSc(re)u K Kc

- ~1 o o / 3 r , ; ~e

3

( r , ,&; ~

, 1

a'

+ rL)

]v'n/3n,

~G~r~;,v r~;~ + ~a~/35~(r,;~v--~ ~

&

u

~

/3--

-~G~F#;~v P~;~ + ~G~/35~(F~;~v ) -

88

+ rL)]~H~--~ },

(2.6.22)

124 where F~;~ = cg~O§

and so on. Analogously

1 ^ ^a e ~Gab6e(rc)u K bK

c

- 7G~F~;~v Fe; ~ + ~Gfi~6~(r~;~v ) %

1 f .--:-~(r~#;~v ~ ~ + - iG~F~;~v

F~;~)Jv~H~H ~

~G~F~;ov F§ 1

f" =-.~

w

~G~z,~(F~;,v )

#

~Ga~F~;~v F~e;,v ]v~H~-H-~

(2.6.23)

7 G ~ F ~ ; , v F~;~ + ~G~5~(F~;~v--Q -7G6BFe;t,v FTa;ov-~v H H + [C~6g(r;%)-I o- p 5t - u

(F~;~v

,

c~ #

+

On the other hand,

( (~( g , ~2)X)~ [ g ~ = ( (f~(X,'-H)x) ~ { g ~ = Re{ [G~6~(F;~) - G~5,(r~,~)]v~H~-~ -~

(2.6.24)

+ [ G ~ 5 ~ ( F ~ ) - G~5~(r~)]-~HZH~}.

We must subtract (2.6.23) and (2.6.24) from (2.6.22). Using (2.3.16), v~F ~- = 0, (2.6.18), v ~F~e;o ~ = 0 and v ~F~a;o ~ = -F~;~, we obtain f

) -76~(F~;~v )]v g g

[ ~)V ~ 3 ~ 17 ( r ~ ; ~ ) ~ fi

%

+

1 a # F~;~v fi =TD" [~G~F~;~v 1

~

~-7-~ fi

~

"

,

t t ~~c;~,

~1 G , ~ x ( r ~ ; ,~v

-

p

)] H ~ -~

89

+ [ - T G e , zP.~;ov r;~ 1

~~ /, ~. , , ~ %~ a; , v

xr:._r,~,

,,,vfi

_ 89 (2.6.25)

The Bianchi identities say that D " r = O. Now, D" r

a

~

- r_%,,;,r

- 5

a

-~

^ dz" A r

| O, + r ~ , , 6 d r ~ ) d z , A d~ ~ A d ~ | 0,;

In particular, fit r a ~7(F,;,~) = F~;~F,;~.

Since F is Ks ~7(r,;ov~v

(2.6.26)

(2.6.26) yields =

5, , u a, e = Fa.~v-FFt.Tv

and the first term in (2.6.25) vanishes.

6~

.

a

= 5~(F.;~v ) v ,

125 Furthermore, since by (2.6.20) 0 =

Ga~re;.r~;~v

v

a~r~;.v.

= o, (2.6.26) and (2.6.18) imply

=

=

and so the last term and the first addend in the second term of (2.6.25) vanish. Now set ~o(W) = r u ( x , W ) , so that

= - r ~ ; ~.~-~ | ~ . Then

vx~ = - x ( r % ) r

- -

| ~ - r~;.r ot

- -

| v~

= - [ x ( r ~ y ) + r-~;.v.r~] r | ~o. The covariant derivative commutes with contractions; so, since VxX = 0, we have

(vxr~)(x,W) = ( v ~ ) ( W ) = -[x(r~;.v ) + r~;.. r ; . ] w ~ : a

it

a

/~

c~

In particular, the second term in (2.6.25) is given by -12{{g, (Vx v~) (X, O - ' (H)))), while the third term in (2.6.25) is given by I ( H , (Vxr~)(X, O-1 ( g ) ) ) . In conclusion, we have proved the following THEOREM 2.6.9: Let F: T I , ~ ~ R + be a strongly convex KMaler Finsler metric on a complex manifold M . Then for any H E 7"l one has

(fi(~, [t)[t1~) = ((f2(H, 2 ) x ) ~

~ - ((f~(X,Y)X)~

~ + l g ( ( ( ~ T x r T ~ ) ( X , O - l ( H ) ) ) ~ 1 7 6 ).

So we have the following situation: the integrands in the second variation formulas Theorem 1.5.4 and Theorem 2.4.4 are different (unless r(x,-) - 0, of course) but the integrals themselves agree (at least for fixed variations). So if we are able (and in the next chapter we shall) to control the integrand in the complex form of the second variation formula then we are able to control the real Morse index form, even though we are still not able to control the real horizontal flag curvature. This is the reason that suggested the unusual formulation of the Cartan-Hadarnard Theorem 1.7.17; and maybe it also suggests that there could be a better connection than the Caxtan connnection on real Finsler manifolds: it seems that the Caftan connection contains pieces useless from a geometric point of view (i.e., disappearing in the computation of the first and second variation formulas).

CHAPTER

3

Manifolds with constant holomorphic curvature

3.0. I n t r o d u c t i o n In order to study intrinsic metrics and distances on complex manifolds Vesentini [V] introduced the notion of complex geodesic for the Kobayashi and Carath6odory metrics and distances. Roughly speaking a complex geodesic is a holomorphic embedding of the unit disk equipped with the hyperbolic metric which is an isometry with respect to the intrinsic metric or distance (or both) defined on the manifold under consideration. The problem of existence of complex geodesics is satisfactorily solved only for convex domains by the work of Lempert [L], who moreover proved uniqueness for strongly convex domains (see also [A, chapter 2.6]). In this case the Kobayashi and Carat6odory metrics coincide and are smooth strongly pseudoconvex complex Finsler metrics. In order to find a different approach to this problem, which may eventually lead to an understanding of it on a larger class of complex manifolds, it is very natural to ask whether it is possible from a differential geometric point of view to find optimal conditions on an abstract complex Finsler metric implying the existence and uniqueness of complex geodesics. In this general framework we shall investigate the problem of finding isometric holomorphic embeddings into a complex Finsler manifold of the unit disk A with Poincar6 metric, of C with the euclidean metric and of P1 with the l~-klbini-Study metric. Not surprisingly the machinery devised in the previous chapters will be used extensively and the necessary and sufficient conditions for the existence of the isometric embeddings will involve Kfil'derianity, symmetries of the curvature and constancy of the holomorphic curvature. As a byproduct of our work we shall indicate how these result may contribute to the understanding of the function theory and the classification of complex Finsler metrics of constant holomorphic curvature. In particular when the curvature is negative constant we shall show that the metric must be the Kobayashi metric of the manifold and that the geometric picture is quite alike to the one of strongly convex domains. When the curvature vanishes we shall show how this may be used to characterize C n. In both cases it is very interesting to discover that it appears very naturally a relation with the homogeneous complex Monge-Amp~re equation. All these applications and remarks suggest the possibility of further investigations.

128 T h e content of this chapter is the following. In the first section we introduce the notion of geodesic complex curves (roughly speaking, totally geodesic holomorphic disks) a n d we describe conditions ensuring their existence and uniqueness: it turns out t h a t one needs constant holomorphic curvature, the weak Kghler condition and a s y m m e t r y p r o p e r t y of the curvature operator. In the second section we focus on Kghler Finsler manifolds with nonpositive constant holomorphic curvature, and we prove t h a t they are foliated by geodesic complex curves, reproducing in this setting the situation described by Lempert for strongly convex domains. The proof of this t h e o r e m requires more or less all the material discussed in the previous chapters.

3.1.

Geodesic

complex

curves

3.1.1. Definitions and properties We begin by giving a precise notion of geodesic complex curves of a complex Finsler manifold. Let us fix some notations. First of all, set

~xr={zECllzI
DEFINITION 3.1.1: For c E R let #c denote the Hermitian metric

1 ~c = (1 + (c/2)lCI2) 2 dC | dC, which is defined over the disk A r with r = X / ~ if c < 0, on the complex plane C if c = 0 a n d on the projective line P1 = C O {oo} if c > 0. In particular, for c = - 2 we have the Poincar6 m e t r i c # - 2 on A 1 # - 2 - (1 - [r

dr | d~;

(3.1.1)

for c = 0 we have the euc//dean metric #0 on C

#o = d( | d~,

(3.1.2)

and for c = 2 we have the Fubini-Study metric #2 on P1

1

#2 = (1 + [(]2)2 dC | d~.

(3.1.3)

129 We recall a couple of facts concerning these metrics. First of all, they have constant Gaussian curvature K(#c) - 2c. Furthermore, if we introduce a complexvalued function Ac defined where #c is defined and given by c~ At(C) = 1 + (c/2)1r

(3.1.4)

then the geodesic equation for #~ is = A~(a)(&) 2.

Now let M be a complex manifold of dimension n and let F : TI'~ a strongly pseudoconvex Finsler metric on M.

(3.1.5)

~ R + be

DEFINITION 3.1.2: A holomorphic map ~: AN ~ M is called a segment of/nt]n/tesimal c-geodesic complex curve if T is an isometry from A r endowed with #c to M endowed with the strongly pseudoconvex Finsler metric F. We say that ~ is

a segment of c-geodesic complex curve if[ T maps geodesics parametrized by arc length for the metric #c restricted to A r into geodesics parametrized by arc length for F o n M. If c < 0 a n d r = X / ~ ( r e s p e c t i v e l y , c = 0 a n d r = + c ~ , o r c > 0 and ~ is defined on P1) we call ~ an (infinitesimal) c-geodesic complex curve. If c = - 2 (respectively, c = 0 or c = 2) we shall sometimes say hyperbolic (respectively, parabolic or elliptic) geodesic instead of c-geodesic. Finally we shall say that is a (segment) of (infinitesimal) c-geodesic complex curve through (p; v) E T l'0M iff ~ ( 0 ) = p ~ d

~ ' ( 0 ) = v.

Our definition of geodesic complex curve is of course related to Vesentini's notion of complex geodesic for intrinsic metrics and in fact it is a bit more flexible. For instance there axe no complex geodesics in the sense of Vesentini for an annulus A C C endowed with the Kobayashi metric (i.e., holomorphic maps T: A --+ A such that for all z, w E A the Kobayashi distance between ~(z) and ~(w) equals the Poincar6 distance between z and w) as suc h a ~ would necessarily be biholomorphic. On the other hand there axe plenty of hyperbolic geodesic complex curves in the sense explained above. Furthermore the definitions given above have the advantage that they implicitly contain a recipe to find differential equations for geodesic complex curves. As a first step towards the solution of the problem of existence of such curves we recover the system of P D E ' s which they must satisfy. Let M be a complex manifold with a strongly pseudoconvex Finsler metric defined on it. Then, if, as usual the prime stands for 0 / 0 ~ and Ac are the functions defined in (3.1.4), we have (as first proved, in a different way, in [AP1]):

130 P R O P O S I T I O N 3 . 1 . 1 : Let F: T I , ~ -+ R + be a strongly pseudoconvex Finsler m e t ric on a complex manifold M , and ~: A r --+ M a holomorphic map. Then ~ is a segmen~ of c-geodesic complex curve, for c 6 R (and r < ~ when c < 0), if/

I

Ct

Dr

~

Itct

I~

) +Ac(~)

ct

!

?#

+F;~(~)(~)

"

=0

(3.1.6)

Nc,o(~) = [a.;~(~; ~') - a~;.(~') + ao~(~')r;~A~') ] (~'). = 0, for all a = 1 , . . . , n, that is iff

{ v(~,)~(~') H = -A~x(~'), < - 0, ro~ aJ1 g e ~ , .

(3.1.7)

Proof: For any ~ G Ar and ~ E C -- T~'~ with #c(r ~) = 1, let a~,e be the unique geodesic for #r parametrized by arc length, starting from ~ tangent to ~. Then ~ is a segment of c-geodesic complex curve iff for all such ar162the curve 7r -- ~ o a~,~ satisfies the geodesic equation (2.4.11) for F, that is iff in local coordinates

~r + r~(~,~)~',~ = c~(~<,~)[G,;~(~<,~) - G~;~(~<,~) + G~Z(;r<,~)r;~(~<,~)] ~',~,

(3.1.8)

fora=l,...,n. Now, being ~ holomorphic, one has

~<,~ = (~' o a<,~)~<,~, ~r

= [(~" + Ar

(3.1.9)

o ar

where we used (3.1.5). In particular,

~r

: ~'(r

and ~r

= [~,"(~) + Ac(()~'(r

2.

Assume now that ~, is a segment of c-geodesic complex curve. Then evaluating (3.1.8) in t = 0 we get (~(~" + A ~ ' ) ~ + F ~ ( ~ ' ) ~ ( ~ ' ) "

= c~(~ for

a = 1,...,

n.

