现代黎曼几何简明教程

当代数学讲座丛书 Lectures in Contemporary Mathematics 1 现代黎曼几何简明教程曹建国王友德著北京目 vi· · 内容简介本书是一本现代 Riemann(黎曼...

Author: 曹建国 | 王友德

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当代数学讲座丛书 Lectures in Contemporary Mathematics

1

现代黎曼几何简明教程曹建国

王友德著

北京

目

vi·

·

内

容

简

介

本书是一本现代 Riemann(黎曼)几何的简明教材, 共分两部分. 第一部分为一至四章, 介绍 Riemann 几何的基础知识, 内容包括多种形式的比较定理、 Calabi-Yau 体积估计、郑绍远最大直径定理和 Cheeger 有限定理的讨论等. 内容新颖且简单明了, 尤其是比较定理的证明采用常微不等式的方法, 不同于经典的变分方法, 新的证明和讨论通俗易懂、简易明畅. 本书的第二部分包括第五、六和七章, 分别讨论测地流、负曲率流形和正曲率流形这三大现代 Riemann 几何研究领域的最新成果, 许多新的研究结果如 Cheeger-Gromoll 灵魂猜想的新证明都是第一次在中外几何教科书中出现. 本书可供从事 Riemann 几何相关领域研究的学者参考, 也可作为高年级本科生和研究生的教材和参考书. 图书在版编目(CIP)数据现代黎曼几何简明教程/曹建国, 王友德著. —北京:科学出版社, 2006 (当代数学讲座丛书; 1) ISBN 7-03-016435-0 Ⅰ.现… Ⅱ.① 曹… ② 王…

Ⅲ.黎曼几何-教材 Ⅳ.O186.12

中国版本图书馆 CIP 数据核字(2005)第 130425 号责任编辑: 吕

虹／责任校对: 朱光光

责任印制: 钱玉芬／封面设计: 王

浩

出版北京东黄城根北街 16 号邮政编码: 100717 http://www.sciencep.com

印刷科学出版社发行

各地新华书店经销

* 2006 年 1 月第一版 2006 年 1 月第一次印刷印数: 1—3 500

开本: B5(720×1000) 印张: 10 字数: 180 000

定价: 25.00 元 (如有印装质量问题, 我社负责调换〈环伟〉)

录

前

iv·

·

言

《当代数学讲座丛书》序近二十年来, 中国数学有了引人注目的发展, 国际学术交流活动也大大增加. 许多大学和研究所都举办了不同层次的现代数学系列讲座或暑期学校 (例如自 1998 年开始举办的北京大学特别数学讲座), 聘请国内外著名数学家讲授课程或研究成果. 这为我国数学工作者和研究生提供了学习数学各学科的基础知识和接触前沿研究问题的极好机会, 大大促进了我国新一代青年数学家的成长. 《当代数学讲座丛书》是在这些学术交流活动以及系列讲座的基础上形成的, 其宗旨是面向大学数学及其应用专业的高年级学生、研究生以及青年数学工作者, 为他们提供高水平的专门教材. 本丛书通过整理优秀系列讲座、暑期学校中的精品课程以及其他各种形式的讲义, 着重介绍国际上前沿数学的研究领域, 使相关学生与年轻数学的工作者能在较短的时间内对数学各个领域的发展有较为深刻的了解, 尽快地掌握这些领域的基础知识和重大研究问题. 我国数学家的共同心愿是, 使中国在不久的将来成为数学强国. 为此必须造就越来越多的立足国内, 并具有国际影响的青年数学家. 我们相信此丛书的出版将为实现这一目标做出项献.

田

刚

2005 年 10 月 16 日

目

vi·

·

《当代数学讲座丛书》编委会名单主

编：田

刚

编

委：（以姓氏笔画为序）王立河

许进超

阮勇斌

林晓松

夏志宏

鄂维南

陈秀雄

录

Æ

Æ Æ Æ $#% ! !""# ', &'("$ )% *&+ () - *.*(/+*,- 0 1 ./ 0 2 1 3 ! 0452"6"3' 7 89# :4$5 6; <=0>0?@'+7 08 $ 5 9: ; % <= & Æ A B C B D>E0? @' # AB 00? CDEFGFG )H KL&M ( 'N *HC ' I 1 JI J ) 67 OKC LM%N *+,

6; P-; Q O0PQRSR?. 8 /TUS$5<=0VTUVQRS?W ? . : 0 O0WX PXYYZ 1 [ Z [\YW0\]^ YW 2 _ ]? . :^ 3 *_``

6;aP ab: 8cbcJ 3dAB# e f ;d & ,(: '8 4 ef G ;g & 8gef jjkkb Æ h 2 F Yh6 *ii 5l blG [ ; <=0mm!]noYn a po> YW YW !a 6 q ] 0h r s omm!at 78 p 0h 52 " VT uv 9 YW 7h qrw : ]stxu 6qr^ 3 :yz v 78 p {w ; x' 6B < :y|Sz { I] = m Æ LM 6'Ws| } >?@ ABC ~ DE}~ ,

30

,

.

,

,

.

Yang-Mills

Ricci

Yau

, Calabi-

.

Riemann(

)

.

Gromov

,

. Gromov

Gromov-Witten . Gromov

.

,

1997

.

,

Riemann

Gromov

,

,

,

Riemann

.

Riemann

,

,

,

Riemann

.

,

,

.

1998

.

,

Riemann

,

.

(Walter Wei)

,

.

,

.

10

.

,

2003

Cheeger-Gromoll

.

,

,

Calabi

Cheeger

.

.

,

.

,

,

.

Riemann

.

2004

12

前

iv·

·

目第一部分

言

录

基础知识和基本定理

第一章 Riemann 流形 ...................................................................................... 3 §1.1 流形、切空间和切丛 ...................................................................... .. 3 §1.2 Riemann 联络和仿射联络.................................................................. 6 §1.3 向量场的平行移动和测地线............................................................ 14 §1.4 第一变分公式 ................................................................................. 18 §1.5 指数映照, 完备性和 Hopf-Rinow 定理............................................ 20 习题一 ..................................................................................................... 26 第二章曲率和比较定理 ................................................................................ 31 §2.1 曲率张量、截面曲率和 Ricci 曲率 .................................................. 31 §2.2 测地线族的变分向量场 ................................................................... 32 §2.3 Jacobi 方程和 Riccati 方程 ............................................................. 34 §2.4 Gromov 引理和经典比较定理的新证明........................................... 35 §2.5 Gromov-Bishop 比较定理 ............................................................... 40 习题二 ..................................................................................................... 44 第三章共轭点和最大直径定理 ..................................................................... 47 §3.1 共轭点、第二变分公式 ................................................................... 47 §3.2 Ricci 曲率和 Myers 直径定理 ......................................................... 53 §3.3 郑绍远最大直径定理的简单证明 .................................................... 54 §3.4 Calabi-Yau 体积线性估计............................................................... 55 习题三 ..................................................................................................... 58 第四章单一半径和有限定理 ......................................................................... 62 §4.1 割点、割迹和单一半径 ................................................................... 62 §4.2 Cheeger 的单一半径估计 ................................................................ 66 §4.3 重心和流形中的离散图 ................................................................... 69 §4.4 Cheeger 有限定理 ........................................................................... 73 习题四 ..................................................................................................... 75

目

vi·

·

第二部分

录

现代理论选讲

第五章 Riemann 流形上的测地流 ................................................................ 79 §5.1 测地流和切丛上的辛结构 .............................................................. 79 §5.2 闭测地线 ....................................................................................... 85 §5.3 无共轭点的流形和 Hopf 猜测 ........................................................ 90 习题五(含未解决的问题)......................................................................... 93 第六章具有非正曲率的流形 ........................................................................ 96 §6.1 测地线、非正曲率和负曲率........................................................... 96 §6.2 基本群、Preissmann 和丘成桐定理 ............................................... 104 §6.3 Gromoll-Wolf 和 Lawson-Yau 分解定理 ......................................... 109 §6.4 Eberlein 正规交换子群分解定理 .................................................... 111 §6.5 Gromov 图形流形和最小体积流形 ................................................. 116 §6.6 测地流的刚性定理和其他刚性定理简介 ......................................... 119 习题六(含未解决的问题)......................................................................... 121 第七章具有非负曲率的流形 ........................................................................ 123 §7.1 具有非负曲率流形的例子 .............................................................. 123 §7.2 基本群和陈省身猜测的反例........................................................... 128 §7.3 Cheeger-Gromoll 理论和开流形 .................................................... 129 §7.4 Cheeger-Gromoll 灵魂猜想的证明 ................................................ 134 习题七(含未解决的问题)......................................................................... 143 参考文献 ........................................................................................................ 145

*

*

*

《当代数学讲座丛书》已出版书目 ......................................................................... 148

Æ

,

,

Cheeger

Calabi-Yau

.

,

,

,

.

,

.

Æ Æ Æ ! Riemann

§1.1

.

.

2

.

,

∇f (x) = 0 2

x∈f

2

x +y ,

f

1

−1

−1

(b) , f (b) 2 (1) = (x, y) | x + y 2 = 1 1).

x ∈ f −1 (b) , f −1 (b) f −1 (b) = x ∈ Rn+1 | f (x) = b ,

.

f (x1 , x2 , x3 ) =

R3

+

x22

− x3 ,

f

−1

1

R

.

f (x, y) =

n+1

.

→R

,

Rn+1

.

Rn+1

n

(0) = (x1 , x2 , x3 ) | x3 = x21 + x22

R3

.

,

2

f R

,

x21

f R2 → R

.

−1

(

∇f (x) = 0

1

.

!

!" # " "!"" $ ! Æ # Æ% # $! 1 1

Mn

1.1

M

n

Hausdorﬀ

p

Mn

Up

n

,

.

Up

p,

R

n

,

.

.

.

Mn

H. Whitney

,

.

&Æ%

!# & ''"#() % ($ $ Æ &$ !" Æ % "$ %& Æ *+ Æ& & ' Æ )' " !) *' " "Æ(" !"" + ","! Æ * *( $ Æ ( (( (( ) # *. ," *, 01,() $ +*, ,/ Æ ( Æ) " $ 2) Æ

+3 ) *(( Æ , ,/ ($ $ ) (-$( $ %*4. +") *" *+%& , & " ) "$ ,4. ) * " * "/ ) * " -+3 ) * " / *, ,/ 5 ! " ($" 0 ! - Æ ' ) 6 ' +5,/

Æ. 7 / -

22 ·4·

Riemann

1.2

M n,

n

Φ M n → R2n+1 ,

Mn

n

R2n+1

M n → R2n+1 .