') [c~;~(~') - c ~ ; ~ ( ~ ' ) + c ~ z ( ~ ' ) r ; ~ . ( ~ ' ) ] ~ ( ~ ' ) . ,

Using the homogeneity of G (and of its derivatives) we get

~ [(~" + Ac~') a + r;~(~')(~o')*']

I~l~G~(~ ') [a~;~(~') - a~;~(~') + G~(~')r~(~')] (~')., for all a = 1 , . . . , n, and, being arg ~ arbitrary and G ~a invertible, it follows that ~o satisfies the required system. Conversely, assume that ~ satisfies the system (3.1.6) of PDE's. Fix ( G Ar and ~ G C with #c(r ~) = 1. Evaluating the system in ar and then adding for

131

all a the first equation multiplied by (&Gr 2 to the conjugated of the second one multiplied by G ~ and I~,dt)lL using the homogeneity of G and recalling (3.1.9) we get exactly (3.1.8). Finally, it is clear that (3.1.6) is the expression in local coordinates of (3.1.7); cf. (2.3.22). [] In particular, then, if ~ is a segment of c-geodesic complex curve, the metric F is weakly K~il~ler in the directions tangent to ~v or, as we shall say, that the metric F is weakly KgJ~ler along qo. In other words we have COROLLARY 3.1.2: Let F: T I , ~ --* R + be a strongly pseudoconvex Finsler metric on a complex manifold M , and fix c E R. Then a holomorphic solution ~: A r - . M (with r < x/2/Icl if c < O) of

V(~,)~ (~') H = - A ~ x ( ~ ' ) is a segment of c-geodesic complex curve iff F is weakly KgJz/er along ~v. In particular if F is weakly Kiihler, then a holomorphic map ~: A r ~ M is a segment of c-geodesic complex curve i f / D ~ ( ~ ) = 0 holds for a = 1 , . . . , n.

At this point it is clear that it is crucial to study the system D ~ ( ~ ) = 0 looking for conditions for its resolubility by holomorphic functions. Recalling Corollary 2.5.4 it is clear that we should look for conditions concerning the holomorphic curvature ofF. DEFINITION 3.1.3: We shall say that a holomorphic m a p ~: A r ~ M realizes the holomorphic curvature at 0 if K(~*G)(0) = KF(~'(0)). More generally, if c E R (and r < X / ~ when c < 0) we shall say that ~ realizes the holomorphic curvature at ~o E A r with respect to I~c if ~ o 7~o realizes it at O, where

r

-

1

-

so that 7rc is an isometry with respect to Pc with 7~o(0 ) = (0 and (7~o)'(0) is a positive real number. Then we have: PROPOSITION 3.1.3: Let F: T I , ~ ~ R + be a strongly pseudoconvex Finsler metric on a complex manifold M , and take c E R. Let ~: A r --* M be a holomorphic solution of D y ( ~ ) = 0 with ~'(0) # 0 (and r < 2v/~V~lif c < 0). Then

(i) ~ realizes the holomorphic curvature at O; (ii) ~o realizes the holomorphic curvature with respect to #c at every point of A t ;

132 (iii) ff F ( ~ ' ( 0 ) ) = 1, then T is a segment of infinitesimal c-geodesic complex curve

for F. Proof: (i) As we have seen in Corollary 2.5.4, a holornorphic map ~: Ar --* M realizes the holomorphic curvature at 0 iff (~")~(0) + r T , ( ~ ' ( 0 ) ) ( ~ ' ) , ( 0 ) = a(~')~ for some a E C and all a = 1 , . . . , n . But this is exactlyD~(~) = 0 evaluated at C = 0 (and with a = 0); hence (i) follows at once. (ii) Take Co e A~, and set r = ~ o 7~o' Then r

= ~(G);

r (1 + (ct2)!Gl~)r =(1 + (c/2)IGI2)2(~"(G) + AdG)~'(G)). =

r

Therefore ~o realizes the holomorphic curvature at Co with respect to #~ iff (~ " ( G ) + A r

~+

r;.~ (~'(r

(~'(r

= a(~'(r

and hence we have (ii). (iii) If D ~ ( ~ ) = 0 holds then, recalling the well known homogeneity properties of G, we get a , ( ~ ' ) ( ~ " ) ~ + A,G@') = a ~ ( ~ ' ) [ @ " ) ~ + A~(~') ~] - -a,,(~, ) L . ( ~ ) ( ~ ) = -a;,(~')(~,'),'. Therefore

0

0-~ [G(C)] = G;.(C)(C)" + a~(~')(~") ~ = -AcG(C). Now, along the curve t ~-~ ei~ we have

0 1 -io d -~ = -~e -~; Thus the function t ~ G(~,'(ei~

is a solution of the Cauchy problem

l f'(t) =

1 t f(O) = 1.

But

-2ct

"'t"

f(t) = (1 + (c/2)t 2) -2 is a solution of the s a m e problem; therefore a(qo((); dqor

and we are done.

= G(~(r

~'(()) =

1

(1 + (c/2)1r

~

=/~c((; d/dt),

[]

133 3.1.2. T h e e x i s t e n c e t h e o r e m In complete analogy with the case of usual real geodesics one likes to solve the Cauchy problem for the operators D~. Here of course there axe difficulties, as we have at hands a system of P D E ' s and we look for holomorphic solutions. In particular one expects compatibility conditions. In fact we shall see shortly that these are also sufficient. Let S I ' ~ = {v ~ T l ' ~ [ f ( v ) = I } denote the unit sphere bundle. Then we can summarize our result as follows: THEOREM 3.1.4: Let F: TI,~ --* R + be a strongly pseudoconvex Finsler metric on a complex manifold M , and take c E R. Then the Cauchy problem { D~(T)

0,

,

(3.1.10)

~ ( o ) = p, ~ (o) = ~0,

has a holomorphic solution for a/1 (p; vo) E S l'~

fit the following condition holds: (3.1.11)

r(X,2) = cG~. b-~rtherrnore, when a holomorphic solution of (3.1.10) exists, it is unique.

Proof: To show the necessity of (3.1.11) is an easy matter. Assume ~2: A , ~ M is such a solution, and apply 0 / a ~ to Dc(qo) = 0. We obtain e

o = (1 + (c/2)1r

~ (~')"(rT")~"(~')(~')"(~')"

+ rj;.(~')(~').(~,,)~

C

= (1 + (c/2)1C1=) = ( ~ ' ) " + ~~ + r~;,(~ 1(~ ) r

(1 + (c/2)1r

~ ( ~ ' ) " + o ( r ; , ) ( ~ )(~ ) ( ~ )

- Acr~;,(~')(~')"(~')#

C

= (1 + (c/2)1~1~) ~ (~'1~ + ~(rT")(~')(~')"(#)"' (3.1.12) where we used De(qo) = 0 artd ~;.

=t.~

by (2.3.10) and (2.3.3). Now, since ~'(0) E SI'~

t.~;~v

+G

G~;~

=0,

Proposition 3.1.3.(iii) yields

1 a ( ~ ' ( { ) ) = (1 + (c/2)1~12) =' therefore - - setting v = ~'(~) - - (3.1.12) becomes

c a(v). - = -~(rT.)~.~,

(3.1.13)

134 that is, by (2.3.19), (3.1.11) holds. So (3.1.11) is a necessary condition for Dc(cp) = 0 to have a holomorphic solution. Now our task will be to show that it is sufficient too; so let us assume that (3.1.11) holds. For any e i~ E S 1, let us consider the Cauchy problem

{

~(t)

= -Ac(t)~(t)

g(0) = p,

- F~ (~(t))~(t),

for a = 1 , . . . , n,

(3.1.14)

~(0) = ~1%0.

The O D E theory gives an ~ > 0 and uniquely determined maps g~e : ( - ~ , r ~ M solving (3.1.14). Define qo: A, - . M by

~(~) =

g
(3.1.15)

Writing ff = te ie we have CO

ie -iO

co-? = -

0

CO

CO

2-7- ~ + i t ~

and

COr

ieiO

CO - it-~

2-7 ~

;

so if qo were holomorphic, it would follow that

~'(r

= ~-~~

~"(r = ,-2~~

and thus qo would be a holomorphic solution of (3.1.10). Therefore, we must prove that, assuming (3.1.11), the m a p qo defined by (3.1.15) is holomorphic. This will complete the proof of the theorem since the uniqueness of holomorphic solutions is immediate. In fact, since a holomorphic m a p on a disk is uniquely determined by its restriction to the real axis, the uniqueness statement for (3.1.14) implies that qo is the unique possible holomorphic solution of (3.1.10). Now let fc: R --~ R be the unique solution of the Cauchy problem f = A c ( f ) f 2, f(0)=0, f(0)=l; explicitly, the function fc is given by

fc(t) =

e(_c/2)1/2 t + e(_c/2)1/~ t =

t

if c = 0,

V/~ tan (V/-{t)

i f c > 0,

where, here and in the sequel, the case c = 0 (where some expressions could make no sense) should be considered as a limit for c ---+0, and we have chosen the obvious determination for square roots of negative numbers. Then, if we define do(t) = g,,o (f~(t)), we have

~o = ],(g,,o o fc)

and

~ro= ]2[(~,,0 + Ac#e,o)o f~];

(3.1.17)

135 so a0 satisfies "'(X__

o"0

--

(X

9

"D

--Fm(o'O) o'0,

ao(O) = p,

for

~

:

1,...,

n,

~o(0) = ei~

If h = G(&e), then

So h(t) - h(0) = 1, and the curve #0 = dl'~ the curve

a c t u i l y lives in SI,~

that is

6to = (&e) ~ = dao(d/dt) lives in S n M = (SI'~ ~ C TaM. Now let us consider the restriction to S I , ~ field X:

of the horizontal r a d i i vector

X = XlSl,OM 6 X ( T I ' ~ 1 7 6 which is locally given by

x = v~,

= v~a~ - r ~ j o ~ .

(3.1.18)

Then, by construction, 6t0 is the integral curve in S a M starting at (p; ei~ of the vector field X ~ the r e i part of X. If we denote by e tx~ the l o c l one-parameter group of diffeomorphisms induced by X ~ on SnM, then we may write

6to(t) = e'x~ W~

and

(3.1.19)

oo(~) : r W x ~

where r : S R M ~ M is the canonical projection, u0 = v~ 6 S ~ M and, for simplicity, we write e{~ for (cos 0 + J sin O)uo. Using the vertical r a d i i vector field ~, we define another vector field on S I , ~ by

Z = i tls,,o M 6 X ( T I ' ~ 1 7 6 so that in l o c l coordinates

Z

=

iv~O~.

(3.1.20)

We observe that d r ( Z ) = O. Since, by direct inspection, one immediately sees that the map (e ia, u) s-~ W~ is the one-parameter group of diffeomorphisms of S a M induced by the vector field Z ~ the r e i part of Z, it follows that (3.1.19) becomes

6to(t)

=

etx~176176

and

ao(t)

=

r(etX~176176

(3.1.21)

Because of (3.1.15), (3.1.16), (3.1.17) and (3.1.21), the holomorphicity of ~ is equivl e n t to

~r

_

o,,__0

~r

where we may choose 0 = 0 because u0 is a generic vector in SRM.

,

136 We start by computing [X ~ J X ~ = i[X + "X, X - -X] =

-2i[x,X],

and

IX ~ z ~ = IX + -s z + -g] = [x, z] ~ + [x,-Z] ~ Working in local coordinates (reealfing that F~;~,va = I'~;u and the Lemma 2.3.3), we have

IX, z] = i [ v ~ , v~&] = i{~%.[~., &] + v % ( ~ ) b o = i{v%ZI'~;,b~ - v,r~b~

- v~}

- v~}

(3.1.23)

= -iX.

Also, using (3.1.13) and again Lemma 2.3.3, we find that [X, Z'-] = - i [ v U 6 , , ~ & ] = - i , , " ~ [ ~ , , b,] = -ivUv~--r~-;,& = O. Finally, using again Lemma 2.3.3 together with (2.3.19), we get

IX, x ] = [ . % , ~ 1 - ~(r;~)0+} + " = - ~ ( x , ~) + ~(x,

= v , ~ [ ~ , , ~1 = ~ , ~ { ~ ( r ~ , ) b .

Therefore using the hypothesis (3.1.11) for vectors in SI,~

~).

we get

[X, X] = - a + c~ = icZ ~

So we can conclude [X ~ J X ~ = 2cZ ~

and

Now for r > 0 define ur = e"X~

. Put

u(t) = d ( e t X ~

Then

IX ~ Z ~ = - J X

(3.1.24)

~

~ G T,,(SaM).

du d tx o o -g(t) = -~(d(e ) , - , , , . ~ , ( Z )) = - d ( ~ ' x ~

o)

= d(~,xoL_,xo..(sxo), where s

is the Lie derivative, and = -d(etX~

= -2~ d(~ 'x~176

(f~xo(JX~

(Z o) = -2cu(t).

In other words, u(t) is a solution of the Canchy problem ii = - 2 c u ,

u(O) = Z~

~(0) = Y X ~

(3.1.25)

137

Therefore

e v':-~t + e-~/:-~t u(t) =

Z~

2

+

eVr:~, _ e-V':'~, 2V:~ sx~

and, in particular, d(~ o ~ r x o ) . 0 ( Z ~

= d ~ . . (u(~)) = e ' / ~ r

- ~ 2v/-L~-r

=

Sd~'(X~

24:~

dTr~,(JX~

Now,

ev,:-~l[,(O = 1 + X/~L-712t I - X/-s-C'~t ; thus we get

0

(j:l(OXOeOZO uo)

- ~ Tr .

=d(~oJ:'(Ox~176

t

= l+(c/2)t2Sd%:,(,)(X~

10 = 0

whereas (9 (e$[~(t)XOeeZOu )

~71"

0-1O

=

(9 (el[t (OX~ -~r

=

0

-

-

= -dfc ~ 1 dru,Z,(,)(X o)

1 1 + (c/2)t 2 d~ru,:~(o(X~

and the assertion finally follows from (3.1.22).

I-1

Theorem 3.1.4 is satisfactory from a PDE point of view since condition (3.1.11) is a natural compatibility condition of F~obenius type (in fact, in the case c = 0 Theorem 3.1.4 could be directly derived as a consequence of Frobenius' theorem), b-hrthermore, condition (3.1.11) on the (1,1)-torsion r of the metric F can be rephrased in a more geometric way, involving the constancy of the holomorphic curvature and suitable symmetries of the curvature operator. But for this we need a couple of preliminary computational lemmas. LEMMA 3.1.5: Let F: TI'~ --4 R + be a strongly pseudoconvex Finsler metric on a complex manifold M. Then ( ( V - ~ ) ( H , K ) x , X) = (r( H, O(K, W)), t) for a/l W E 1; and H, K E 7~. /n particular, ((VW~)(H, X)X, X) = 0

for all W E V and H E 7-l.

138 Proof: Since we are interested only in the horizontal part, we may replace fl by ~ n = ft~ | dz ~ | 5o.

Since V w d z ~ = 0 and V W 5 a = 0, we have vwn

~ = (vwn~)

|

dz~ | ~o.

Again, we only need the horizontal part, that is 9 (VW~B o ) = W ( R ~ ; ~ ) dz ~"A d2 ~ R ~o; . ~ w ~p ( W ) d z " A d2 ~. pH

Recalling (2.3.21), taking H, K , L E 7-/we get H"-K--~L~

((V-WFt~(H, K ) L , X) = Go [ W ( R ~ ; ~ ) - n~;,~ ~

- ~(r~;,)r~ + &(rL~(r~,) ) -

rL6~(rT,)r~]H,~-~L~W-~

=-Go[~,&(r~.,)

~

9

o

9

o

~

~

o

(where we used L e m m a 2.3.3), --

"

-

o

~ ( G o r ,o; , ) r ~,o - G ~ . ~ ( r ;a, ) r ~ ] H'-K-~L/~W--~

(where we used Gor L = V ~ , (2.3.1~), a o 0 4 r L ) = G ~

- Go~r L = 0

and L e m m a 2.3.4), o

-

"~

~ ( G o r z ;o. ) r ~ ~~ -

o

or

G ~ ( r ; , ) ra ~ ]

"~

a

H'-K-~L~W-~

(where we used GoFT;. : 0), o -

~

_5~(GoFz;,)F~ ~

H"-K~L~W-~.

v~(rT,)r~]

Hence < ( V W a ) ( H , K)~(, X) -

- [-~(~o~r~;.) -

o

o

+ G~(r~;~)

H,-K-~vZW---~,

v~(r[,)r~]

= ~ ( G o F ; ~ ) F 0 ~ H K WT, (where we used (2.3.3) and v ~ r ~ ; , = F~,), = G~(r;.)r0~H

a

because 8 ( g , W) = - F ~ Finally,

w ~ =
and the last assertion follows.

v ~ W ~ = 0, []

139 In the sequel we shall need some symmetry properties of the curvature operator, which are particular instances of (2.5.3) and hold under weaker hypotheses. LEMMA 3.1.6: Let F: T I , ~ ~ R + be a strongly pseudoconvex Finsler metric on a complex manifold M . Then (i) (fl(H,)()X, X) = (f~(X, ~)H, X) for a//g 6 7-/iff

(OHO(H,X, )~), X)

= 0

(3.1.26)

for all H 6 Tl; (ii) (f~(H, K)X, X) = (f~(X, K)H, X) for M1 H, K 6 7"l iff (OHO(H, x , K ) , X) = 0

(3.1.27)

for M1 H, K 6 7"l. Proof: First of all, OH 0 = p*H(D"O), by (2.2.6). Corollary 2.3.7 says that D"O = q~ Afl;

the assertion then follows immediately from (2.5.4).

[]

The next result begins to establish a relationship among constant holomorphic curvature, the symmetry condition (3.1.26) and the compatibility condition (3.1.11): PROPOSITION 3.1.7: Let F: T I , ~ ~ R + be a strongly pseudoconvex Finsler metric on a complex manifold M . Then r ( x , ~) = c G,.

(3.1.28)

iff F has constant holomorphic curvature K F =-- 2c and (OHO(H, X, X), X) = 0

(3.1.29)

for all H 6 TI. Furthermore, they both imply ( ~ ( x , - K ) x , X) = c G(X, K)

(3.1.30)

for all K 6 ~ .

Proof: If (3.1.28) holds, we have x) = (r(x,

= cG 2,

and so K F =-- 2c. So we may assume that F has constant holomorphic curvature 2c, and prove that (3.1.28) and (3.1.29) are equiva/ent under this assumption.

140 For W E V, let K -- O(W) E "H; note that VWX = 0 and V w X = O ( W ) = K. Then W ( c G 2) = 2cG{x, K);

W(~(X,)~)X, x} = ((VWf~)(X,X)X,X) + (~(X,-K)X,X) + (12(X,~)X,K) (3.1.31) = (~()/,K)x, X> + ( r H ( x , x ) , K ) , where we used Lem_mas 3.1.6 and 2.3.8. Since F has constant holomorphic curvature 2c, we have (~(X, X)X, X) = c G ~, and hence (3.1.31) yields

(~(x, K)x, x) = 2ca(~, K> - (r(x, ~), K).

(3.1.32)

Subtracting (r(X, YC),K) = (~(X, X)X, K) to both sides, we find that (3.1.28) holds if and only if (~t()i.,"K)x, X) = <~(X, X)X, K> for all K E H, that is, recalling (2.5.2), iff (~(K, ~)X, X) = (fl(X, 2)K, X), and thus, by Lemma 3.1.6, iff (3.1.29) holds. Finally, if (3.1.28) holds, (3.1.32) yields (3.1.30).

[]

And now we are able to give the more geometric version of the existence theorem for holomorphic solutions of De(W) = 0: THEOREM 3.1.8: Let F: TI,~ ~ R + be a strongly pseudoconvex Finsler metric on a complex manifold M. Then the Cauchy problem (3.1.10)

{ Do(~) = 0, v ( 0 ) = p, v ' ( 0 ) = v0,

has holomorphic solution for all (p; v0) E SI'~ curvature KF = 2c and either

iff F has constant holomorphic

= 0

(3.1.33)

(D(H, X)X, X) = <~()~, )~)H, X).

(3.1.34)

or, equivalently VH E T-I

~trthermore, when a holomorphic solution of (3.1.10) exists, it is unique. Proof: It follows from Proposition 3.1.7 and Lemma 3.1.6.

[]

141 It must be remarked that in [AP2] condition (3.1.33) was expressed in an equivalent way in terms of a tensor defined in local coordinates. Before going on let us state what we got so far in terms of the main characters of this section, the c-geodesic complex curves. Putting together Corollary 3.1.2, Proposition 3.1.3 and Theorem 3.1.8, we have the following: THEOREM 3.1.9: Let F: T I ' ~ --* R + be a strongly pseudoconvex Finsler metric on a complex man/fold M, and take c E R. Then (i) There exists a segment of infinitesimal c-geodesic complex curve through (p; v) for any p E M and v E S~'~ if either (3.1.11) holds, or KF -- 2c and (3.1.33) holds, or K F -- 2c and (3.1.34) holds; (ii) There exists a unique segment of c-geodesic complex curve through (p; v) for all p E M and v E S~'~ i f f F is weakly Kiihler and either (3.1.11) holds, or KF = 2c and (3.1.33) holds, or KF = 2c and (3.1.34) holds. We are left with a last question: the existence of "complete" geodesic complex curves, i.e., defined on the largest possible disk (or on the plane, or on P1), and not just on a segment. As it may be expected the key ingredient is of course to assume in addition the completeness of the metric. Precisely we have the following: THEOREM 3.1.10: Let F: TI'~ M ~ R + be a complete stronglypseudoconvexFinsler metric on a complex manifold M , and take c E R. Then (i) There exists an infinitesimal c-geodesic complex curve through (p; v) for any (p; v) E S I ' ~ if either (3.1.11) hdds, or KF = 2c and (3.1.33) hdds, or K F = 2c and (3.1.34) holds; (ii) There exists a unique c-geodesic complex curve through (p; v) for any padr (p; v) E S I ' ~ if[ F is weakly Kg2aler and either (3.1.11) ho/ds, or KF = 2c and (3.1.33) hdds, or KF -~ 2c and (3.1.34) ho/ds. Proof: By Proposition 3.1.3, it is enough to prove (ii) and in fact to show that the holomorphic solution of (3.1.10) q0 is defined on A , with r = X / L - ~ if c < 0, on C if c = 0 and on P1 if c > 0. Consider the distribution ~ = C X ~ @ C Z ~ C T(S•M), where X ~ and Z ~ are the vector fields defined in the proof of Theorem 3.1.4. The distribution :P is involutive, since we have (3.1.24) and the remaining brackets are easily computed from t:he definitions:

[X ~

~ = x ~ = [ J x ~176

pX ~

~ = Jx ~

[Z ~

~

If L is the maximal integral manifold of :D passing through (p; v), then, the proof of Theorem 3.1.4 shows that L = it(L) C M is a Riemann surface locally parametrized by the holomorphic solutions of Dc(~0) = 0. Since F restricted to L is a complete Hermitian metric of constant Ganssian curvature 2c, because of Proposition 3.1.3.(iii), using the Riemann uniformization theorem we have that there exists a unique simply connected Pdemann surface T~ endowed with the metric /~c and a unique isometric holomorphic covering map r 7~ + L such that r = p and r = v, where 7~ = A r with r = y/'k--2/c if c < 0, 7~ = C if c = 0 and T~ = P1

142 if c > 0. But, for some r > 0, the holomorphic solution ~: A r --+ L of Dc(~) = 0 with ~(0) = p and ~'(0) = v is another holomorphic isometry from Ar endowed with the restriction of the metric Pc to L endowed with the restriction of F; it follows that ~ = r and necessarily then r is the unique solution of the Canchy problem (3.1.10) - - and it has the required properties. [] For further reference, we explicitly state here some properties of geodesic complex curves which were not underlined before to simplify statements but nevertheless will turn out useful later. The remarks axe obvious consequences of the construction of geodesic complex curves (cf. the proof of Theorem 3.1.4) and of the smooth dependence on initial data of solutions of ODE Cauchy problem. PROPOSITION 3.1.11: In the hypothesis of Theorem 3.1.10, if we denote by ~v the ~hen unique c-geodesic complex curve through the point p tangent to v E S~'~ ~(ei~

-- ~c,~,(().

(3.1.35)

Fhrthermore the mapping ((, v) ~ ~ ~(~) is of class C ~176

3.1.3. A characterization of the Kobayashi metric As an application of our results we present a characterization of Kobayashi metric among the (smooth) strongly pseudoconvex Finsler metrics (cf. [AP2]). Related, though a bit weaker, characterizations were obtained by Faran [F] and Pang [P1]. We need to introduce some additional tools to get our aim; in particular we shall need a suitable version of Ahlfors' Lemma. DEFINITION 3.1.4: A pseudohermitian metric pg of scale g on A is the upper semicontinuous pseudometric on the tangent bundle of A defined by #g = g d~ | d~,

(3.1.36)

where g: A ---+R + is a non-negative upper semicontinuous function such that the set Sg = g-Z(0) is a discrete subset of A. If #g is a standard Hermitian metric on A, i.e., g is a C 2 positive function, we have already defined (in section 2.5.2) the Gaussian curvature of #g by setting

g(pg) = -2~Z~ log g,

(3.1.37)

w h e r e / k denotes the usual Laplacian

02u A u = 4 0r

(3.1.38)

143 To define a Gaussian curvature for pseudohermitian metrics we need a weak Laplaclan. DEFINITION 3.1.5: The (lower) generalized Laplacian of an upper semicontinuous function u is defined by /ku(~) = 41iminf ~ { 2 ~

~o2'r[u(~+rei~ - u ( ~ ) ] d O } .