.

1.2,

Mn

.

M n → R2n+1

.

Mn

p,

n

,

Tp M .

Tp M

n

R

Ap

,

R

p

,

F U → M

n

→ R

F∗

O

. ∂(F1 , · · · , Fm ) = ∂(x1 , · · · , xn ) O m = 2n + 1.

,

F

p

n . p ∈ Mn , ∂(F 1 · · · , Fm ) F∗ = ∂(x1 , · · · , xn ) O O

.

Mn

.

O

Mn

Rm

F (O) = p

O

p

,

M n ⊂ Rm

F U →Mn

M

n

Mn

Ap (Rn ) = F∗ |O (Rn ).

1.3

F

W

F (O) = p.

Ap = F∗ |O Rn → Rm , Tp M n

→ R

m

.

2n+1

.

n

.

.

F

Ap R

m

Rn

U

.

.

Image(Ap )

p

2n+1

,

,

R2n+1

p

Tp M n = Image(F∗ |O ) = F∗ |O (Rn ). Tp M n

,

,

.

.

1.4

Tp M

M

n

R

m

(F, U )

U =F F∗ |O

−1

(W )

.

Image(F∗ |O ) ⊂ Image(G∗ |O ).

p

,

n.

F (U )

,

−1

V =G

G∗ |O

p

.

Mn

(G, V )

F (O) = G(O) = p. ∂(F1 , · · · , Fm ) ∂(G1 , · · · , Gm ) ∂(x1 , · · · , xn ) ∂(x1 , · · · , xn ) . W = F (U ) G(V ) Mn .

Mn

,

n

(F, U )

p

(F, U )

Mn

F (U )

G(V )

F (U ) = G(V ) (

(W ). ,

G(V )

F∗ |O (Rn )

.

G∗ |O (Rn )

2).

n

F |U = G ◦ (G−1 ◦ F ),

Image(G∗ |O ) ⊂ Image(F∗ |O ).

./12.// Æ/8/ 0 * 0 §1.1

,

(F, U )

(G, V )

·5·

Tp M n = Image(F∗ |O ) = Image(G∗ |O ) .

.

4. ) *"

2

Æ$ ()Æ& Æ * 1 "1 9

.

(i)

p = (1, 0)

M 1 = S 1 ={(x, y) | x2 + y 2 = 1}.

.

, Tp S 1

p

x

3 2 . + ,

(

3).

3 ( Æ&4 Æ $ + 1,2 &2) Æ&) *(( 2 ! (ii)

f R2 → R

3

.

2

M 2 = {(x, y, f (x, y)) | (x, y) ∈ R2 } ⊂ R3 .

M2

0, f (0, 0)).

3

R3

f

(

4).

p = (0,

&Æ%

·6·

Riemann

4

F R2 → M 2 ,

(x, y) → (x, y, f (x, y)),

0 3-2 : "$ Æ , ;<5$4

$$4 ⎞

⎛

F∗ |O = ⎝

1, 0, fx

⎠.

u = (1, 0, fx(0, 0))

0, 1, fy

F∗ |O (R2 )=Span{u, v},

Span{u, v}

u

v = (0, 1, fy (0, 0)).

v

.

,

1.5

M

n

Tp M 2 =

.

n

,

T Mn =

Tp M n .

603160 Æ )$ # *5 ) $ # *5 <5$4 ( & Æ () Æ% & #**'45 ) -6#*22 )$ p∈M n

§1.2 Riemann ,

.

.

M 1 = S 1 = {(x, y) | x2 + y 2 = 1}, σ [0, 2π] → S 1 , t → (cos t, sin t).

.

σ

.

X(t) = σ (t) = (− sin t, cos t)

.

73.7473 +3%= §1.2

Riemann

t, X(t) ∈ Tσ(t) S 1

(

·7·

5).

5

- # *. +3 )$ # *5 ) $#*5! .% $#*5 /7 80 0& >$ # *5 6, $6 , 7 $ # *5 ( $' )$ # *56 ,7 85$ 6 Æ9 1 & ) # *5 -Æ7 6 +3'$ .!), 98 Æ 24 % ) 9 +3 0# ::;8 Æ () < "##2 6 , 7 # Mn

1.6

n

,

X Mn → T Mn

X(p) ∈ Tp M n ,

M

n

Mn

X

,

Γ (T M n ) = X | X M n → T M n

.

X ∈ Γ (T M n )

,

,

.

X

.

Mn

.

X(t) = σ (t) = (− sin t, cos t),

σ(t) = (cos t, sin t).

X (t) = σ (t) = (− cos t, − sin t) R

,

, X(t)

X (t) ∈ Tσ(t) M .

m

n

M (n < m),

n

m

σ [a, b] → M → R , X(t) = σ (t),

X (t) = σ (t) ∈

Tσ(t) (M n ).

M n → Rn+k

,

Civita

.

Levi-

&Æ% =

() ;:' "# " +- > 1 #* " 2

> Æ &( ' ) 02? % - !/$* #*. Æ -9 # , - )/#*5 #3), ·8·

Riemann

M n → Rn+k

Rn+k

,

p

Tp Rn+k = Rn+k

p ∈ Mn

,

, Rn+k = Tp Rn+k

M n → Rn+k

Rn+k = Tp Rn+k = Tp M n ⊕ Np M n ,

Np M n = v ∈ Rn+k | v

Tp M n

M n → Rn+k

p

.

R

n+k

Tp M n

Np M n ,

Rn+k

Tp M n

,

)T Rn+k → Tp M n ,

(

v → vT ,

vT

v = vT + vN

R

Y

.

n+k

X

X

Y = (Y 1 , · · · , Y n+k ).

,

M n ⊂ Rn+k

1.7

M n ⊂ Rn+k

DX Y = (XY 1 , · · · , XY n+k )

Y,

n

,Y

,

X

Y

X

∇X Y = (DX Y )T ,

-Æ $* )#*5 )3), . ) ?:< ?: Æ () ;:'?: @%= (DX Y )T

DX Y

.

Γ (T M )

M n → Rn+k

M

Levi-Civita

∇

.

Riemann

∇

.

∇ Γ (T M ) × Γ (T M ) → Γ (T M ), (X, Y ) → ∇X Y,

∇f X Y = f ∇X Y,

#, 3- )?:;)

=4 =/= 56; A < 6* /=@ ∇X (f Y ) = (Xf )Y + f ∇X Y,

f ∈ C ∞ (M n )

, Xf = df (X)

Mn

f

, C ∞ (M n )

X

.

Riemann

Riemann

,

.

.

,

.

73.7473 )/$#*5 Æ0 Æ . B> )$4 )?(?: Æ& §1.2

Riemann

Mn

X, Y

·9·

,

[X, Y ]f = XY f − Y Xf,

f ∈ C ∞ (M n ).

M

n

[X, Y ] = XY − Y X

TM

X

Y

.

∇

,

)* #*. @ +3 2)

72 #*.C A 9 @ $ Æ, 6 0+3 τ∇ (X, Y ) = ∇X Y − ∇Y X − [X, Y ].

X

Y

τ∇ Γ (T M ) × Γ (T M ) → Γ (T M ). f ∈ C ∞ (M ), τ∇

,

τ∇ (f X, Y ) = f τ∇ (X, Y ) = τ∇ (X, f Y ).

, τ∇ (X, Y )|p (

X(p)

)

Y (p)

,

.

C ∞ (M n )

α(X1 , X2 , · · · , Xk )

,

α(f X1 , X2 , · · · , Xk ) = α(X1 , f X2 , · · · , Xk ) =···

= α(X1 , X2 , · · · , Xk−1 , f Xk )

?: A+:* +:* Æ Æ& = 0 . # # , Æ( 8 9 3 ), * # *5 B!' = f α(X1 , X2 , · · · , Xk )

f ∈ C ∞ (M n ) ∇

Xi ∈ T M

α

k

.

M n = Rn

n

.

∇X Y = DX Y

Y X .

∂ ∂ ∂ , ,···, Rn . f ∈ C ∞ (Rn ) , ∂x1 ∂x2 ∂xn ∂ ∂ ∂2f ∂2f , − = 0, f= ∂xi ∂xj ∂xj ∂xi ∂xi ∂xj ∂ ∂ , , = 0. ∂xi ∂xj ∂ ∂ ∂ ∂ ∂ ∂ τ∇ , −∇ ∂ − , =∇ ∂ = 0. ∂xi ∂x ∂xj ∂x ∂xi ∂xj ∂xi ∂xj j i

Æ

τ∇

.

.

,

B5< Æ 7

∇

τ∇ .

&Æ% - )6*., Æ 7 2 > Æ

) ) 6* Æ 4 0 ' . Æ Æ $ # $ Æ * B!' ($ C )7 D % · 10 ·

Riemann

Rn

∇=D

,

1.8 (Riemann

Tp M n

Mn

τ∇ ≡ 0.

,

)

M

n

, g(p) = ·, · p ,

Riemann

g = { ·, · p }p∈M n = {g(p)}p∈M n

(M n , g)

.

Riemann

M

n

→ R

Riemann

R

.

Rn+k

,

m

Riemann

m

.

R

.

·, · Rm .

ξ = (ξ1 , ξ2 , · · · , ξm )

p ∈ Mn

,

m

Rm

.

η = (η1 , η2 , · · · , ηm ),

ξ, η Rm = ξ1 η1 + ξ2 η2 + · · · + ξm ηm .

T p M n ⊂ T p Rm = Rm ,

·, · Rm

,

Tp M

g(p) = ·, · p = ·, · Rm Tp M n .

Æ )2"# 1, 6* ÆÆD0 $ - "# ?: $2' => #

) 6* # E"# $B

C @ EFD ; E?ÆF C F )( GG -@A "# 2H Æ ?: , ;+3 % = )B$#*5 .