(3.1.39)

This definition is quite nice as it shares all the important features of the definition for regular functions. First of all, when u is a function of class C 2 in a neighborhood of the point C0, (3.1.39) actually reduces to (3.1.38). In fact, for r small enough we can write

~(r + ~r

= ~(C0) + ~(C0)re

+

(r

-~~

102u ~. 2 2io

O~u

[~(Co+ ~o) _ ~(r

dO =

2

+ ~y~T(c0)r e + 0-~(r

102u

+ ~T(Co)

r2

~-

2io

+ o(~1;

hence 1 i ~

Jo

~ O--~-~(Co) 0~ + o(r~),

and the claim follows. Second, if u is an upper semicontinuous function, it is easy to see that A u > 0 is equivalent to the submean property; so Au _> 0 iff u is subharmonic. Finally, if C0 is a point of local m a x i m u m for u, then clearly Au(~0) < 0. DEFINITION 3.1.6: Let ~ be a pseudohermitian metric on A. Then the Gaussian curvature K ( p g ) of ~9 is the function defined on A \ Sa by (3.1.37) - - using the generalized Laplacian (3.1.39); clearly, if #g is a standard Hermitian metric, K ( # g ) reduced to the usual Gaussiau curvature. We are able to prove the classical Ahlfors lemma in this more general situation: PROPOSITION 3.1.12: Let ~ = gd~ | d~ be a pseudohermitian metric on A such

that K(#) <_ - 4 on A \ Sg for some a > O. Then g <_g-2, where g_~(C)

-

1

(1 -lr

2

is the scale of the Poincaxd metric.

Proof: T h e proof is just an appropriate adaptation of the classical one due to Ahlfors. For the sake of completeness we report it here.

144 For0
define g-2. r . A r ~ R + by 1 9~-2(r

= (1 - I C p / r 2 ) 2

= g-2(C/r),

r . A~ ~ R + . and set f~ = g/g-2. Being upper semicontinuous, g is bounded on At; since we clearly have g~-2(~) -'* + c r as [([ ~ r, there is a point ~0 E A , of maximum for f t . Clearly, ~0 ~ Sg; hence

0 ->- A l o g f r ( ( 0 ) ->- Alogg(ff0)

r g-2((0). (3.1.40) By assumption, K ( # ) ( ( 0 ) _< - 4 ; therefore (3.1.40) yields g((0) _< g[2((0) and thus, by the choice of C0,

vC e A,.

--

Alogg:2((0 )

~

-2g((0)K(#g)((0) -

8

9(C) -< g-2(Cl~').

Letting r ---+1 - we obtain the assertion.

[]

To get our characterization, we shall now need some considerations which are meaningful without any change in the more general contest of "weak" Finsler metrics - - and thus we shall present them in this setting. DEFINITION 3.1.7: An upper semicontinuous complex Finsler metric F on a complex manifold M is an upper semicontinuous map F : TI'~ --~ R + satisfying 10 (i) F(p; v) > 0 for all p E M and v E T~, M with v # 0; (ii) F(p; Xv) = [AiF(p; v) for all p E M, v E T~'1 0 M and X E C.

Typical examples of upper semicontinuous complex Finsler metrics are the Carathdodory and Kobayashi metrics defined at the beginning of section 2.3. Using a variational approach, we can easily define a notion of holomorphic curvature for an upper semicontinuous complex Finsler metric F following ideas of Royden [Ro], Wong [W2], Suzuki [Su]. As usual, we shall denote by G: TI'OM ---+R + the function G = F 2. 10 DEFINITION 3.1.8: Take p E M and v E T~' M, v # 0. The holomorphic curvature K f ( v ) of F at v is given by

KF(v) = sup{K(~*G)(0)}, where the supremum is taken with respect to the family of all holomorphic maps ~ : A --, M with ~(0) = p and ~'(0) = Av for some A E C*, and K(~*G) is the Gaussian curvature discussed so far of the pseudohermitian metric ~*G on A. Clearly, this holomorphic curvature depends only on the complex line in T~,~ spanned by v, and not on v itself. Furthermore, by Corollary 2.5.4 if F is smooth we recover our previous definition of holomorphic curvature.

145 This notion has been used for studying intrinsic metrics and, for instance, it is known that the Carath@odory has holomorphic curvature bounded above by - 4 , whereas the holomorphic curvature of the Kobayashi metric is bounded below by - 4 ; see, e.g., [W2] or [Su]. It is worthwhile to remark that, though apparently on the unit disk we have given two notions of curvature, they (fortunately) coincide. If F is an upper semicontinuous complex Finsler metric on A - - and so G = F 2 is a pseudohermitian metric G = g de | d~ on A - - , a priori we have defined two curvatures for F: KF(r 1) and K(G)(~). As a consequence of the following Lemma we have at once that g f ( ~ ; 1) = g ( a ) ( r LEMMA 3.1.13: Let #g = g d(| be a pseudohermitian metric on A, and ~: A ~ A a holomorphic ~e~:-map of ~ . Then on/', \ [~-~(S,) O Sr we have

K(~*#g) = K(#9) o V. Proof:" Let {gn} be a sequence of C 2 functions such that gn > g,+l and with gn(x) ~ g(x), and let #n = g,, d~ | d~ be the corresponding metrics. Then on /k \ [~o-l(Sg,,) U S•,] D A \ [r U S.,] we have

K(~*#,.) =

1 2]~'12(9., o ~,) A log(lW'l=g,, o ~) 2

O=logl_~'l=]

[021og(g,,oT)

=

I,P'l=(g,,o:) L

=

1~'12(g, o w ) 1 : ' 1 2

=-

2(g. o : ~ ( A l ~

1

+

aCoC

J

021~162 .] o :

o ~ = g(.,.) o :.

Letting n --~ +co and applying the dominated convergence theorem we get the assertion, r-l The holomorphic curvature defined in this way is clearly invariant under holomorphic isometries. More generally, if f: M --+ N is holomorphic and F is an upper semicontinuous complex Finsler metric on N, we have

KF (f(z);dA(v)) >_ that is f * K F > K f . F . Yhrthermore, as it is well known, this notion of holomorphic curvature have a built-in Ahlfors' Lemma, and so it is suitable for producing criteria of hyperbolicity (see for example Granert-Riekziegel [GR], Kobayashi [K], Abate-Patrizio [AP2]). Bounds on the holomorphic curvature allow to compare a complex Finsler metric with the Kobayashi metric and to give conditions for a given Finsler metric to be the Kobayashi metric. We now provide the right set-up for the problem in the general case. We need a last auxiliary notion to formulate our observation.

146 DEFINITION 3.1.9: Let F: T ] , ~ ~ R + be an upper semicontinuous Finsler metric on a complex manifold M, and take (p; v) E T l ' ~ We say that F is realizable at (p; v) if there is a holomorphic m a p ~: A ~ M such that ~(0) = p and A~'(0) = v with [A[ = F(p; v). In other words, ~ is an isometry at the origin between the Poincar~ metric of A and F. Because of its very definition, the Kobayashi metric is realizable in any taut manifold; on the other hand, as a consequence of the next result, the Caxath~odory metric is realizable iff it coincides with the Kobayashi metric. PROPOSITION 3.1.14: Let F : T ] , ~ ---+ R + be an upper semicontinuous Finsler Then: metric on a complex manifold M , and choose (P0; v0) E T l ' ~ (i) I r E is realizable at (P0; v0), then F(p0; v0) > ~M(P0; Vo); (ii) I.f g F ~ - 4 , then F ~ t~M; (iii) /.fF is rea//zable at (P0; Vo) and g F <_ --4, then F(p0; v0) = ~M(P0; v0). Proof: (i) Let ~: A --+ M be a holomorphic map with ~(0) -- P0 and v0 = A~'(0) such that F(p0; v0) = [A[. Then

F(p0; v0) = I~1 > ~M(;0;v0). (ii) Take (p; v) E T I ' ~ and let ~: A -+ M be a holomorphic map with ~(0) = p and v = A~'(0). Then ~ * G is a pseudohermitian metric on A with Gaussian curvature bounded above by - 4 (by L e m m a 3.1.14); it follows from Proposition 3.1.12 that ~ * G < #-2. Thus F(p; v) = F(~(0); A~'(0)) = ~*F(0; A) _< [h I.

Since this holds for all such ~, we get F _< ~M. (iii) This is just a consequence of (i) and (ii).

[]

Let us turn to the case of smooth Finsler metrics. In this case we have given the differential geometric conditions which guarantee the same good behavior that Lempert [L] discovered for the Kobayashi metric of strongly convex domains. It should not be surprising that we can now obtain the following: TtIEOREM 3.1.15: Let F: T I , ~ ---+ R + be a strongly pseudoconvex smooth complete Finsler metric on a complex manifold M. Assume that one of these equivalent conditions hold:

r(x, ~) = - 2 G~, or

K F = --4

and

(OHO(H,x,~),x}=O

f o r a 1 1 H 67-l,

or

K F =-- --4

Then

F is the

' and

(~(H, 2)~, X> = (~q(X, ~)H, X) for MI Kobayash/metr/c of M.

H 6 7"I.

147

Proof: Under the hypotheses, using also Lemma 3.1.6 and Proposition 3.1.7, the holomorphic curvature of F is - 4 . Furthermore, by Theorem 3.1.10 F is realizable. Then Proposition 3.1.14 complete the proof. []

3.2. M a n i f o l d s w i t h c o n s t a n t n o n p o s i t i v e h o l o m o r p h i c curvature 3.2.1. A complex Cartan-Hadamard theorem Now we shall take a closer look to K/~hler Finsler manifolds of nonpositive constant holomorphic curvature, with an eye toward the very natural problem of their classification. Examples show that in the Finsler setting this problem is much more complicated than in the Hermitian one; for instance, Lempert's work ILl and Theorem 3.1.10 imply that all strongly convex domains of C n endowed with the Kobayashi metric are weakly Kghler Finsler manifolds with constant holomorphic curvature - 4 . As a step toward the classification, in this section we shall prove that a simply connected K~hler Finsler manifold of nonpositive constant holomorphic curvature is Ediffeomorphic to an euclidean space. Furthermore, in the case of constant negative holomorphic curvature our results show that the Finsler geometry of the manifold is pretty much the same as the one of strongly convex domains endowed with the Kobayashi metric. The crucial point in our argument will be to estimate the curvature terms Re [(~/(X,-if)H, X)v

-

(fl(H, fc)H, X),, + ((r~( H, "2), H)},, - ((rn(X, H ) , H))~] (3.2.1)

appearing in the second variation formula for complex Finsler metrics, using as much as possible the constancy of the holomorphic curvature. Let F: T I , ~ --+ R + be a strongly pseudoconvex Finsler metric on a complex manifold M. Then we know that F has constant holomorphic curvature KF ~ 2c 6 R iff (~(X, X)X, X) - cG2. (3.2.2) In this kind of problems tradition advises that the key idea it is to differentiate (3.2.2) in such a smart way to get an expression of (3.2.1) whose sign may be estimated without much effort. This scheme, which works without great pain in usual K/~ler geometry, is much more complicate to carry out in K~hler Finsler geometry, as the reader will notice in a while. In subsection 1.2 of this chapter we have already proved a few technical result needed here (Lemma 3.1.5 and Lemma 3.1.6). Furthermore, in Proposition 3.1.7 it was shown that, if KF ~. 2c and either VH 9 7-I

(OHO(H, X, Yr X) = 0

(3.2.3)

r(X , ~) = c G~,

(3.2.4)

or, equivalently,

148 then it follows that < ~ ( x , K ) x , X) : c a ( x , K)

(3.2.5)

for all K 6 7"/. This is a first step on the way of giving a nice expression to the terms of (3.2.1). To proceed, we need a further computational result 9 LEMMA 3.2.1: Let F: T I , ~ --~ R + be a strongly pseudoconvex Finsler metric on a complex manifold M . Then ((Vvf~)(H,-K)x, X) = (7(0(H, V), K---),~) for a11 V 6 1; and H, K 6 ~ . In particular,

( ( V v ~ ) ( x , A')x, x) = 0 for all V 6 1; and K 6 ~ . Proof: As in Lemma 3.1.5 it suffices to consider ~ t = 9t~ | dz ~ | 5o; so O

(Vv~)|174174174174

Vv~ ~

~

f

We are interested only in the horizontal part. Taking H, K 6 7-/we get

= G o [V(R~;.~) - a L o ~ ; ( v ) ]H.k-~

ao(Vva~)(H,~) --a~[o~(r~;.) _

"

- ~(r~;~)r~

o

O

Gr

+ 0"~ ( r , ~oo-~ ( r ~ . )-)

tO

- r ~ . 5 0 ( r ; p ) r . ~ ] VXH~-K-'~ =-ao[6~&(r~;,)

- r x~; o 09. ( r eo; . ) - ~ ( r ~ ,,) r . ~

+F~6~(FX;u)

o

-

,,

~

o

+ o"~ ( r ~o . ) ~ o ( L . ) ~,

p

~, ~, ,,

(where we used Lemma 2.3.3), = -[~.(aoo~(r~.)) 9

(~

+ ao0~(r~)~(r;~) -

~ " o r~;~ao0~(r~.) - ~ ( a o r L )9 r #% ~

+ a~(rx;,,)

-

G ~ . r , ; . r ~ ;"~

VaHU-K--;

a~.~(r~)r~]

(where we used Lemma 2.3.4 and G o F ~ = G ~ ) ,

- -[-~(ao~r~;~) +

~(b~,,(a~))

+

r~,,co~r,;~ - ~(a~r~)r~

+ G o O x ( F ~ ) 6 ~ ( F ; , ) + G~6~(F~;,, ) - G~F~;~,F~;~

V~H.-~

- a~(r~'.)r~]

(where we used G o r ~ ; . = 0 and a o r ~ ; . = 6.(G~)), =-[-~(G~r~;.)+6~(Oo6.(a~))

+ a ~ . 6 . ( r ~ , . )o"

- a ~ . ~ . ( r ; . ) rO"

p

6

' r ~

~]

y~g.TCv.

o

9

~

.

Furthermore, o

0"

~"

A

~

v

149

[~(r~;.) + r~j~(r;.)]

Go w ~ ( V ) ~ ( H , K ) = -a~s

V~H~-K--~

= -a3"~ [~(r~;.) + r ~ . ~ ( r T . ) ] V ~ H ~ K v. Summing up we find ( ( V v a ) ( H , K)X, X)

= - [ - ~ ( G . ~ r ~ ; ~ ) + ~(o~6~(G~)) - ~ ( G ~ r ~ ; , ) r ~ + G~Z)~(r~.)~(r~'.) - G#~5~(F;p)s

+ G~(r~;.) = - [~(~b~.(a~)) + v~(rT.)]

+

+ G3"~r~J~(%)].4V~H"-~

~

- ~(Gov~r~;.)r.~ ~ ~ + Gov~0~(r~.)~(r~.) V ~ H . - R -~

= -[-$~(G;~)F~x - Gx~6~(r;~) + G~3"~(F~)] V~H~'K ~ (where we used v~O~.(G~) = v~(G~;~ - s

-

r~'.a~.~)

= 0

and Gav~0~(r~a ) = - G x ~ ) , =

~(G;~)F~V~H.R

--~ = ~ ( G . F T ~ ) r ~ V ~ H " R

= V~(rT~)r~y~g.-R

-; =

~

(~(0(g, Y), g--), ,).

The final assertion follows from O(X, V) = O.

[]

Now we are in condition to give the desired expression for one of the terms of (3.2.1) involving the Hermitian product: PROPOSITION 3.2.2: Let F: T I , ~ --~ R + be a strongly pseudoconvex Finsler metric on a complex manifold M with constant holomorphic curvature 2c E R. Assume that

VH 6 "H

(0HO(H, X, 2), X) = 0.

(3.2.6)

Then

(a(H,-K)X,X) + <s

= c{(H,x)(x,K) + <X,X>
(3.2.7)

for all H, K e TI. In particular, if

VH, K 9 7-/ then

( 0 . 0 ( H , x , K ) , x ) = 0, C

<~(x,'K')H, X> = { { + (X, X)
(3.2.8) (3.2.9)

for all H , K 9 H.

Proof: Take V, W 9 12 such that O(V) = H and O(W) = K and extend them as sections of V in any way (and thus extend H and K as sections of 7"l via O). We have

Y ( c G(X, g ) ) = c[(H, X)(X, K) + G(H, K) + G(X, V v K ) ] ,

150 and, using Lemma 3.2.1,

V(gt(x,K)x,X) = ((Vv~)(X, K)X, X) + (~(H, K)X, X) + (~(X, VvK)x, X) + (s = <~(H,K)x, X> + <~(X, V----~)X, X> + <~(x,K)H, X>,

K)H, X).

Since (3.2.6) holds, we can use Proposition 3.1.7 (that is, (3.2.5) applied both to K and to V v K ) to get exactly (3.2.7). Finally, (3.2.9) follows from Lemma 3.1.6.

[]

The other Hermitian product term of (3.2.1) is obtained in the following: --* R + be a strongly pseudoconvex Finsler metPROPOSITION 3.2.3: Let F: T l ' ~ ric on a complex manifold M with constant holomorphic curvature 2c 6 R. A s s u m e that (3.2.6) holds. Then

(~( H, y<)K, x> = c ( (H, x) (K, x) + (X, X) ((H, K>) } for all H, K 6

7-{.

Proof: First of all, we have

(ft(H, X)X, X) = (f~(X, H)X, X) = c G{H, X),

(3.2.10)

where we used (2.5.2) and (3.2.5). Now let W 6 V be such that O(W) = K; then

W(c G(H, X) ) = c { (K, X)(H, X} + G(VwH, X)}, W<~(H, )~)X, X> = ((Vw~)(H, X)X, X> + (~(VwH, Y<)X,X> + <~(H, )~)K, X>, so that (3.2.10) yields (~2(H, 2 ) K , x ) = c ( K , x } (H, x) - ((Vw~2)(H,~)X,X). Now Lemma 3.2.1 gives

((Vw~)(H,)~)X, X) = (r ( 8( H, W), y<), ,) = (~ ( 8( H, W), :~)X, X) = c G(O(H, W), X}, again by (3.2.10). But

{O(H, W), X> = -G~,F'~zK ~H" = -<>, and we are done.

[]

Now that we have got the Hermitian product terms, we turn our attention to the symmetric product terms. The first one is almost immediate:

151 PROPOSITION 3.2.4: Let F: T I , ~ --~ R + be a strongly pseudoconvex Finsler metric on a complex manifold M wi~h constant holomorphic curvature 2c E R. A s s u m e that (3.2.8) holds. Then

v~(g, ~) =

C

-~ {(K, x)X +

(X,x)K }

(3.2.11)

for all K E 7-i. In particular, c <> = ~<X, X><(H , K))

(3.2.12)

for all H , K E 7-l. Proof: As we remarked in L e m m a 2.3.8, we have

~n = a(., ')x. Thus, using again (2.5.2), we get = (12(x,-K)H, X> for all H , K E 7-/. Then (3.2.9) yields (3.2.11), and (3.2.12) follows immediately. [] For the other symmetric product term we need the weak Kiihler condition: PROPOSITION 3.2.5: Let F: T I , ~ ---* •+ be a weakly K6Jder Finsler metric on a complex manifold M such that (3.2.8) holds. Then ((H, T~(x,K)>> = 0 for all H, K E 7-l. Proof: The weak K~aler condition (8(H, X), X> = 0 for all H 6 7-I implies VH, g E ~

( ( V ~ 0 ) ( H , X), X> = 0

(3.2.13)

because V ~ X = 0 = VKX. Now, writing O = 0 a | 5a, we have V ~ O -- (V~-O a) | 5a and

v~e" = g~(r~;.)

dz" ^ d z V +

K~5~(r~) r ^ d z ~.

Therefore (3.2.13) implies

G.[~(r~;.) - 6~(r~;~)]g.-k--Cv ~ = 0

(3.2.14)

for all H , K E 7-/. Writing the curvature in local coordinates we find (~(X, g ) g , X) = -G~[5~(F,;~) +

r,~(r;.)]g

( ~ ( g , g ) x , x) = - G ~ [ ~ ( r . ; . ) + r . ~ ( r ; . ) ] g

K v ,

g v .

152 So (3.2.14) yields (~(x,K)H,x)-

(fI(H,K)x,X) =

Ot O" D -G~r.~(r;~)H g TvV

= ((H, ru(x,-K))),

[]

and the assertion follows from (3.2.8). We can finally collect all our computations in

~ R + be a weakly K ~ d e r Finsler metric on a THEOREM 3.2.6: Let F: T I , ~ complex manifold M . Assume F has constant holomorphic curvature 2c 6 R and that (3.2.8) holds. Then Re[

-(a(H, ft)K,x>

+ (
-

<>]

C

= -~ R e [ G { ( H , K ) - <(H, K)>} + (H,x>{<x,K> - 2 ( K , x ) } ]

forM/H, K

6

H.

Proof: It follows from Propositions 3.2.2, 3.2.3, 3.2.4, 3.2.5.

[]

The computations made above allow us to prove the main result of this subsection: THEOREM 3.2.7: Let F: T I , ~ ~ R + be a complete strongly pseudoconvex Finsler metric on a simply connected complex manifold M. Assume that: (i) F is Ks (ii) F has nonpositive constant holomorphic curvature 2c < O; (iii) (OHO(H,x,-K),x) = 0 for a11 H, K 6 ~ ; (iv) F is strongly convex. Then expp: T] ,~ ~ M is a LipscMtz E-diffeomorphism at the origin for any p 6 M. fhrthermore, M is foliated by isometric totally geodesic holomorphic embeddings of the unit disk A endowed with a suitable multiple of the Poincar4 metric if c < O, or by isometric totally geodesic holomorphic embeddings of C endowed with the euclidean metric if c = O. In particular, if c = - 2 then F is the Kobayashi metr/c of M , and if c = 0 then the Kobayashi metric of M vanishes identically.

Proof: Fix p 6 M, and let a0: [0, a] --* M be a radial normal geodesic in M with a0(0) = 0, and P, a regular fixed variation of a0 such that the transversal vector U of ~ satisfies Re(UH, TH)~ o -- 0 (note that, by Corollary 2.6.5, we single out the same variations of a0 by requiring orthogonality with respect to the Riemannian structure and the horizontal lift induced by the real Finsler metric F~ in other words, the space 2(o[0, a] is the same

153 both working in the complex setting and in the real setting). Then Theorems 2.4.4 and 3.2.6 yield

d2g~ ( . .

c G(bo) Re [(U H, uH)~o

-- ((V H, uH))dro] } d t .

Now, the curvature is nonpositive, and Proposition 2.6.1 yields

Re[(UH, U H) - ((uH, uH))] = ((iuH)~ [ (iuH) ~ >_ 0;

SO , u ) > O.

Fhrthermore, the second variation may vanish iff VTH+'~guH -- O, that is iff U is parallel along a0. But U(0) = op; so the second variation may vanish if and only if the transversal vector is identically zero. Recalling (1.7.5), we have just proved that the Morse index form I s is positive definite on 2(0[0, a]. But a0 was any radial normal geodesic in M; so we can apply the Cartan-Hadamard Theorem 1.7.17, proving that expp is an E-diffeomorphism at the origin. The remaining assertions follows from Theorems 3.1.10, 3.1.15 and (for the case c = 0) from standard properties of the Kobayashi metric. [] We recall that a complex geodesic in the sense of Vesentini [V] is a holomorphic map T: A --. M which is an isometry with respect to the Poincard distance on A and the Kobayashi distance on M. As a consequence of the last theorem we get existence and uniqueness of complex geodesics:

COROLLARY 3.2.8: Let F: TI,~ ---* R + be a complete strongly pseudoconvex Finsler metric on a simply connected complex manifold M. Assume that: (i) F is Kh~hler; (ii) F has constant holomorphic curvature -4; (iii) (OHO(H, X, g ) , X) = 0 for M1 H, K e 7-l; (iv) F is strongly convex. Then every geodesic complex curve in M is a complex geodesic in the sense of Vesentini. In particular, t'or any p E M and v E T1,0M -p _._ - - or/'or any pair of distinct points p, q E M - - there exists a unique complex geodesic passing through (p; v) - respectively, passing through p and q. Proof: By Theorem 3.2.7, the exponential at any point p E M is injective; in particular, then, the geodesics are all globally length-minimizing. Since, by definition, a geodesic complex curve sends geodesics with respect to the Poincar~ metric in geodesics with respect to F (which coincides with the Kobayashi metric because c = - 2 here; Theorem 3.1.15), it follows that every geodesic complex curve is a complex geodesic in the sense of Vesentini. Finally the last statement follows from the analogous statement for real geodesics, and from the fact that every real geodesic is contained in a geodesic complex curve. []

154 We explicitly r e m a r k t h a t the conditions (i)-(iv) in T h e o r e m 3.2.7 are not necessarily independent. For instance, the results of [P2] seems to suggest t h a t conditions (i)-(iii) directly imply the convexity of the indicatrices of F - - but perhaps not the strong convexity. Finally, an advice: condition (iii) is not a consequence of the KKhler condition, unless F comes from a Hermitian metric. In fact, differentiating the weakly Ks condition one gets

K<~(', x), ~> = <0.~(, ~, ~), x} + <<',r~(x, ~)>>.

3.2.2. The Monge-Amp~re exhaustion T h e o r e m 3.2.7 suggests that in case the Kobayashi metric on a complex manifold M is p a r t i c u l a r l y nice then the geometric picture is quite the same as the one Lempert ILl described for strongly convex domains in C n. There is still an i m p o r t a n t p r o p e r t y t h a t in this class of domain is available: in terms of the Kobayashi distance it is possible to construct Monge-Amp~re potentials with logarithmic singularity at any given point. Our next task is to show that the same can be done in a Finsler setting. A n analogous results will be achieved for metrics with holomorphic curvature constantly zero. Of course in this case we are in a parabolic environment r a t h e r t h a n in a hyperbolic one, and the model that one should keep in mind is a complex Minkowski space. Let set up the necessary notations and state our assumptions. Let M be a simply connected complex manifold of dimension n and let us suppose t h a t on M is defined a strongly pseudoconvex Finsler metric F such t h a t F is complete KKhler;

(3.2.15)

KF -- 2c < 0, where either c -- - 2 or c = 0;

(3.2.16)

(~HS( H, X, K ) , X) : 0 for all H , K 6 7"/;

(3.2.17)

F is strongly convex.