Mn

g = { ·, · p }p∈M n n

n

(M , g) ⊂ (R , g0 )

R

M

m

Rm

Riemann

? J. Nash

)

Riemann

Riemann

Riemann

M

g0 = ·, · Rm .

,

1957

n

.

n

,

2

m 2n + n

g,

F M n → Rm ,

1

v ∈ Tp M n

F∗ vRm = v, v g2

(Rm , , Rm )

p ∈ Mn

,

Nash

X Y

.

,

.

,

Z

M

n

, (M n , g)

.

Riemann

J. Nash

.

(M n , g)

Riemann

1.9 (Nash

n

M n → Rm

1994

.

Riemann

, (M n , g) ⊂ (Rm , ·, · Rm ),

X Y, Z g = DX Y, Z Rm + Y, DX Z Rm = (DX Y )T , Z Rm + Y, (DX Z)T Rm = ∇X Y, Z g + Y, ∇X Y g .

73.7473 %&H "# H < ! < = . 1% += 6H< $4 )>?(?: +3 §1.2

· 11 ·

Riemann

Nash

,

Riemann

.

1.10 (Levi-Civita

M

n

Levi-Civita

TM

(M n , g)

)

n

Riemann

,

∇

X Y, Z g = ∇X Y, Z g + Y, ∇X Z g ,

(1.1)

- # )$Æ" # *5 ) * "

# ) * " ," $

! : 0 0 ,/ C,/ ># ?(?: # ) * " ) -9 .G %> 2 ) ?:?;+3*8 ?: # # %> %> +3% /*8 τ∇ (X, Y ) = ∇X Y − ∇Y X − [X, Y ] ≡ 0,

X Y

Mn

Z

.

n

p ∈ M ,

{(x1 , · · · , xn )}x∈U .

p

O. M n

p

∂ ∂ ∂ , ,···, ∂x1 ∂x2 ∂xn ∂ . ∂xn

n Tp M ∂ ∂ Tp M n ∼ , ,···, = TO (Rn ) = Span ∂x1 ∂x2

,

gij (x) =

(g ij )

g = {(gij (x))}x∈M n .

p

∂ ∂ , ∂xi ∂xj

, g x

(gij )

.

∇

,

(1.2)

Christoﬀell

.

,

{(x1 , x 2 , · · · , xn )} ∂ ∂ ,···, ∂x1 ∂xn

,

n

∇

∂ ∂xi

, Γkij (x)

∇

∂ ∂ = Γkij . ∂xj ∂xk k=1

∇

Christoﬀell

(1.1)∼(1.2)

.

Christoﬀell

k Γij

Γkij = Γkji ,

(1.3)

n

∂gjk = Γlij glk + Γlik gjl . ∂xi

@

7 # Æ(;H:*IA )Æ6 J2 B> - *8 #

∂ ∂ , ∂xi ∂xj

(1.4)

l=1

,

=0.

,

τ∇

∂ ∂ , ∂xi ∂xj

.

=0 n

0=∇

∂ ∂xi

∂ ∂ ∂ Γkij − Γkji −∇ ∂ = . ∂x j ∂xi ∂xj ∂xk k=1

,

· 12 ·

-

&Æ%

Æ72 # &

∂ ∂ ,···, ∂x1 ∂xn ∂ ∂ X= ,Y = ∂xi ∂xj

,

Z=

∂ ∂ gjk = ∂xi ∂xi = ∇

22 2 (1.4).

=

(1.2)

,

∂ ∂ , ∂xj ∂xk

∂ ∂xi

(1.3).

(1.1)

∂ ∂ , ∂xj ∂xk

+

∂ ∂ , ∇ ∂ ∂xi ∂x ∂xj k

n l Γij glk + Γlik gjl . l=1

(1.3)∼(1.4)

>2 n

Γkij

∂ ∂xk

Riemann

1 kl = g 2

∂gil ∂gjl ∂gij + − ∂xj ∂xi ∂xl

.

(1.5)

- 0 %>2 >I # #>?:+3 ÆÆ ?: # / 7 I Æ$4 7 I' ) # *4 A 6*J?(?:

)B4 l=1

Christoﬀell

(1.1)∼(1.2).

(1.5)

,

g,

.

Riemann

.

,

.

Mn

1.11

B #*.! #*+3= ) # *4 ) . J 0 C π

Rk −→ E −→ M n

Ex = π −1 (x)

k

,

E

n

σ M →E

E

Mn

.

π(σ(x)) = x,

σ

Mn

.

Γ (E) = σ | σ M n → E,

π(σ(x)) = x

x

.

∇ Γ (T M ) × Γ (E) → Γ (E),

+3

(X, σ) → ∇X σ,

∇f X σ = f ∇X σ,

(1.6)

∇X (f σ) = (Xf )σ + f ∇X σ,

(1.7)

73.7473 ?(?: & . J Æ # *4)?(?: >" Æ& $ ' # *4 ) # *4 0 #*5 #80 D., K> )?: $#*5 -(-9ÆJ2 # * * # ) # *4 5# D K 3 ( = =Æ!EF# # *4 ) ?(?:! G # *4 . KLG 5 ML EF A HN )**) B Æ ?(?:H & )/ ?(?: Æ/86/?(?:CD # *4 :*7 @ Æ0 87 > ) 2 72 ($L1( :* 0 #*4 )M ?(?: - +3 :* 8 §1.2

· 13 ·

Riemann

∇

E

f ∈ C ∞ (M n )

,

, σ ∈ Γ (E)

.

.

n

M → R

n+k

x ∈ M n,

,

Nx M n = v ∈ Rn+k | v ⊥ Tx M n .

NMn =

,

Nx M n

Mn

.

x∈M n n+k

E = N M n, D

R

σ ∈ Γ (E)

X ∈ Γ (T M )

.

,

NM

,

n

∇X σ = (DX σ)⊥ ,

ξ⊥

ξ

.

Mn

E,

,

.

20

80

E

n=4

E

2

.

,

,

Yang-Mills (

-Mills)

Gromov-Witten

.

.

π

E −→ M n

∇

∇

,

.

,

Φ(X, σ) = ∇X σ − ∇X σ.

Mn

,

f ∈ C ∞ (M n ),

(1.6)∼(1.7)

Φ(f X, σ) = f Φ(X, σ) = Φ(X, f σ),

Φ

,

Γ (E) −→ Γ (E)

C ∞ (M n )

Φ(·, ·)

,

∇

π

E −→ M

(1.8)

(1.8)

.

n

,

∇Φ Γ (T M ) × Γ (E) → Γ (E) (X, σ) → ∇X σ + Φ(x, σ)

,

Φ Γ (T M )×

&Æ%

· 14 ·

+3()7-2 ?(?:! # *4 ) G ?: ! +3 DKA; ÆÆ H:* - G # *4 Æ( 3I!E @ H &H?: ?: ?: 0 !E"I ?:!H @H&H H:* 0
5 )* J J)* 5# D K 3I

AE )* # 2F )**) B < 3 K LM&Æ" K NGHM L ONIJ M OÆ # * ) N PK Æ ) * ) # *5 $ $ Æ45,N O Q ∇Φ

(1.6)∼(1.7).

E

Riemann

.

,

M(E) = Φ Γ (T M ) × Γ (E) → Γ (E) | Φ ,

20

.

n

4

E

80

.

(1.8) . 2

,

,

, Donaldson

Yang-Mills

Y M(E, M 4 ) = ∇ | ∇

= ∇|∇

.

Ω∇

.

Y M(E, M 4 )

Donaldson

, E. Witten

.

Yang-Mills

,

.

Seilberg-Witten

20

90

Donaldson

Gromov-Witten

.

Riemann

.

§1.3

.

3

R3

.

σ [a, b] → R3 σ

,

σ(t) = (t, 0, 0),

6.

X(t) = (0, cos t, sin t).

6

>#*5 Æ &$POO* ,

)O*H#*5+3 E(X) =

σ

σ

a

b

∇σ X2 dt.

∇σ X ≡ 0.

X

PLMPRRP.NOS

* ) # *5 0 ) N# * 5 * ) N# *5 ($N#*5 =.#*5 * PE #* #> )N#*5 $ )N#*5 ) *" §1.3

Q

· 15 ·

(M n , g)

1.12

σ

(

, σ [a, b] → M n

Riemann n

X(σ(t)) ∈ Tσ(t) M ),

∇σ X ≡ 0,

X

,X

σ

.

(M n , g) = (Rn , g0 )

X(t) = (x1 (t), · · · , xn (t))

, σ [a, b] → Rn

n

σ

,

,

0 = ∇σ X = Dσ X = (x1 (t) · · · , xn (t)),

X (t) = 0, X = (a1 , · · · , an )

.

.

n

1.13

(M , g)

n

, σ [a, b] → M n

Riemann

,

X0 ∈ Tσ(a) M n ,

(1)

σ

X

X(a) = X0 . (2)

V

W

σ

,

d V, W ≡ 0. dt

(1)

F U → Mn

X(t) =

∇σ X = 0

-C

n

σ(a)

fi (t)

i=1

0 = ∇σ X =

0 # PE-*8

n

∂ , ∂xi

n

fi (t)

i=1

fi (t) +

∇σ

n ∂ ∂ + fi (t)aij (t) , ∂xi i,j=1 ∂xj

n

Q > > ) N# *5 Æ$' fj (t)aji (t) = 0.

j=1

fi (a) = ci , i = 1, · · · , n,

(2)

0

{V (t)}

∂ ∂ = aij (t) . ∂xi ∂x j j=1

{W (t)}

(1.9)

σ

.

,

d V (t), W (t) = ∇σ V, W + V, ∇σ W

dt = 0.

.

(1.9)

&Æ% ),&L( 7 * )# N DQ # *5 DQ ($ 2,& $2#S N#*5 $ +-=+3 0ÆJ2 R $' 0$#*5 S N Q %& 0 3 ), Q)*

RH #* < S-6- -9 R , ÆR Æ(& T RQ & R -C * R · 16 ·

Riemann

σ : [a, b] → M n

1.14

(M n , g)

Riemann

,

{Ei (t)}.

σ

Tσ(a) M n

{e1 , · · · , en }

,

ei , ej g = δij .

1.13

{Ei (t)}

σ

Ei (a) = ei ,

Ei (t), Ej (t) = Ei (a), Ej (a) = δij .

.