(3.2.18)

Of course, in light of Theorem 3.1.15, when c = - 2 we are considering the Kobayashi metric of M and we are assuming t h a t it has special properties. For simplicity of exposition, for the moment we shall nevertheless treat the negative curved case together with the flat one and we shall underline the differences only later when it will be necessary. We recall some more notations. Let

Sp = {v e T~'~

] F(v) = 1}

(3.2.19)

be the unit sphere in the tangent space at p and

I, = I , ( M ) = {~ e T;,1 0 M I r(~) < 1}

(3.2.20)

155 he the indicatrix o f f at p. For v E Sp let ~ be the c-geodesic complex curve through (p; v), which exists because of Theorem 3.1.t0. We define maps H - 2 : A x Sp -+ M in case c = - 2 and H 0 : C x Sp -+ M when c = 0 by He((, v) = ~ . ( ( ) .

(3.2.21)

Because of Proposition 3.1.11, the maps Hc are C~ Finally we define maps E - 2 : Ip ~ M in case c = - 2 and E0: T I ' ~ ~ M when c = 0 by Ec(v) = H c ( F ( v ) , v / F ( v ) ) = ~,,/F(,,)(F(v)).

(3.2.22)

Again because of Proposition 3.1.11, for A 9 C with [A[ = 1 and v 9 Sp ~(~)

= ~(A~);

(3.2.23)

therefore for any choice of A 9 C with [A[ = 1 and v 9 Sp we have E~(Av) = ~ . ( A F ( v ) ) .

(3.2.24)

Choosing any system of coordinates around p we can identify C '~ ~- T ] , ~ so that E - 2 will be defined on a strongly convex complete circular domain Ip C C n and E0 will be defined on all C ~. Let us collect some immediate properties of 'the maps Ec. If expp: T ~ , ~ --* M is the exponential map of F at p, we have:

PROPOSITION 3.2.9: Let F: T I ' ~ -+ R + be a strongly pseudoconvex Finsler m e t ric on a simply connected complex manifold M satisfying (3.2.15)-(3.2.18). Then for v E Ip and c = - 2 we have E - 2 ( v ) = expv

(=1 F(v)

v

(3.2.25)

while [or v E C n and c = 0 Eo(v) = exp,(v)

(3.2.26)

~-hrthermore, as a consequence,

(i) Ec is an E-diffeomorphism at the origin; (ii) i f L is any line through the origin in T~,~ the restrictions o f Eo to L and o f E - 2 to the disk L • Ip are holomorphic; (iii) i f M is a Stein manifold, the m a p Er is biholomorphic iff it is of class C ~ at the origin. Proof: The equalities (3.2.25) and (3.2.26) are consequence of the definition of Ec in terms of c-geodesic complex curves. (i) follows immediately from Theorem 3.2.7, Theorem 1.6.2 and the fact that the function atanh t f ( t ) -t is an analytic function.

156 (ii) is a direct consequence of the definition of E~. Finally if M is Stein, then it may embedded in C N for some suitable N. Then it is well known - - for instance it is a consequence of a theorem of Forelli (see [Ru]) - - that the components of the composition of Ec with the embedding are holomorphic functions because are smooth and have holomorphic restriction to the complex lines through the origin. []

We shall now introduce on M a nice exhaustion function. For simplicity we set R-2 = 1

R0 = +oc;

DEFINITION 3.2.1: The Monge-Amp~re exhaustion ar M --~ [0, Re) of M is defined as follows. If q E M then, for some ( E C and v E Sp, q = Er and we set ar

= ac(Er

= I~1 ~

(3.2.27)

Because of (3.2.23) and (3.2.24), the definition of a~ is well posed. We gather now the important properties which the function ac enjoys. THEOREM 3.2.10: Let F: T I , ~ --~ R + be a strongly pseudoconvex Finsler metric on a simply connected complex manifold M of dimension n satisfying (3.2.15)(3.2.18), and let p E M. Then the function ac is an exhaustion of M with the following properties: (i) if p is the distance from p relative to the metric F, then a - 2 = (tanh p)2 and cr0 ~ p2; (ii) G = F 2 = a c o E c ; (iii) ac E C ~ A C o o ( M \ {p}); (iv) U M is the blow-up at p, then o e Coo(M); (v) ddCac > 0 on U \ {p}; (vi) dd c log ac >_ 0 on M \ {p); (vii) (dd~logac)" = 0 on M \ {p); (viii) log at(z) = log [[z[[2 + O ( 1 ) with respect to any coordinate system centered in p. In particular M is a Stein manifold. Proof: Some of the properties are immediate from the definition of ctc. In fact the statement that ~7c is an exhaustion function, (i), (ii) and (iii) are direct consequence of the definition of ac and of (3.2.25), (3.2.26) and (3.2.27). For (iv), as we noticed above the map Hc is of class C ~ and hence ac o ~r E Coo (.AT/) iff p 0 Hc is of class C ~ and this is obvious from the definition. For (v) we must work more carefully. The proof follows closely the line of Theorems 2.2 and 2.3 of Semmes [S]. In fact our map Ec is essentially what Selnmes calls a Riemann map. We give the necessary details here. We start recalling some notations. If r > 0, we set Bp(r) = {q E U [ p(q) < r},

Sp(r) = {q E U I p(q) = r}

157 and

[~p(r) = {v e T~'~

] F(v) < r},

Sp(r) = {v 9 T~'~

I F(v) = r}.

Thanks to (ii), we know t h a t Ec sends Sp(r) onto Sp(r'), where r ' = r if c = 0, and r ' = atanh(r) if c = - 2 . For v 9 Sp(r), let q = E~(v) 9 Sp(r'); we denote by HvSp(r) the holomorphic tangent space to Sp(r) at v, and by HqSp(r') the holomorphic tangent space to Sp(r') at q. Inspired by [L] and [P3], we shall say that a holomorphic embedding ~ of the disk A (if c = - 2 ) or the plane C (if c = 0) in M is stationary if the Al'~ m a p ~ given by 1 extends to a holomorphic m a p defined on A (or C). In particular, then, setting p -- ~(0), the holomorphic tangent spaces to Sp(r) yield a holomorphic fiber bundle along the image of ~. Then the main point in proving that our Ec is a Riemarm map is contained in the following LEMMA 3.2.11: Let F: T I , ~ --+ R + be a strongly pseudoconvex Finder metric on a simply connected complex manifold M satisfying (3.2.15)--(3.2.18), and t7x p 6 M. Then: (i) every geodesic complex curve ~v is stationary; (ii) for every r > 0 and v0 6 Sp(r) the differentia/ dEc maps Hv0Sp(r ) into HqoSp(r'), where qo = Ec(vo).

Proof: (i) Recalling Theorem 3.2.10.(i), we immediately get = 2 ( ( 1 + (c/2)1r

The Gauss L e m m a 1.6.10, together with Corollary 2.6.5, says that

dp = Re< .H, TH)T, where T(q) is the unit tangent vector in q to the unique geodesic joining p to q (see section 1.6.3). Therefore 1 Op =

-~('H,TH)T,

(3.2.28)

8xld so

@~(() = (1

+

(C/2)I(12)(.H, TH>T =

(1

+

(c/2)](12)2(.H,~'v(()H)~,(() ,

(3.2.29)

where we used the fact that 1

~'~(() = 1 + (c/2)I(I2 T(~(()), which follows from Proposition 3.1.3.(iii).In particular, then, =

(3.2.30)

158 is well-defined. To prove that ~3v is holomorphic, it is enough to prove that the function ( ~ ~ . ( ( ) ( ~ ( ( ) ) is holomorphic for any holomorphic vector field ~ along ~ . But indeed, since O/0~ is just the tangent vector to the disk ~ . - - and thus it is T" by (3.2.29) we have -

-

o-~[4~(r

=

(1 + (c/2)I~ID~<~(O/~,~,'~(O'/<(<)

= (1 + (~/2)1~1~)~(~(0 ~, ~ (' ~ ) ~/)~'(
+ (~ + (~/2)M~)~'-~-~<,(O ", , = (1 + (c/2)1r

=

"

~ [Ar

H

!

, ~(~)

H

>~,(<)+ ( v ~ ( O / ~ ,

~ (, ~ )

)~,(<)

(I+(c/2)KI2)2[Ac(C)_Ar

, = o, <~(r n , ~ v,( r g )~(<)

where A~ is given by (3.1.4), and we used the holomorphicity of~ and Corollary 3.1.2. (ii) Since G is a defining function f o r / ~ ( r ) , it is clear that H,0S~(r ) = ker OGv0 = {V 9 l;, 01

.0

= o}

On the other hand, (3.2.28) says that HqS/r')

= {v e

T~'~

= 0}.

(3.2.31)

So we have to prove that if Y ~ Y,0 is such that (V,~),0 = 0 then d(Er H belongs to Hqo S~(#). Let v: ( - ~ , ~) ~ S~(r) be a smooth curve with v(0) = v0 and ~)(0) = V, and set et = ~,(,)/~ for t e ( - ~ , e ) . Since E~(v(t)) = e , ( r ) , it follows that

d(Z~),o(Y ) = ~r

; t=0

so if we define t

0

it will suffice to prove that ~(~) belongs to the holomorphic tangent space to Sp(r') at cpv0/r(~) for all 4, where r' = i
2,CC. ' ,

.r

= ~(0(1+6c/2)1~1D 2( m'0 / d O ~ , ~ 0' / ~ ( r

~ )~oo.(<) , = ~(r

by (3.2.31) and (3.2.30). Since ~ 0 / r is stationary, it follows that r1 is holomorphic, and then also g is holomorphic.

159 Since ~(O) = op, we can then write ~ ( ( ) ~- ( ~ l ( ( ) ~ ' v 0 / r ( ( )

"F ( g l ( ( ) ,

(3.2.33)

with rh and gl still holomorphic. The rest of the argument now is similar to the one described in Lemma 5.3 of [P3]. Since r = E c ( ( v ( t ) / r ) , from Theorem 3.2.10.(ii) it follows that adr = a ( O ' ( O / r ) = 1(12; (3.2.34) differentiating this identity with respect to t and computing for t = 0 we get 0 = do'c(~(()) = 2 Re[(~l((~)cOac(~'vo/r(())]"

(3.2.35)

Now setting t = 0 in (3.2.34) and differentiating with respect to ( we get

therefore (3.2.35) yields 21(I 2 R e , h ( ( ) -= 0. Being ~h holomorphic, it follows that 7/1 - ia for some a E R; if we show that a = 0 we are done, by (3.2.32). But indeed differentiating v(t)/r = r with respect to t (using (3.2.33), of course) and then evaluating at t = 0 we get

!vr = ( ( 0 ) =

=

(gl(0)).

Now, by assumption (gl(() H, T H ) T =_ 0; for ( = 0 then (3.2.30) yields

0 = (gl(~)H, TH)T '=O = (gl(O)H,X)vo/r'-~ X(gl(O)H,X)v o = X(tvo(gl(O)),t)v o. So

ia 0 = -(V,e)vo = (~'(O),~>vo = --(t,~)vo = ira, r

-

and we are done.

[]

Now we can resume the proof of Theorem 3.2.10. By Lemma 3.2.11.(ii), dEc maps at every point the kernel of cOG onto the kernel of cOac; therefore there exists a never zero function g such that (E~),cOa~ = gcOG. Now, for v E Sp let j~ denote the inclusion in T~,~ of the line (if c = 0) or the disk (if c = - 2 ) through the origin in the direction of v. By construction, E~ o j~ is holomorphic; hence, being ac o E c = G and so a~o ( E c o j ~ ) = G o jr, we have

( j , ) , O G = (j,),(Ec),Oac = (j~),(gOG) = (g o j , ) ( j , ) , O G and thus g o j , = 1. Since this is true for all v E Sp it follows that

( E~),Oac = OG.

160 This is a crucial observation since, as a formal consequence, it implies that ddCG = (i/27r)O-OG = (1/2rri)-OOG = (1/27ri)dOG = (1/27ri)d(Ec).Oac = (1/27ri)(Ec).dOac = (Ec).ddCac.

But then, since ddCG is a positive definite form, ddCac must be non degenerated. On the other hand from (3.2.25), (3.2.26) and Theorem 1.6.4.(iv) it follows that Ec is differentiable at the origin and dEc(O) = id. Thus in a neighborhood of p, with respect to a coordinate system centered at p, we have a t ( z ) = a ( z ) + o(IMI ~)

and hence ac is strictly plurisubharmonic nearby p. As M is connected, (v) follows. Now we shall prove (vi) and (vii). We set uc = log ac and we keep all the notations introduced in the proof of (v). We have i

(ac)2ddCuc = acdd%rc - dec A d%rc = a c ~ ( O O a c - O a c h Oat),

and hence, arguing as before, it follows that ( Ec).ddCuc = dd c log G.

As a consequence (Ec).(ddCuc) n = (dd c log G)n.

Since from the homogeneity properties of G follow at once that (ddClog G) n - 0 on T I , O M \ {p}, we have also (ddCuc) n ~ 0 on M \ {p} and hence (vii) holds. Let us prove (vi). Let q E M \ {p} be any point and suppose that q 9 Sv(r' ). Then 10 q = Ec(~vo) = ~v0(r for some ~ 9 C and v0 9 Sp. Let ~ 9 T~, M. Then for some A 9 C and w 9 H q S p ( r ' ) one has =

A

i qOvo(( )+ w

----

(dEc)r

(~Levo(VO)) + w.