σ [a, b] → M n

σ

,

σ

∇σ σ

σ

1.15

∇σ σ

σ [0, 1] → Rn , σ(t) =

.

∇σ σ ≡ Dσ σ = σ (t) = −(cos t, sin t) = 0.

(cos t, sin t),

, ∇σ σ

Riemann

σ [a, b] → M

σ

(

∇σ σ ≡ 0,

.

.

n

Riemann

σ

σ

n

M

(M , g)

),

n

,

σ (t)

σ(t)

.

.

(1)

(Rn , g0 ).

n

Rn

σ(t) = (x1 (t), x2 (t), · · · , xn (t)).

,

0 = ∇σ σ = σ (t) = (x1 (t), x2 (t), · · · , xn (t)).

σ

Rn

σ(t) = (a1 t + b1 , a2 t + b2 , · · · , an t + bn )

Rn

(2)

.

(M 2 , g) = S 2 (1)

& S

R3

(

7)

S 2 (1) = {(x, y, z) | x2 + y 2 + z 2 = 1}.

σ

[0, 2π] → S 2 (1), t → (cos t, sin t, 0).

σ

§1.3

Q

PLMPRRP.NOS

7

8-

-

#*

) *" 0 +3RQ

∇σ σ = (Dσ σ )T = (σ )T = 0,

σ (t) = −(cos t, sin t, 0) {x1 , · · · , xn }

· 17 ·

M

n

S (1) * 2

σ(t)

,

(σ )T = 0.

,

n

∇

∂ ∂xi

∂ ∂ = Γkij . ∂xj ∂xk k=1

∇σ σ = 0, n ∂ 0 = ∇σ σ = ∇σ xi (t) ∂xi i=1 n ∂ ∂ = + xi (t)∇σ xi (t) ∂xi ∂xi i=1 n ∂ ∂ n = + xi (t)∇ xi (t) ∂ xj (t) ∂x ∂xi ∂xi j i=1 j=1 n n ∂ k = Γij (x(t))xi (t)xj (t) , xk (t) + ∂x k i,j=1

σ(t) = (x1 (t), · · · , xn (t))

($) *"

k=1

{(x1 , x2 , · · · , xn )}

RQ

n d2 xk dxi dxj = 0. + Γkij (x(t)) dt2 dt dt i,j=1

(1.10)

# > R $ * R ") * " 0 RQ >>+3 / 1.16

,

(M n , g)

n σv [a, b] → M dσ ≡ . dt {(x1 , · · · , xn )}

, F (O) = p.

Riemann

σv (a) = p

Mn

p

(1.10)

, p ∈ M n . v ∈ Tp M n ,

σv (a) = v.

σ [a, b] → M n

,

F U → Mn

&Æ%

· 18 ·

⎧ ⎨ xk (0) = 0,

Riemann

⎩ x (0) = v k k

*R Æ;:' k = 1, 2, · · · , n

.

σ [a, b] → M n

,

d σ , σ = ∇σ σ , σ + σ , ∇σ σ = 0, dt dσ d 2 = (σ (t) ) ≡ 0, . . dt dt

($ 0 TTURSS ÆRTET -$AT3/"Q U V6 §1.4

.

→ Mn

(M n , g)

Riemann

,

b

L(σ) =

σ [a, b]

σ

dσ dt. dt

0 26* 1,EU >Q = Æ& ) +3 ) $ )) # *5 V ' ) a

,

g

dg (p, q) = inf{L(σ) | σ [0, 1] → M n , σ(0) = p, σ(1) = q}. ,

β

.

[a, b] × (−ε, ε) → M n , (t, s) → β(t, s),

β(t, 0) = σ(t),

βs (·) = β(·, s)

σ

V (t) =

σ

,

∂β (t, 0) ∂s

.

.

1.17

σ, β, V (M n , g) , σ (t) = l. ∂ 1 ∂ ∂β (t, s) σ , V − V, ∇σ σ . = ∂s ∂t l ∂t s=0

(1.11)

&ÆWWXU Æ;:'

§1.4

s=0

,

∇ ∂β ∂s

· 19 ·

∂β ∂β ∂ ∂ , = β∗ , = 0, ∂t ∂s ∂t ∂s

$-

∂β ∂β = ∇ ∂β , ∂t ∂t ∂s

(

s=0

)

1 ∂ ∂ ∂β ∂β 2 ∂β (t, s) , = ∂s ∂t ∂s ∂t ∂t s=0 ∂β ∂β , 2 ∇ ∂β ∂s ∂t ∂β ∂β 1 1 ∂t , = = ∇ ∂β ∂s ∂t 2 ∂β ∂β 12 l ∂t , ∂t ∂t ∂β ∂β 1 , = ∇ ∂β ∂t ∂s l ∂t ∂β ∂β 1 ∂ ∂β ∂β , , ∇ ∂β = − ∂t ∂t l ∂t ∂s ∂t ∂s

=

1 ∂ V, σ − V, ∇σ σ . l ∂t

0)V .)V V 37

T ' Q +3 D R T ' Q Æ&/Y"M ) +3 - #)#*5 +3 .

(1.11)

(M n , g)

1.18

p

.

Riemann

σ ≡ l.

q

.

σ [a, b] → M n

Mn

σ

p

, p, q ∈ M n ,

q

σ [a, b] → M n

.

.

β [a, b] × (−ε, ε) → M n , (t, s) → β(t, s),

β(t, 0) = σ(t)

β(a, s) ≡ p,

β(b, s) ≡ q.

,

V (a) = 0 = V (b).

{V (t)}

· 20 ·

-

σ (t) ≡ l,

&Æ%

#)VÆ72

Riemann

, s=0 b ∂ d ∂β L(βs (·)) (t, s) 0= = dt ds a ∂s ∂t s=0 s=0 b 1 ∂ V, σ − V, ∇σ σ dt = a l ∂t 1 1 b = V, ∇σ σ dt. V (b), σ (b) − V (a), σ (a) − l l a

- )#*5 Æ72 0 * R 0 UUVW VXW YV Æ)O #EU ÆD 0 V * 8 6* X Z6*[ 0& R X Z Æ&J () * R +3 0 , & \ H # * R+3 * +3 R PE * 8 37 0 - $ ) 9 2 , & Æ$'# R X Z

,

a

b

V (t), ∇σ σ dt = 0

V

,

∇σ σ = 0,

Mn

σ

.

.

§1.5

,

Hopf-Rinow

(M n , g),

Riemann

dg .

n

,

(M , dg )

(M n , g)

Riemann

1.19

,

n

(M , g)

→ Mn

.

n

, p ∈ M , v ∈ Tp M n , σv [0, 1]

Riemann

σv (0) = p, σv (0) = v.

Expp v = σv (1).

Expp

1.20

v

σw [0, δ] → M

σw (0) = w,

σw (0) = p

σεw

.

∇σw σw

n

(M , g)

Bε (O) .

Expp

(M n , g)

ε > 0, δ 0, → M n, ε t → σw (εt).

= 0. B1 (O) = {v ∈ Tp M v 1}

ε = ε(p) > 0

=ε

2

.

Riemann

σεw (0) = εw

σεw (0) = p

σ ∇σεw εw

n

(1.12)

.

,

.

.

1.20,

Expp Tp M n → M n

,

XYYZ Z][. ^W

XZ6* %X # # $ ) [" % /"

* QR T3W( #2 ) " &2# 7 2 9\ 6 _ )6* & XZ - 80 L Y X$ %0 !Y 6* ' X Z ($

- \ /"

T - A 4 ÆJ2 J () BZJ () 9[?" +/ +Z ,/ Æ 0 §1.5

,

· 21 ·

Hopf-Rinow

1.21 (Hopf-Rinow)

.

n

(a) (M , dg )

;

n

Expp0 Tp0 M n → M n

(b)

p0 ∈ M ,

(c)

n

n

p ∈ M , Expp Tp M → M

(d) M n

p

(a)

n

.

(d)

(d)

.

g, (M , dg )

,

1}

(a)

,

.

(a)

,

D2 (1)

,

(d)

' ] ,/

⎜ ⎜ ,I=⎜ ⎝

eA = I + A +

ξ

p

Expp

⎛

m

M2

,

.

⎞

1

0

..

.

0

.

(M 2 , dg0 )

, Riemann

A = (aij )

.

M 2 = D2 (1) = {(x, y)|x2 +y 2 <

.

n = m2 − 1.

M

g0 = dx2 + dy 2 ,

.

,

(a)∼(c)

n

(a)

.

g0

q

0

,

n

(d)

,

.

,

,

M

;

q

1.21

,

n

⎟ ⎟ ⎟ ⎠

m

.

1

A3 Ak A2 + + ···+ + ···. 2! 3! k!

$ Æ

M n = SL(m, R) = {A|det(A) = 1},

Mn

I

TI M n = sl(m, R) = {ξ|ξ + ξ T = 0}. , det(eξ ) = etrξ = e0 = 1.

TI M n = sl(m, R),

- ) )6* Z],/ A )PK Æ # Z *R ξ, η I = tr(ξη T ),

ξ, η ∈ sl(m, R)

.

SL(m, R)

g.

SL(m, R) ,

ϕξ (t) = A(t) = etξ

(M n , g)

.

,

Mn =

ξ ∈ sl(m, R) = TI M n

&Æ% < ` < 1 1,"* 0 ]^ & & · 22 ·

Riemann

1.21

1.22 (Gauss

)

n

n

Rn = T p M n

ρ(t) = tv

n

,

w ∈ Tρ(t0 ) (Tp M ) = Tρ(t0 ) R = R , w

ρ (t),

(Expp )∗ |ρ(t0 ) ρ (t0 ), (Expp )∗ |ρ(t0 ) w = 0.

w = 0, (1.13)

(1.13)

w = 1 v coss + wsins , v .

v(s) = v

(1.13)

.

v(0) = v, v (0) = ||v||w.

β(t, s) = Expp [tv(s)] v = Expp t v coss + wsins . v

-

w = 1, v(s) = v ,

2)V$

t0

L(β(·, s)) = 0

∂β (t, s) dt ≡ v t0 . ∂t

∂β (t, 0), σ(t) = β(t, 0) ∂s t0 ∂ ∂ 1 L(β(·, s))|s=0 = V, σ − V, ∇σ σ dt 0= ∂s v 0 ∂t V (t) =

= V (t0 ), σ (t0 ) − V (0), σ (0)

0 </ 37 "# > %

= (Expp )∗ |ρ(t0 ) w, (Expp )∗ |ρ(t0 ) v .

.

Gauss

.

1.23

Bε0 (O) = {v| v ε0 , v ∈ Tp M n }, Expp Bε0 (O) → M n

> +3 % + Y"V6 1 _ )'( # $ .

(1)

Bε0 (p) = Expp (Bε0 (O)).

v ∈ Bε0 (O), σv (t) = Expp (tv), σv [0, 1] → M n

,

L(σv ) = d(p, σv (1)) = v.

,

ψ [0, 1] → M n

σv

,

, ψ = σv .

(2)

q∈ / Bε0 (p),

q ∈ ∂Bε0 (p)

d(p, q) = ε0 + d(q , q) = d(p, q ) + d(q , q).

(1.14)

XYYZ Z][. 0

§1.5

,

Hopf-Rinow

, d(p, q) ε0 .

^W

· 23 ·

+3

r(q) = d(p, q), ψ [0, 1] → M n (1) ∂ ∇r = ∂r = 1,

-C t1 < 1,

ψ (t) ψ (t), ∇r = ψ (t) = λ(t)∇r, λ(t) 0.

ψ(0) = p, ψ(1) = Expp v.

d [r(ψ(t))]. dt

&

t1 = sup{t|ψ([0, t]) ⊂ Expp (Bε0 (O))}.

ψ(t1 ) ∈ ∂Bε0 (p),

t1

ψ (t)dt +

L(ψ) = 0

t1 0

1

t1

ψ , ∇r dt =

ψ (t)dt t1

0

d [r(ψ(t))]dt dt

= r(ψ(t1 )) − r(ψ(0)) = ε0 v.

-C )7, Æ

t1 = 1, ψ (t) = λ(t)∇r = λ(t)

L(ψ) > v.

t1 = 1,

1

L(ψ) = 0

-C 2) 72

T ' 0

ψ (t) = λ(t)∇r = λ(t)

ψ [0, 1] → M n

= ,

,

ψ (t)dt r(ψ(1)) − r(ψ(0))

= d(p, σv (1)) = v.

p

q

1 σ (t). v v ,

(1).

t0 = inf{t|ψ(t) ∈ ∂Bε0 (p)}.

(1)

($

1 σ (t), v = ε0 . v v

,

(2)

-

L(ψ) = 0

t0

ψ (t)dt +

1

t0

ψ (t)dt,

ε0 + d(∂Bε0 (p), q),

d(p, q) ε0 + d(∂Bε0 (p), q).

(1.15)

&Æ%

· 24 ·

L1( 2 B[ $2 ,

Riemann

(1)

d(p, q)

inf

q ε∂Bε0 (p)

{d(p, q ) + d(q , q)} d(q , q).

(1.16)

d(ˆ q , q) = d(∂Bε0 (p), q).

(1.17)

- 9 $ ( $2 0 <A# 1Q7

Æ [ ) ( 7! `I # *QR $ 7 2 7 $2 ' \ H a$ "# # +3 = ε0 +

∂Bε0 (p)

inf

q ∈∂Bε0 (p)

∃ qˆ

,

(1.15)∼(1.17)

d(p, q) = d(ˆ q , q) + ε0 .

.

Gauss

1.21

1.21

1.24.

(c)⇒(d),

(b)⇒(a)⇒(c)⇒(b).

(c)⇒(d).

Expp0 Tp0 M n → M n

1.24

σv [0, l] → M n

q ∈ M n,

,

q = σv (l), L(σv ) = d(p0 , q),

σv (t) =

Expp0 (tv), v = 1.

1.23

Expp0 Bε0 (O) → M n

ε0

q ∈ ∂Bε0 (p0 )

,

0

,

d(p0 , q) = d(p0 , q) + d(q, q).

& % $

v=

Exp−1 p0 q

Exp−1 p0 q

, l = d(p0 , q)

t0 ε0 > 0.

ÆÆ

σv (l) = q.

b'. Æ ,

7 L Æ,2\Æ ' Æ# 7 '2

E = {t|d(σv (t), p0 ) = t,

t0 = sup{t|t ∈ E}.

t0 = l,

t0 < l, q = σv (t0 ) = q.

d(p0 , σv (t)) + d(σv (t), q) = d(p0 , q)}. .

, t0 < l

1.23

d(q , q ) + d(q , q) = d(q , q).

.

q ∈ ∂Bε (σ(t0 )),

XYYZ Z][. ["I §1.5

,

Hopf-Rinow

^W

· 25 ·

E

d(p0 , q) = d(p0 , q ) + d(q , q) = d(p0 , q ) + d(q , q ) + d(q , q)

- EU B[ a0 2 $2 0 ($2R T ' QR D *Q - > Æ72 d(p0 , q ) + d(q , q).

d

(1.18)

,

d(p0 , q) d(p0 , q ) + d(q , q).

(1.19)

d(p0 , q) = d(p0 , q ) + d(q , q)

(1.20)

d(p0 , q ) = d(p0 , q ) + d(q , q ). ψ , σv

(1.21)

(1.18)∼(1.19)

ψ

q

q

[0,t0 ]

8).

(

,

σv (t0 + ε) = q .

2

(1.20)∼(1.21)

8

$2

d(p0 , q) = d(p0 , σv (t0 + ε)) + d(σv (t0 + ε), q),

($7 \Æ 7 ^]$> 0 d(p0 , σ(t0 + ε)) = t0 + ε.

t0 + ε ∈ E,

.

1.24, (b)⇒(a)

1.24

.

.

(1.21))(

&Æ% $ a%a% 27 72 #

· 26 ·

Riemann

Mn

{qi }

S

n−1

(1)

Cauchy

, li = d(p0 , qi ).

1.24

vi ∈

,

{ij },

qi = Expp0 (li vi )

lij → l0 , vij → v0 ∈ S n−1 (1)

2 T1 Æ$2 0 XZ - a% ($ #

0 Æ,2\Æ +3 - #($ # < a8% 7_-2 %XZ ($ !*R \Æ ' 0 \^ c bÆ\]^_M]^_ Y b _MÆ\ `P \ c ^ QPL b ` da]MÆ\^ef b ba_b^ef c Mc` d` b ae Maaeb cdW bba_^efc Mc` ^f cegQPL fhg MÆ\QPL dh cegQPL ib .

Expp0

,

qij = Expp (lij vij ) → Expp0 (l0 v0 ) = q0 , {qi }

lim d(qij , q0 ) = 0.

j→+∞ (M n , dg )

Cauchy

lim d(qi , q0 ) = 0,

i→+∞

.

v ∈ S n−1 ⊂ Tp M n ,

(a)⇒(c),

},

t0 < +∞,

σv (t) = Expp (tv), t0 = sup{t|Expp (tv)

ti → t0 ,

.

qi = Expp (ti v) = σv (ti ),

d(qi , qj ) |ti − tj |.

ti → t0 ,

{qi }

lim σv (ti )

.

Expq0 (tw0 ).

σv

i→+∞

(c)⇒(b),

1.

(gij (x)) (ii)

(iii)

q0

Gauss

,

[0,t0 +ε]

n

p

i=1

w0 = σv (t0 ), ψ = ,

.

ai

Riemann

,

∂f ∂xi

(g ij (x))

?

.

g x

?

X=

df (X) = X, Y g

.

q0 =

.

ij

n ∂f df = dxi ∂x i i=1 n

Mn

, g

∂f ∂ ∇f = g ∂x i ∂xj i,j=1 ! " ∂ ∂ gij (x) = , ∂xi ∂xj

X

,

.

n

df (X) =

q0 = σv (t0 ). σv

.

f Mn → R M

(M n , dq )

.

ψ

.

{(x1 , · · · , xi )} (i)

Cauchy

ai

∂ , ∂xi

Mn

Y

Æ \^cd _e\l b \jb Mk Y b i M`Pc . f . jbbmU ib \^73cf_Y Mb\]^_kY b _MÆ\]^ `P bciM d` k Ml` d gh ee b M]^_Y mhmU bbib f bbkÆ\jP gi^W mhmUbbib gi fkf d` .\^ kc _MghÆc.QPLe_ f d` nhiocpU Mj ee k MÆljk mkf c _MgÆ\]^ QPL ni^focb`P M 73 lb Mj`k `ghÆ\^efc k ee fmp . ee boiM no m n kf gh

· 27 ·

(iv)

c > 1

1

f (x) = x1 . Rn n ∂f ∂ ∇f = g ij ∂x i ∂xj i,j=1

g = dx1 ⊗ dx1 + · · · + dxn ⊗ dxn ∂f ∂ ˆ ˆ = gˆij . ∇f = ∇f ? ∇f ∂x i ∂xj i,j=1 n

f Mn → R

2.

Riemann

M n = Rn = {(x1 , x2 , · · · , xn )|xi ∈ R}

,

.

f

Mn

, g

gˆ = c2 g.

,∇

Riemann

Hessian

Hess(f )(X, Y ) = X, ∇Y (∇f )g ,

∇f

, ∇f =

f

n

g ij

i,j=1

(i)

Mn

h

∂f ∂ ∂xi ∂xj

df (X) = ∇f, X(

1).

,

Hess(f )(hX, Y ) = hHess(f )(X, Y ) = Hess(f )(X, hY )

?

Hess(f )

(ii)

?

1.10,

?

X, ∇Y (∇f ) = Y Xf − (∇Y X)f.

(ii)

(iii)

Hess(f )(X, Y ) = Hess(f )(Y, X),

X

3.

Mn

Y

f M

n

.

→ R, g

X, ∇Y ∇f , trS (i)

∇

S

{e1 , · · · , en }

Tx M n

.

Hess(f )(X, Y ) =

, n (trS) = S(ei , ei ). x

(ii)

∆f = tr[Hess(f )],

.

Mn

i=1

Y,

g

Riemann

∇,

divY = tr(X → ∇X Y ).

{(x1 , · · · , xn )}

(iii)

{Γijk }

,

n i=1

Christoﬀell

divY =

n i=1

G =det(gij ),

Y =

n

bi

∂ , ∂xi

,

#

i ∂bi + Γki bk ∂xi n

k=1

1 ∂ √ Γiki = √ ( G). G ∂xk i=1

$

.