On the other hand as dEc maps at every point the kernel of OG onto the kernel of 0ac we have w = (dEc)r for some W 9 Hr ) - - where r = r' if c = O, and r = tanh(r ~) if c = - 2 . Thus, as log G is plurisubharmonic, ddC~(~,~) = d d % ((dE~)~v0 (~,~o0(v0) + W),(dEc)r

+ W))

= (Ec).(ddCuc)(,ke;vo(VO) + W, A~i.0(v0) + W) = ddClogG(,k~r

+ W, A~r

+ W) > 0,

and hence also (vi) is proved. Finally, again using the fact that Ec is differentiable at the origin and dEc(O) = id, we see that in a neighborhood of p with respect to a coordinate system centered at p the function uc has the same singularity as log G in a neighborhood of 0, and hence (viii) is immediate. [] Thanks to this result we can introduce the machinery of the complex MongeAmpere equation in the study of complex Finsler manifolds with constant non positive holomorphic curvature. In fact Theorem 3.2.10 in the language of [Pa2] states

161 that under these assumptions the manifold M is a manifold of circular type, of finite radius if the holomorphic curvature is constant negative, of infinite radius if the holomorphic curvature vanishes identically. It is also interesting to note that, in the hyperbolic case, Theorem 3.2.10 gives precise conditions in terms of complex Finsler geometry for the existence of regular pluricomplex Green functions (cf. [D])

3.3. Characterization

o f C~ a n d o f c o n v e x c i r c u l a r d o m a i n s

3.3.1. Geometry of the Monge-Amp~re foliation The geometric theory of the complex homogeneous Monge-Amp@re equation gives techniques and criteria useful to characterize and classify the manifold M. In the remaining of this chapter we shall illustrate how this can be done, trying to give an exposition adapted to our setting and as self-contained as possible. We shall now recall some facts about the geometry of the Monge-Amp@re equation. It is worthwhile to do so in the general setting, although we shall limit ourselves to the notions which are relevant for our aims. Details and complete treatments are easily found in the literature. We refer for the basic properties for instance to [BK], [BB], [Su], [Pa2], [PW], [St], [Wo]. Let M be a complex manifold of dimension n and let p E M. Suppose that there exists an exhaustion function ~: M --* [0, R) - - here R may be a real number or +oo - - with the following properties: a E C~

VI C~176 \ {p});

(3.3.1)

if 7r: 2~/--* M is the blow-up at p then a o or E C~176

ddCa > 0

(3.3.2)

on M \ {p};

(3.3.3)

ddclog a _>0 on M \ {p};

(3.3.4)

(ddCloga) n = 0 on M \ {p};

(3.3.5)

log a(z) = log [[zll2 + O(1),

(3.3.6)

with respect to any coordinate system centered in p. Because of (3.3.4) and (3.3.5), it follows that associated to a there is a rank 1 distribution in TI,~ \ {p}) defined by the eigenspaces relative to the zero eigenvalue of dd ~log a. Formally this distribution is defined by

= {v E T a ' ~

{p})ldd~logo(v,w) = 0 for all w E T I ' ~

\ {p})}.

(3.3.7)

Since dd ~log a is a closed form, the distribution E is integrable. Furthermore, as ddCloga is a (1, 1)-form, the distribution E is invariant under the almost complex structure and hence each leaf of the associated foliation is a complex submanifold of M \ {p} of dimension 1.

162 DEFINITION 3.3.1: This foliation in Riemann surfaces generated by E is called the M o n g e - A m p b r e foliation associated to a.

By definition the leaves of the Monge-Amp~re foliation are the maximal submanifolds of dimension one along which log a is harmonic. We shall use later this observation. It is useful to introduce a further notion. Let h be the Ks metric - - this time is a usual K ~ l e r Hermitian metric - - whose fundamental form is ddCa. A vector field f of type (1,0) called the c o m p l e x gradient of a is defined as the dual with respect to the metric h of the form 0r i.e., as the unique vector field f such that for all vector field ~ of type (1, 0) h ( f , ~) = Oa(~).

In local coordinates then we have the following expression for f in terms of a: f = a~ a ~' oOzu.

(3.3.8)

It is worthwhile to list the properties of f. Before doing so we remark that as (a)2dd c log a = addC a - da A dCa,

(3.3.9)

taking the n-th exterior product and using (3.3.5), we have a(ddCa) '* = n d a A d~a A (ddCa) n-1

which in local coordinates is equivalent to a = ao a ~ a u,

(3.3.10)

or in terms of the complex gradient f a = f(a) = h(f, f).

(3.3.11)

Using again (3.3.9), we have (a)2dd c log a ( f , f ) = addCa(f, f ) - d a ( f ) A dCa(f) = a 2 - a 2 = O.

Since ddCloga > 0 it follows that f is a global section of E. Since from (3.3.11) it follows that f never vanishes on M \ { p } , then E is a trivial subbundle of T l , ~ spanned by f . In particular it is possible to express the properties of the distribution E and of the associated Monge-Amp~re foliation in terms of f. DEFINITION 3.3.2: The Monge-Amp6re foliation is holomorphic iff for any point q E M \ { p } there exist an open neighborhood U of q, some r > 0 and a biholomorphic map ~ : U ~ A r x A~ -1 such that if L is a leaf of the foliation with L N U ~ q~, then ~ ( L fl U) = Ar x {y} for some y E A~ -1 and conversely if y E An "l there exists a leaf L of the foliation such that ff~(L N U) = Ar x {y}.

163 This is the case, as it is easy to see, iff the distribution associated to the foliation is a holomorphic subbundle of TI,~ \ {p}). Thus, because of the considerations made above, we get the following obvious, but important consequence: PROPOSITION 3.3.1: The Monge-Amp~re foliation associated to a is holomorphic iff the distribution ~, is a holomorphic subbundle of Tl,~ M \ {p}) iff the complex gradient vector field f is a holomorphic section of T l , ~ \ {p}). We notice that examples show that very seldom f is a holomorphic section of TI,~ and hence the Monge-Ampbre foliation associated to a is in general not holomorphic. Nevertheless the vector field f , as a consequence of (3.3.11) is always holomorphic when restricted to a leaf of Monge-Ampbre foliation. As we shall not need these considerations, we refer the interested reader to the afore mentioned references. On the other hand, for our applications we shall need some handy geometrical criterion to establish when the Monge-Amp~re foliation or - - as it is the same - - the vector field f is holomorphic. There are in the literature two main results in this regards. We end these general remarks on the geometry arising from the Monge-Amp~re equation quoting them. The first and most general one is related to the Pdcci curvature of the leaves of the Monge-Amp~re foliation ([BB], [St]): THEOREM 3.3.2: Let S be the Pdcci form of the metric h associated to the form ddCa. Then i S(f, f ) -- ~-~nOfHh (3.3.12) where I1" lib is the metric induced by h on (T~ \ {p}) | T I ' ~ \ {p}). In particular the Monge-Amp~re foliation is holomorphic iff all the leaves are Ricci 1tat with respect to the metric h. As a very clever and deep application of Theorem 3.3.2, Burns [Bu] proved the following THEOREM 3.3.3: /f a// the leaves of the Monge-Amp~re foliation associated to a are parabolic, i.e., are uniformized by C, then the foliation is holomorphic. We end this subsection by illustrating how the general theory of Monge-Amp~re foliations fits in our context of complex Finsler manifolds of constant nonpositive holomorphic curvature. To start we have the following

164

PROPOSITION 3.3.4: Let F: T l'~ ~ R + be a strongly pseudoconvex Finsler metric on a simply connected complex manifold M of dimension n satisfying (3.2.15)(3.2.18), and let p E M so that the function c% has all the properties listed in Theorem 3.2.10. Then the leaves of the Monge-Amp~re foliation associated to ac are exactly the images of the c-geodesic complex curves through p (with the point p removed). Furthermore if He is the map de6ned in (3.2.21) and f is the complex gradient vector field of a, then f ( H c ( ( , v)) = ~H'c((, v) = (~'v((),

(3.3.13)

where the prime denotes again differentiation with respect to r Proof: In this proof we set a = ac. By definition a(H~((,v)) --I([ 2.

(3.3.14)

Hence log a is harmonic along the image of ~v and this is enough to prove the first part of the claim. More formally, from (3.3.14) follows that a(Hc(r v))a~,~ (Hc(~', v))H~'((, v ) H ~ ( (, v)

- a~,(H~((, v))g'~'((, v)a~(Hc((, v))H'~(r v) -- 0 which is equivalent to dd c log a(gc((, v)) (H:((, v), H~(~', v)) = 0.

This, together with the fact that dd ~ log a _> 0, implies that ddCloga(g~((,v))(H'~((,v),~) = 0

(3.3.15)

for all vector field ~ of type (1,0). Thus H~e((,v) is tangent to the foliation for every v and hence the claim. For the rest of the statement we proceed as follows. From (3.3.14) it follows that a,(H~(r

= ~.

Since a 2 dd ~log a = a dd~cr - da A dr

from (3.3.15) we get that a ( g c ( r v))a,~ (H~((, v ) ) g : ' ( ( , v) = a,, (He((, v))g'~'(~", v)a~ (Hc((, v))

and hence [C[2a~w(He(C, v))H:"((, v) = ~a~(gc(r v)). We can conclude that r

v))gtc'((, v) = a~(g~(r v)).

This, together with the expression in local coordinates (3.3.8) of f, implies f ' ( H c ( ( , v)) = a~ (H~(r v))a ~" (H~((, v)) = ~g'c'(( , v)

165 []

and the proof is complete. The last result that we need is the following

THEOREM 3.3.5: Let F: TI'~ M ~ R + be a strongly pseudoconvex Finsler metric on a simply connoted complex manifold M of dimension n satisfying (3.2.15)-(3.2.18) and let p E M. Then the map Ec relative to the point p is biholomorphic iff the Monge-AmI~re foliation associated to ac is holomorphic.

Proof: If Er is biholomorphic, as the leaves of the Monge-Amp~re lobation associated to a are images via Ec of the intersections of the domain of Ec and the complex lines through the origin, in one direction the result is obvious. Since we have shown in Theorem 3.2.10 that M is a Stein manifold, because of Proposition 3.2.9.(iii) to prove the other implication it is enough to show that Er is of class C ~ in a neighborhood of the origin. In fact we shall show that Ec is holomorphic in a neighborhood of t h e origin. Our assumption is equivalent to saying that the complex gradient f is holomorphic on U \ {p} for some open set U containing p. Defining f(p) = 0 we extend holomorphically f to all of U. As we have seen that f(Hc(~,v)) = ~H~((,v) = (T~(~) and with respect to any system of holomorphic coordinates ~ ( ~ ) = ~v + O([~[2), it follows that dfp = id. We conclude that in a coordinate neighborhood centered at p, which we denote again by U, we have f = id +Q.

(3.3.16)

Let us denote by D a small ball centered at the origin in the domain of definition of Ec, and define for some r > 0 a map A: A~ x D ~ U by

A ( ( , v ) = Ec(@) = Hc

(v)r

.

Thus in particular f ( A ( ~ , v ) ) = ffA'(ff, v). For every j >_ 0 there exists Aj: D ~ U so that A(r v) = ~ A~(v)r j>l We can conclude that

Q(A(r

= iA'(r

-A(r

= ~--~(j - 1)A3(v)( j. j~2

(3.3.17)

Define P ( ( , v ) = ~ j > l Aj(v)r j-l" If Q = E k > 2 Qk is the development of Q in homogeneous vector polynomials, then for some vector polynomials Qkr we have

Qk(P(r v)) = ~ Ok~(Al(v) . . . . , Ar+~(v))r ~ r>O so that

Q(A(,,?))) = E k~2

Qk(P(~'v))= E ( E Qkr(Al(~))""'Ar~l(V))l~j" j>2 \/=k+r

]

166 Comparing the two expressions of Q(A((, v)), we get (j-1)Aj(v)=

E

Qkr(Al(v),...,Ar+l(v)).

j=k+r

Since k _> 2 we have r + 1 <_ j - 1, and there exist vector polynomials Rj so that for all j > 2 (j - 1)Aj(v) = R j ( A I ( V ) , . . . , Ar+l(V)). As A1 = dfp = id is holomorphic, by induction, it follows that A j is holomorphic for all j. For e > 0 small enough 7]j>l d j ( v ) ( j converges uniformly for [[vii < 2~ and [([ < 1/2. Then we can conclude R a t , if [[v[[ < 2e, Ec(v) = A(1/2, 2v) = E

Aj(v)(1/2)J

converges uniformly and hence, as claimed, Ec is holomorphic in a neighborhood of the origin. []

3.3.2.