∇

∂ ∂xk

Y

divY .

Y

&Æ%

· 28 ·

mU bbib n

Riemann

(iv)

∆f = div∇f

?

∇f =

(v)

n

g ij

i,j=1

._Æ

∂f ∂ ∂xi ∂xj

(iii)∼(iv)

kf

n √ 1 ∂ ∂f . ∆f = √ G g jk ∂xk G j,k=1 ∂xj

o bba_b^ef c Mc` \^ c f k _lM]^_nYh hgb _MÆ\]^. `Pp 2Mleec bbgmhmU ib \^ . c k qMeq]^qlS eer`Æ\]^Y4 c dh . mpe q l Si m snee b Mtlcdql S mp kkrsM S12 MÆqltWr f c imbqlS . d` b _MÆ\]^`P \^ b\m f f imb fj cbbr`Æq ql S imb bdh Mu` fv kmnbb o\imnpr`Æ q qql S `ÆuhqlS imb b ee pgkv` f ceg ib j bbriÆ! c m`lwYZ (vi)

fj Laplace p

4.

η M

n

∆

→ R

2

gˆ(X, Y ) = η g(X, Y ),

?

M

n

Mn

, g

2

X, Y gˆ = η X, Y g ,

Riemann

jk

ˆ ∆} ˆ {(ˆ g ), G,

3

,

ik

{(g ), G, ∆}

.

n = 2,

ˆ = 1 ∆f ∆f η2

?

5.

σ0

[0, 1] → M n M

n

,

σ1 S 1 → M n

Mn

H(·, 0) = σ0

H(·, 1) = σ1 ,

σ1

σ0 (t) ≡ p0

.

.

1

(i)

H S1 ×

.

n

C(S , M )

,

Ω[σ0 ] = {σ S 1 → M n |σ

L(σ) =

1 2 dσ = dσ , dσ ,g dt dt dt g g

σ0 }

dσ dt, dt S1 g

Mn

.

l[σ0 ] = inf L(σ) | σ ∈ Ω[σ0 ] , σ,

σ0

σ

σ

Mn

,

σ0 .

Mn

L(σ) = l[σ0 ] ?

?

{σj }+∞ j=1 ,

(ii) dσ j ≡ cj dt

t ∈ S1

,

σj

σ0

L(σj ) → l[σ0 ] .

{σj }+∞ j=1

σj S 1 → M n

?

σj

,

Æ v mn jbbr`Æ\ouh robÆq qlS \^gic c m`_Ye_sW pn&ÆWWXUkfs x lS bÆqqNOS \^ bm b qtqgÆqlkrsMqlS f lkrs d` ^ bnhql S c Mu` . pblrmYp buY vw Y . riMP y ts mlmUp Msu bblt cd u z 12 qM! vs .r {w d` bvcpY4 s c jiMixw mlmU p bbib ` xqvYyqy"fkfrcr {w12 _gh`P pglmU wbmUibnvbsv b_Ætx`Py_Æ\k Y zo g ` xu y"f^W\^|xbyb12\Z]`P g k _lw_YM12 v kmn ^f cdemz{c c d` k _Mvw_Y gh

· 29 ·

(iii)

Ascoli

Mn

,

{σji }+∞ i=1

,

σ∞ S 1 → M n ?

σ∞ S 1 → M n

(iv)

n

6.

(M , g)

,

σ∞

Riemann

M

.

Sys(M n , g) = inf L(σ) | σ

σ S1 → M n

L(σ)

a

(i)

2

}.

a

b

2

(T , ga,b ) = R /aZ ⊕ bZ.

Sys(T , ga,b ).

2

(ii)

,

.

, aZ = {an | n ∈ Z

b

2

.

n

(T , ga,b )

.

(iii)

Area(T 2 , ga,b ) [Sys(T 2 , ga,b )]2

?

7.

S 2 (1) = (x, y, z) | x2 + y 2 + z 2 = 1

R3

3

RP 2 = S 2 (1)/Z2 ,

Z2

φ R3 → R3 , v → −v

.

Sys(RP 2 ).

(i)

(ii)

Area(RP 2 )

?(

2 [Sys(RP 2 )]2 π

RP 2

1962

Area(RP 2 , g)

2 [Sys(RP 2 , g)]2 , π

g

,

,

8.

[Pu]. Gromov

1983

[Gr2].)

n

, C(M n )

(M , g)

Mn

f = maxn |f (x)|. x∈M

Gromov

Φ (M n , g) → C(M n ), x → dx ,

dx (p) = d(x, p)

g

(M n , g)

.

.

Mn

,

&Æ%

· 30 ·

n|xlmU kf zw bf{vwMz{ c k^W zgM z{ yz}fgn z{ lÆ ^b{vwM ~hc|{ qM{S. qMNOS \^ b\Z]M b _{^MÆt | q]^ cfmp b d`MÆquWl}pS fee c fjb\]^b \^_Y}#`Æ\|| bee bbÆq^ MNOb\SZ]M k qM]^o \^ b M 73 c qMoÆt /12 g\ }~W}b d`` qb tM~12 c _Mghe\/ b ` QPL qMlj}12 f .f Riemann

|d(x, z) − d(y, z)| d(x, y)

(i)

Φ(x) − Φ(y) = d(x, y),

Φ

.

Ψ (M n , g) → Rm

(ii)

R

. (

n

9.

(M , g)

f

U

(M , g)

Np M

n

M

, p0

M

n

x ∈ U.

h Mn → R dσ = ∇h|σ(t) , dt

,

,

.

, Mn ⊂ M

Mn

m Np M n = v ∈ Tp M | v ⊥ Tp M M

m

p

.

M

p, n

m

M

Tp M n

1.10.)

.

∇

m

σ

Tp M

m

,

X

g = g|T M n

∇

.

= Tp M n ⊕ Np M n

n

d`hccdin^W bQP M/WP j bbbv~~`P (·)

m

Tp M

∇X Y = (∇X Y )T ,

(·)T

Ψ

, f (x) = d(x, p0 ).

∇h

Riemann

Riemann

n

Nash

?

(M , g)

,

.

.)

∇f (x) ,

,

m

m

M

σ R → Mn

n

(M , g)

Nash

n

Riemann

∇h ≡ 1,

10.

1.9

m

~^M

Y

Riemann

73

? (

~

, , . Jacobi . Jacobi Riemann

. Cheeger Ebin

Riemann Æ( [ChE]). Æ Æ.

Ricci , g) Riemann , ∇ Riemann , §2.1

(M

n

R(X, Y )Z = −∇X ∇Y Z + ∇Y ∇X Z + ∇[X,Y ] Z,

Z M . Riemann R , !. 2.1 M f M → R X Y Z, X, Y

n

n

n

(1) R(f X, Y )Z = f R(X, Y )Z = R(X, f Y )Z = R(X, Y )(f Z), (2) R(X, Y )Z = −R(Y, X)Z, (3) R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0,

(4) R(X, Y )Z, W = R(Z, W )X, Y .

(1), "

R(X, Y )(f Z) = f R(X, Y )Z.

, Riemann #, $ Y Xf −XY f = −[X, Y ]f (∇ (∇ Y )f = −[X, Y ]f,

Y

X

R(X, Y )(f Z) = − ∇X ∇Y (f Z) + ∇Y ∇X (f Z) + ∇[X,Y ] (f Z) = f R(X, Y )Z − (XY f )Z + (Y Xf )Z + [X, Y ]f = f R(X, Y )Z.

X)f −

% !

· 32 ·

"! (2) &'!". (3), ""## $ Bianchi "!() ∇ #%". (3) Æ $, #* +, (3) !% X = ∂x∂ , Y = ∂x∂ Z = ∂x∂ $ &. &-., ' (1)

i

(3)

Æ$ = − ∇

j

k

∂ ∂ +∇ ∂ ∇ ∂ ∂xj ∂xi ∂x ∂xk k ∂ ∂ −∇ ∂ ∇ ∂ +∇ ∂ ∇ ∂ ∂xj ∂xk ∂x ∂xk ∂xj ∂x i i ∂ ∂ −∇ ∂ ∇ ∂ +∇ ∂ ∇ ∂ ∂xk ∂xi ∂x ∂xi ∂xk ∂x j j ∂ ∂xi

∇

∂ ∂xj

$ τ ≡ 0). "! (4) %(/, &' ' ( [ChC] p141∼155). $ )( ) () **) (4), *++ ,- (4) . ,. . X, Y ∈ T M +# , '/& X, Y 0 =0

(

∇

n

p

P = Span{X, Y }.

(M

n

,

P ⊂ Tp M

, g)

R(X, Y )X, Y K(P) = KXY = X, X X, Y X, Y Y, Y

.

) 2.1 % P , K(P) )0-1. {X, Y } 12. Ricci , 2 T M 2/. {e , · · · , e }. p

n

1

Ric(X, X) =

n

n

R(X, ei )X, ei .

i=1

!013 . -.3 /0142 Riemann (M , g) ,

§2.2

2. {β (·)} s

β(t, s),

s∈(−ε,ε)

n

β

[a, b] × (−ε, ε) → M (t, s) → β(t, s).

n

,

βs (t) =

345567638

§2.2

· 33 ·

Æ4 s, β t → β(t, s) (M , g) , $ 9 {β(·, s)} (M , g) . 2.2 . σ [a, b] → M (M ∂β, g) 4 , {β(·, s)}

, β(t, 0) = σ(t) 5 J(t) = ∂s (t, 0), {J(t)} 9 σ Jacobi . !: J 65 6. 2.3 . {J(t)} 7 σ [a, b] → M Jacobi , J 7 !7 Jacobi ; n

s

n

n

n

n

t∈[a,b]

J (t) + R(σ (t), J(t))σ (t) = 0.

∂t

= ∇ ∂β ∂t

!

(2.1)

89< #=#* 0 = ∇ ∂β

s∈(−ε,ε)

∂β ∂β ∂β ∂β − ∇ ∂β − , ∂s ∂t ∂s ∂t ∂s ∂β ∂β − ∇ ∂β , ∂s ∂s ∂t

=

∂β (t, 0) {β(·, s)} β(t, 0) = σ(t), J(t) = ∂s ∂β ∂β ∂ ∂ ∂β ∂β , , = ∇ ∂β . ∇ ∂β = β∗ =0 , ∂s ∂t ∂t ∂s ∂t ∂s ∂t ∂s

)"

% s = 0 $= 8><

∂β ∂β ∂β , R(σ , J)σ = R ∂t ∂s ∂t ∂β ∂β ∂β + ∇ ∂β ∇ ∂β + ∇[ ∂β , ∂β ] = −∇ ∂β ∇ ∂β ∂t ∂s ∂t ∂s ∂t ∂t ∂t ∂s ∂t ∂β = −∇ ∂β ∇ ∂β ∂t ∂s ∂t ∂β = −∇ ∂β ∇ ∂β ∂t ∂t ∂s

= −∇σ ∇σ J = −J (t).

)!7$Æ β(·, s) , #* ∇ !" Jacobi ; (2.1). ,. 9) 89

∂β ∂t

∂β ≡ 0. ∂t

)&"

% !

· 34 ·

?@ Riccati ?@ 0:: Jacobi , ";;!7< . 2. (M , g ) (R , g ) :; §2.3 Jacobi

2

3

1

0

S 2 (1) = {(x, y, z)|x2 + y 2 + z 2 = 1}.

:;* K ≡ 1. ;< σ [0, 2π] → S (1), 2

t → (cost, sint, 0)

4 , J(t) = (0, 0, c cost + c sint) σ Jacobi , c c *. . {J , · · · , J } A' σ <# (n − 1) Jacobi , {E , · · · , E } 7 σ 0=2/.*' E (t) = σ (t). >==!. 1

1

2

1

n−1

1

n

Ji (t) =

n−1

2

aji (t)Ej (t),

n

Rij (t) = R(σ , Ei )σ , Ej .

j=1

$ J

i

+ R(σ , Ji )σ = 0,

7

#*>?@ A(t) = (aij (t)) A (t) + R(t)A(t) = 0,

R(t) = (Rij (t))

>?.

(2.2)

§2.4

Gromov

A ?B ! 5BC>

$ {J , · · · , J D

n−1 }

1

#, @ A

−1

(t)

· 35 ·

C . A II(t) = A (t)A

−1

(t).

) (2.2)

II (t) = [A (t)A−1 (t)] = A A−1 + A (A−1 ) = −RAA−1 − A A−1 A A−1

)

= −R − II2 .

'<

II + II2 + R = 0.

(2.3)

;9 Riccati ;. ! Gromov-Bishop Æ$ () < Riccati ;. D!BÆ$, **)< Riccati ; C Riccati )"!. &D! §2.4 §2.5. DEFEFGGE/EF? 0) H GÆ. H.!. 2.4A (Gromov) f f˜ I7 §2.4 Gromov

⎧ ⎨ f + Kf 0,

⎩ ˜ ˜ f˜ 0, f +K f˜(t) ˜ < K(t) K(t). f (0)f˜(0) f˜ (0)f (0), f˜(t), f (t) 0, % f (t) > 0 $ t f (t) JH. ff˜ = f˜ f f− f f˜, A h(t) = f˜ f − f f˜,

2

h = (f˜ f − f f˜) = f˜ f − f f˜ ˜ f˜f + Kf f˜ −K ˜ f˜ 0. = (K − K)f

&K34I h(0) = f (0)f˜(0) − f˜ (0)f (0) 0, " h(t) 0,

h f˜ = 2 0, f f

(t) $ ff˜(t) JH. ,.

% !

· 36 ·

"J@B (K c) Æ. 2.5(LÆ, Rauch) . (M , g) AMRiemann, σ[0,+∞) → M 4 , {J(t)} σ Jacobi 7 J(0) = 0, J (0), σ (0) = 0,

J (0) = 1 = σ . !7K &. 0, J(t) t; (i) B K e −e ; −1, J(t) sinht = (ii) B K 2 1, t ∈ [0, π], J(t) sint. (iii) B K A f (t) = J(t) , 8>< Mn

n

n

Mn

t

−t

Mn

Mn

1

f (t) = J(t) = [J(t), J(t) 2 ] =

1 2J, J J, J , 1 = 2 J, J 2

J

f (t) = (f (t)) =

J, J

J

J, J J − J, J J

J 2 J, J

J (J , J + J, J ) − J, J

J

=

J 2 J, J 2 −J, R(σ , J)σ J + J J , J −

J

=

J 2 1 = −K(t) J + ( J 2 J 2 − J, J 2 )

J 3

=

−K(t)f (t), , J)σ , J K(t) = R(σ J

, J(t), σ (t) ≡ 0, σ = 1, f (t) = J(t) 7 C)"!

2

f + Kf 0.

$ J(0), σ (0) = 0 = J (0), σ (0), @J(t), σ (t) ≡ 0.) ˜ (i) % K ≡ 0, f˜(t) = t. ) 2.4A ' 0 $, (A K(t)

(

Mn

t f˜(t) = f (t) f (t)

§2.4

Gromov

A ?B ! 5BC>

· 37 ·

JH, 89< lim

t→0+

f (t)

J(t)

= lim+ t t t→0

1 J(t) J(t) 2 , = lim+ t t t→0 = J (0) = 1.

$ f (t) JH, #*' t f (t) 1 t

9 t 0 &, L f (t) t. K (ii)∼(iii) MD, @,-. ,. HI) ÆD. 2.6(LNÆ, Berger) .(M , g)AM Riemann , σ[0, +∞) → M 4 , {J(t)} σ Jacobi 7 J (0) = 0 = J(0), σ (0), J(0)

= 1 = σ . !7K & (i) % K 0 $, J(t) 1; e +e ; −1 $, J(t) cosht = (ii) % K 2 (iii) % K 1 $, J(t) cost. Berger Æ H Rauch Æ OE JK, ) . ,. ! '/!B= (K˜ c). % K˜ c $, Æ )< Riccati ;9>?@Æ. " 89< Riccati ;"N Jacobi ;. $, 2.4 A 5 PQ !!. 2.4B A f, f˜, K K˜ 2.4A. (2. λ = (logf ) λ˜ = (logf˜) < ⎧ n

n

Mn

t

−t

Mn

Mn

Mn

Mn

⎨ λ + λ2 + K 0, ⎩ λ ˜2 + K ˜ + λ ˜ 0,

˜ λ(0), S R. λ(0) ˜ λ(t). λ(t)

2.4B, LN.! II(t) Æ.

% !

· 38 ·

˜ 2. {S(t)} (n − 1) × (n − 1) 9>?@, <

2.7

t>0

S˜ (t) + S˜2 (t) −cI,

I

PF:>?. S

⎧ 1 ⎪ I, ⎪ ⎪ ⎪ ⎨ t ˜ S(t) (cott)I, ⎪ ⎪ ⎪ ⎪ ⎩ (cotht)I,

A {λ˜ (t), · · · , λ˜ λ˜ (t)
n−1

c = 0,

c = 1, t ∈ (0, π),

c = −1. ˜ ˜ (t))} S(t) GL@. S(t) 29,

i

˜ n−1 (t)}. ˜ = max{λ ˜ 1 (t), · · · , λ λ(t)

˜ MH, λ(t) Lipschitz , OTT%C. ˜ + λ˜2 + c 0 λ

OTT &. ˜ ˜ ) GL@GL , E(t ) = ˜ ) * λ(t 2. λ(t) t T%C, E(t ) S(t ˜ ˜ ˜ 1. 2 ϕ(t) = E(t ), S(t)E(t ). ⎧ ϕ(t) λ(t) # t 0 &< ϕ(t ) = λ(t ). ⎨ ϕ(t) λ(t), ˜ $ ϕ λ˜ t T%C, & ⎩ ϕ(t ) = λ(t D ϕ (t ) = λ˜ (t ). ) " ˜ ) 0

0

0

0

0

0

0

0

0

0

0

0

0

0

˜ 2 (t0 ) = ϕ (t0 ) + ϕ2 (t0 ) ˜ (t0 ) + λ λ = E(t0 ), (S˜ (t0 ) + S˜2 (t0 ))E(t0 ) −c

#%CU t &. )NO 2.4B K@4I7, %P) 2.4B '" &. ,. 2.8 (K˜ c Æ) 2. (M , g) !B (K˜ ˜ c) AM Riemann , σ [0, +∞) → M 4:I , {J(t)} σ ˜ = 0, J˜ (0), σ (0) = 0, J˜ (0) = 1. Jacobi <7 J(0) ⎧ ⎪

c = 0, ⎪ ⎨ t, ˜

J(t)

sint,

c = 1, ⎪ ⎪ ⎩ sinht, c = −1. 0

n

Mn

Mn

n

§2.4

Gromov

A ?B ! 5BC>

· 39 ·

;<

˜ = (log J )

˜ J(t) J˜ (t) , ˜ ˜

J(t)

J(t)

.

27 σ 0=:2/. {E˜ (t), · · · , E˜ (t)} *' E (t) = σ (t), E (0) = J˜ (0). A J˜ (t), · · · , J˜ (t) 7 σ Jacobi ,
1

n

n

1

n−1

J˜i (0) = Ei (0),

J˜i (0) = 0,

∀i = 1, 2, · · · , n − 1.

J˜i (t) =

n−1

˜j (t), R ˜ ij = R(σ , E ˜i )σ , E ˜j . a ˜ji (t)E

j=1

˜ = (˜ a R A(t)

˜ ˜ ij (t)), J(t) = (J˜1 (t), · · · , J˜n−1 (t)), = A˜ (t)A˜−1 (t), R(t) = (R ˜ = (E˜1 (t), · · · , En−1 (t)). ˜ E(t), ˜ E(t) J(t) = A(t) ˜

ij (t)), II(t)

S

(log J(t) ) =

$ K˜

Mn

c,

J (t) J(t) ,

J(t) J(t)

=

J(t) J(t) ˜ , II(t)

J(t)

J(t)

S ˜ (t) + II ˜ 2 (t) + cI II ˜ (t) + II ˜ 2 (t) + R(t) ˜ = 0, II

L ˜ (t) + II ˜ 2 (t) −cI. II ˜ = II(t)) ˜ &" 2.7 ' (A S(t) ⎧ 1 ⎪ ⎪ , ⎪ ⎪ ⎪ ⎨ t (log J ) cott, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ cotht,

$ 2.8 '. ,. !0 Ric

Mn

−(n − 1)c

c = 0,

c = 1, 0 t π,

c = −1. .