A characterization

of C n

In this and the next subsections we shall apply the results of the previous section to characterize special complex Finsler manifolds. We start with the case of manifolds with vanishing holomorphic curvature. It is easy to provide examples of manifolds satisfying the hypotheses of Theorem 3.2.10 with c = 0, just by considering complex Minkowski spaces. It should not be a surprise that the existence of such metrics is a distinct feature of C n. In fact our result is the following: THEOREM 3.3.6: Let M be a simply connected complex manifold of dimension n and let us suppose that on M is defined a strongly pseudoconvex Finsler metric F such that F is complete Kiihler, (3.3.18) K F -- 0

(3.3.19)

<3HO(H,x,-K),x) = 0 for a/l H, K E ~ ;

(3.3.20)

F is strongly convex.

(3.3.21)

Then/'or every p E M the map E0:C n ~_ T~'1 0 M --* M is biholomorphic. Proof: At this point this result follows immediately from Theorem 3.2.10 and Theorem 3.3.5 since in this case the leaves of the Monge-Amp~re foliation associated to a0 axe, by construction, all biholomorphic to C* and hence one may apply Theorem 3.3.3. []

It should be remarked that Theorem 3.3.6 is satisfactory from the point of view of geometric function theory but leaves open a problem in Finsler geometry; in fact it is not yet a characterization of complex Minkowski metrics, as it should be expected.

167 3.3.3.

A characterization

of convex

circular domains

Let us turn to manifolds of negative constant holomorphic curvature. In this case most of the time the map E - 2 is not biholomophic, as the the examples of strongly convex domain in C" with the Kobayashi metric show. In fact most of them have trivial automorphisms group, whereas a domain biholomorphic to a circular domain has a rich automorphism group. Thus one must expect some additional geometric assumption in order to get the holomorphicity of E - 2 . Some criteria are available in the literature (cf. [LPW]) and they may be applied in this context. For instance we may prove: THEOREM 3.3.7: Let M be a simply connected complex manifold o[ dimension n

and let us suppose that on M is defined a strongly pseudoconvex complex Finsler metric F such that F is complete Kiihler, (3.3.22) KF -- --4

(3.3.23)

(OHm(H, x,-K), XI = 0 [or a/1 H, g E 7"/;

(3.3.24)

F is strongly convex.

(3.3.25)

Then F is the Kobayashi metric o[ M. Furthermore assume that there exists a point p E M such that for some r0 > rl > 0 there exists a biholomorphism @: Bp(ro) ~ Bp(rl) fixing p. Then the map E-2 is biholomorphic and hence M s biholomorphic to a smooth strongly convex circular domain in C n. More precisely, M is biholomorphic to the indicatrix Ip(M) of the Kobayashi metric at p. Proof: According to Theorem 3.3.5 it is enough to show that the Monge-Arnp~re foliation associated to a = a - 2 is holomorphic in a neighborhood of p or, that is the same, if f# is the complex gradient of a, then f# is holomorphic on U \ {p} for some open set U containing p. If we denote by S# the Ricci form of the metric ddCa, we must show that for some neighborhood U of p we have S#(f#, f~) = 0 on U \ {p}. Iterating the map g2, we may assume that both Bp(ro) and Bp(rl) are contained in any given neighborhood of the point p. At p, as in any other point, the indicatrix of F is strongly convex. If p: C n = "" T~, 1 0M ~ R is defined by ~(v) = G(p; v), then # is a strictly convex function which - - we recall it here for further reference d(Ec)p = id, we have that in a small enough coordinate neighborhood centered at p,

- - satisfies (3.3.1), (3.3.2), (3.3.3), (3.3.4), (3.3.5), (3.3.6) on Ip. Since

=,

+ o(1),

and hence in particular it follows that, with respect to the given holomorphic coordinates, a is strongly convex in a neighborhood of p. As a consequence we may assume that both Bp(ro) and Bp(r~) are contained in a coordinate neighborhood U of p and are biholomorphic to a strongly convex domain in C '~. With respect to the

168 given system of coordinates, if # is as above, using the homogeneity of # and the fact that # = a o E we have that for all fixed z E U \ {p} and small It] r 0, we have

a~(tz) = I~a~(tz) + o(1) = tt~$(z) + o(1), and hence log det(a=~(tz)) = log d e t ( # ~ ( z ) ) + o(1). Differentiating again O2

O2

t2~(logdet(a~$(tz)))-

0z~026

(log detI.o Iz)/) + or

so the Ricci forms S~ and S~. are related by

t2S~(tz) = S,(z) + o(1). Now, the complex gradient f~ of/~ is given by

OzC~ Therefore, again using the fact that # = a o E

fa(tz) = s

+ o([tl) = tf.(z) + o([t[),

and thus

S.(f~, f~)( tz) = Si,(f~, , f.)(z) + o(1). Since the Monge-Amp~re foliation induced on Ip(M) by # is holomorphic, it follows that sAY.,/.)(z)

= o

and hence

S,(f~, f,)(z) = o(1). (3.3.26) Now we shall use the remark that Bp(ro) and Bp(rl) may be assumed to be biholomorphic to strongly convex domains, and so we can apply the very special properties that the Kobayashi metric and distance enjoy in this case. For j = 0, 1 let us denote by pj the Kobayashi distance of Bp(rj) from the point p. Then, as Bp(r/) is biholomorphic to a strongly convex domain, the function (tanhpi)2: Bp(rj) ~ [0,1) is the unique exhaustion of Bp(rj) which satisfies (3.3.1), (3.3.2), (3.3.3), (3.3.4), (3.3.5), (3.3.6) on Bp(rj) (cf. [L], [D]). Since also the exhaustion defined by 1 (rj) 2 a: Bp(ri) ~ [0, 1) satisfies (3.3.1), (3.3.2), (3.3.3), (3.3.4), (3.3.5), (3.3.6) on Bp(rj), then necessarily it must be 1 (ri) 2 ~ = ( t a n h p~)~

and hence, as k~ being a biholomorphic map is an isometry for the Kobayashi dis-

tances, 1

1

(rx)~ o ~ = ~ .

(3.3.27)

169 Let A = r l / r 0 9 (0, 1), and define 7" = A-2a. Then evidently 7"E = A-2o'~,

ra~ -- A-2o'a~,

r aft --- A2o"af.

(3.3.28)

Because of (3.3.27) it follows that a o r = A-2a = T. Then 9 is an isometry for the Ks metrics defined by ddCa and ddCT. Furthermore, it is obvious that 9 , ( f ~ ) = f~. On the other hand, (3.3.28) implies that

and

Sr = dd c log det(v~$) = dd c log det(aa$) -- Sa. Thus

Sa(f~,,f--~)(z) = S~(p,(f~,), p,(-f~))(p(z)) = S~(f~,'-F)(p(z)) = Sa(fa,-/'~a)(p(z)), and hence for all positive integer k we have

S , ( f , , f-~)(z) = S,(f~,-f'~) (pk (z)).

(3.3.29)

Since g2(Bp(ro)) is relatively compact in Bp(rj), a theorem due to Herv4 (see [H, p. 83]) implies that the sequence of iterates {g2 k} is converging, uniformly on compact subsets, to p; therefore (3.3.26) and (3.3.29) yield Sa(f~,,-f'~)(z) - 0 on Bp(ro) \ {p}. This, as observed before, complete our proof. [] We end by characterizing the most special circular domain: the unit ball. Since we do not want to make more restrictive the assumptions on the Finsler metric - it is easy to fall into the case of Hermitian metrics and hence recover well known results! - - we shall use the very special properties of the automorphisms group of the ball as it was done for instance in [Pal]. We have the following COROLLARY 3.3.8: Let M be a simply connected complex manifold of dimension n with a strongly pseudoconvex Finsler metr/c F. Then the following statements are equivalent:

(i) M is biholomorphic to the unit ball in Cn; (ii) the assumptions of Theorem 3.3.7 hold for two distinct points p, q E M ; (iii) the assumptions of Theorem 3.3.7 hold for a point p E M which is not kept fixed by Aut(M). Proof: It is clear that (i) ==* (ii). Let us prove that (ii) ==#, (iii). By Theorem 3.3.7, there exist two bounded strongly convex circular domains D1, D2 C C n and biholo= p r q = r In particular, D1 morphic maps C j : D j ---* M such that r and D2 are biholomorphic. By a result of [BKU], there exists a linear isomorphism L 9 GL(n, C) such that L(D1) = D2. Then r = r o L o r is an automorphism of M with r = q. Finally we show that (iii) ==~ (i). By Theorem 3.3.7, M is biholomorphic to a bounded strongly pseudoconvex circular domain D C C n via a biholomorphic

170 map sending the origin into p. Since the orbit of p under Aut(M) has at least two points, the origin in D is not a fixed point of Aut(D). Then it is known (again by [BKU]) that the orbit of the origin under Aut(D) is the intersection of D with a non-trivial complex subspace of C n. In particular, Aut(D) is not compact and thus, by B. Wong's theorem [Wl], D is biholomorphic to the ball in C n. []

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[Bu] [C] ICE] [Chl]

[Ch2] [D] [DoC] [F]

Tu]

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[K]

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Index

Curvature (continued): - horizontal holomorphic flag curvature, 106. negative curvature, 31. - operator, 22, 23, 31, 81, 92, 93. - positive curvature, 31. - symmetries, 105. tensor, 22, 81. Double, 42. Exponential map, 42, 49, 53. Exterior differential, 20. E-diffeomorphism, 44. Finsler manifold, 24. - complete Finsler manifold, 50. Finsler metric, 24. coming from a Riemannian one, 24-26, 29. coming from a Hermitian one, 84. - complete Finsler metric, 50. - complex Finsler metric, 84. - distance induced by, 41. - strongly convex Finsler metric, -

Ahlfors' lemma, 143. Bianchi identities, 23, 83, 92. Bonnet's theorem, 61. Canonical form, 21, 80. Carath~odory metric, 84. Cartan-Hadamard theorem, 60, 147. Change of coordinates, 3-5, 67, 75, 78. Christoffel symbols, 12, 13, 16, 74, 78. Complex gradient, 162. Conjugate point, 57. Connection, 7. Cartan connection, 27, 28, 111. - Chern-Finsler connection, 87, 88, 111. complexified non-linear connection, 71. - complexified vertical connection, 76. - complex non-linear connection, 72. - complex vertical connection, 76. - good complex vertical connection, 77. - good connection, 16, 18. - homogeneous complex non-linear connection, 73. - homogeneous connection, 10, 11, 29. linear connection, 7, 18. non-linear connection, 7-9, 12, 18. on tensor spaces, 19. - vertical connection, 15. Covariant derivative, 7, 21, 23. Covariant differential, 7. Curvature, 22. Gaussian curvature, 109, 143. - holomorphic curvature, 108-110, 144. horizontal flag curvature, 30. -

-

-

-

-

-

-

-

-

-

111.

- strongly pseudoconvex Finsler metric, 85. - upper semicontinuous complex Finsler metric, 144. First variation formula, 36, 99. Fubini-Study metric, 128. Fundamental form, 95. Gauss' lemma, 46, 50. Generalized Laplacian, 143. Geodesic, 14, 33, 37, 41.48. c-geodesic complex curve, 129. complex geodesic, 153. equation of, 37, 101. - geodesic complex curve, 140, 141. - infinitesimal c-geodesic complex curve, 129. - normal geodesic, 55. radial geodesic, 41. segment of c-geodesic complex curve, 129. Hermitian product, 113. Hopf-Rinow theorem, 49. -

-

180 Horizontal bundle, 6, 8, 9, 12, 13, 16. - complex horizontal bundle, 72. - homogeneous complex horizontal bundle, 73, 75. - homogeneous horizontal bundle, 10, 11. Horizontal form, 93. Horizontal lift, 18, 91. Horizontal map, 6, 8, 9. - complex horizontal map, 72. - homogeneous complex horizontal map, 74. - homogeneous horizontal map, 11. Horizontal part of a form, 93. Horizontal vector field, 14. Indicatrix, 24, 85. Injectivity radius, 45. Jacobi field, 52, 53, 57. - proper Jacobi field, 55. Kobayashi metric, 84. Kglller Finsler metric, 94, 96. - strongly Kgl~_ler Finsler metric, 94, 95. - weakly KSl'tler Finsler metric, 94, 96, 131. Length, 32. Minkowski space, 24. complex Minkowski space, 84. Mixed part of a form, 94. Monge-Aml~re exhaustion, 156. Monge-Aml~re foliation, 156, 162. Morse index form, 56, 153. Parallel transport, 40. Parallel vector field, 14, 40. Poincar~ metric, 128. Pseudohermitian metric, 142. Radial horizontal vector field, 12, 75. - complex radial horizontal vector field, 75. Radial vertical vector field, 6, 69. - complex radial vertical vector field, 69. Regular curve, 32. Riemann map, 156. Scale, 142. -

Second variation formula, 38, 56, 102. Stationary map, 157. Symmetric product, 102. Torsion, 21, 30, 92. (1,1)-torsion, 80, 93. (2,0)-torsion, 80. - horizontal (1,1)-torsion, 102. Transversal vector, 34, 97. Variation, 33. - fixed variation, 33. geodesic variation, 51, 52. regular variation, 33. Vertical bundle, 5, 69. - complexified vertical bundle, 68. - complex vertical bundle, 69. - real vertical bundle. 68. Vertical lift, 18, 91. Zero section, 3, 6. -

-

-

-

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