.

% !

· 40 ·

§2.5 Gromov-Bishop FGGE Gromov-Bishop mann

, A

ÆPJ$Æ. . (M

n

, g)

AM Rie-

Br (p) = {q ∈ M n |d(p, q) < r}, Sr (p) = {q ∈ M n |d(p, q) = r},

K

Br0 (O) = {x ∈ Rn ||x| < r}.

Bishop

1964

K Ric

M

0,

voln [Br (p)] voln [Br0 (O)]

(2.4)

# r 0 &. Gromov VQ 8><$ Area(∂Br (p)) Area(∂Br0 (O))

(2.5)

JH, R<J$RLÆI&S.. T UWU "#, MOX. (;< Gromov GY)VSÆ Bishop K (2.4), <&WZ). < ) (2.5), N %*" Calabi-Yau [ (M , g) \6AM Riemann 7 Ric 0, p ∈ M , C c = c > 0 *' n

n

M

1

voln (Br (p)) cr

# r 0 &.” L X) Gromov-Bishop Æ O' U. T Gromov-Bishop Æ, ,GS]#. . {(r, Θ)} n OPQY R Z ^, n

r0 Θ∈S n−1 (1)

Expp

Rn → M n , (r, Θ) → Expp (rΘ).

R (M

TJ$RL ∂B (p) $RL, 2 {e , · · · , e } ⊂ T (S (1)) 2/., Θ ∈ S (1). A σ (t) = Exp (tΘ). σ '/ Jacobi n−1

Θ

n

, g)

n−1

Expp (rΘ)

r

n−1

Ji (t, Θ) = (Expp )∗ |tΘ (tei ).

1

Θ

p

Θ

§2.5

!

Gromov-Bishop

· 41 ·

gij (t, Θ) = Ji (t, Θ), Jj (t, Θ), ϕ(t, Θ) = det(gij (t, Θ)),

dA∂Br (p) = ϕ(r, Θ)dΘ, dvol(M n ,g) = ϕ(r, Θ)drdΘ,

:; S MH dΘ

n−1

(1)

:$R.

Area(∂Br (p)) = voln (Br (p)) =

r

ϕ(r, Θ)dΘ,

Θ∈S n−1 (1)

ϕ(t, Θ)dΘdt. 0

Θ∈S n−1 (1)

$=J$R ϕ(t, Θ) U . 2.9 (Gromov-Bishop, Æ) . (M , g) AM Riemann , p ∈ M , {(r, Θ)} M p T Z ^, dvol = ϕ(r, Θ)drdΘ dA = ϕ(r, Θ)dΘ , dΘ S (1) ^ J$RL, !7K & ϕ(t, Θ) (i) Ric 0, JH, "! ϕ(t, Θ) ≡ t # Θ ∈ t S (1), t ∈ [0, t ] %
n

n

∂Br(p)

(M n ,g)

n−1

M

n−1

n−1

n−1

0

0 t0

t0

M

n−1

n

M

n−1

p ∗ tΘ

Θ

i

Ji (t) =

n−1

aji (t)Ej ,

j=1

Rij (t) = R(σ , Ei )σ , Ej . n−1

Ric(σ , σ ) = R

n

n−1

1

i

n−1

n−1

ii (t).

i=1

A A(t) = (aij ), R(t) = (Rij (t)).

n+1

0

% !

· 42 ·

& Jacobi ; Ji + R(σ , Ji )σ = 0,

' A + RA = 0.

(2.6)

&>? A(t) , ϕ(t, Θ) = detA(t). (i)

% tr(R) =Ric(σ , σ ) 0 $, +,

detA(t) tn−1

JH. +, JH. A Ψ(t) = [detA(t)] . _Æ 1

1 n−1

[detA(t)] n−1 t

1 1 [detA(t)] n−1 −1 tr(A A−1 )det(A) n−1 1 1 [detA(t)] n−1 tr(A A−1 ) = n−1 1 = Ψ(t)tr(A A−1 ). n−1

Ψ (t) =

(2.7)

(A II(t) = A A , & (2.6) ' Riccati ;

−1

II + II2 + R = 0.

#*% tr(R) =Ric(σ , σ ) 0 $,

tr(II ) = −tr(II2 ) − tr(R) −tr(II2 ).

(2.7) ::SD'

1 Ψ(t)tr(A A−1 ) n−1 1 [Ψ (t)tr(II) + Ψ(t)tr(II )] = n−1 1 1 2 2 Ψ(t)(tr(II)) − Ψ(t)tr(II ) . n−1 n−1

(2.8)

Ψ (t) =

(2.9)

§2.5

Gromov-Bishop

!

· 43 ·

$ II(t) ∂B (p) L.!>?PF, #* II(t) %*\], R1G L@ t

λ1 (t), · · · , λn−1 (t),

$

1 1 [tr(II(t))]2 = (λ1 + · · · + λn−1 )2 n−1 n−1 λ21 + · · · + λ2n−1 = tr(II2 ),

$ (2.9) D

Ψ (t) 0.

!H

89<

Ψ(t) t

=

Ψ (t)t − Ψ(t) , t2

[Ψ (t)t − Ψ(t)] = Ψ (t)t + Ψ (t) − Ψ (t)

Ψ (0) · 0 − Ψ(0) = 0,

= tΨ (t) 0,

'

Ψ (t)t − Ψ(t) 0,

&

Ψ(t) t

0,

Θ) Ψ(t) JH, IT ϕ(t, JH. t t

"! ϕ(t, Θ) ≡ t # t ∈ [0, t ] Θ ∈ S ⎧ n−1

n−1

0

n−1

(1)

&,

⎨ λ (t) = · · · = λ 1 n−1 (t), ⎩ R(t) ≡ 0,

' II(t) = 1t I, A(t) = tI,

I

^">?. g %

g = dt2 + t2 dΘ2 .

% t ∈ [0, t ] $, g PQ0U, B &. 0

t0 (p)

"[ B

0 t0 (O).

K (i)

% !

· 44 ·

K (ii) (iii) VK (i) D, $++ , V. ,.

" #

_WWX`aX5). `Y {(t, θ)} Z[X R aYX5Zbbc$d. c\ Riemann e3 g = dt + f (t)dθ . (i) d!]e ε = θ , f σ (t) = (t, ε). [ σ fR → M ZgZZ^345?(_`fC > d(σ (t ), σ (t )) = |t − t |.)∂F (t, θ ) ZgZ\345 σ a5 Jacobi 8? (ii) f F (t, θ) = (t, θ). [

∂θ

∂ ∂F ∂F ∂ ∂F ∂F (iii) _W ∂t ∂t , ∂θ , [ ∇ ∂θ∂ Zg638 ∂θ∂ []? , ∂t ∂θ ∂θ ∂ ∂ (iv) `Y ∇ = λ(t) , ^h λ(t). ∂θ ∂θ ∂F (v) _b = J(t) Z σ (·) a5 Jacobi 8, c`d % Jacobi ij J (t) + ∂θ K(t)J(t) = 0, e_W J (t) C> 2

1(

2

2

2

0

ε

1

ε

2

1

ε

2

ε

2

0

θ0

∂ ∂t

∂ ∂t

θ0

(t,θ0 )

K(t) = −

2(

f (t) . f (t)

kafX5). bgc\kafX S (1) a5f$dc 2

⎧ ⎪ ⎪ ⎨ x = sint cosθ, y = sint sinθ, ⎪ ⎪ ⎩ z = cost.

(i)

[ S (1) → R afX?Be3ZgZ 2

3

g = dt2 + (sint)2 dθ2 .

hiZ S (1) ?Be35?(_`fdj 1 5gh.) 3 (_Wkl[X5). mkl[X R 5d &$ d 2

(ii)

2

⎧ ⎨ x = tcosθ, ⎩ y = tsinθ.

e^h?Be3 g = dx + dy = dt + f (t)dθ e5 f (t) l_W R 5. 4 (nij H 5). fgo D = {(x, y)|x + y < 1} 5 Poincar´e e3 g = 4(dr + r dθ ) 4(dx + dy ) = , hk r = x + y . (1 − x − y ) (1 − r ) 1 + r e −1 , C> a t = log (i) f r = e +1 1 − r 2

2

2

2

2

2

2

2

2

2

2

2 2

2

2

2 2

2

2

2

2

2

t t

g = dt2 + (sinht)2 dθ2 ,

2

ij

· 45 ·

hk sinht = e −2e . (ii) dj 1 _W Poincar´e e35. ¯ , g¯) Zbqm5 Riemann pj, M ⊂ M ¯ bM ¯ 5 (Riemann ' pj5). Y (M ¯ ¯ e5lm'pj. `! ∇ Z (M , g¯) 5 Riemann nr. ¯ Y, η = ∇ ¯ X, η (i) n X Y Z M a5s638t η Zbo638u, [Zgk ∇ po? ¯ v∈T M ¯ . bgf v q`63 v 5o73. !l63mr (ii) v p ∈ M ⊂ M jw t

−t

m

n

m

m

m

n

n

X

m

m

p

Y

⊥

¯ X Y )⊥ . II(X, Y ) = (∇

e[sw II(X, Y ) = II(Y, X) `ck M a5s638po? ¯ , g¯) pno X Y pp[X5, K(X, Y ) b (M ¯ Y ) b (M (iii) f K(X, 5qX. sCvt5 Gauss tw. n

m

¯ K(X, Y ) = K(X, Y)+

, g)

qr

II(X, X), II(Y, Y ) − II(X, Y )2 , X, XY, Y − X, Y 2

hk X u[]n Y . 6 (klije5 Riemann ' pj5). `Y M pj, Ff D → M , 2

n

→ Rm

Zklij5 Riemann '

n

(u, v) → F (u, v)

Zbrxuy, X = F Y = F . [ u

n

v

K(X, Y ) =

(Fuu )⊥ , (Fvv )⊥ − (Fuv )⊥ 2 Fu 2 Fv 2 − Fu , Fv 2

Zgvsnrx F 5wm, hk (ξ) vq ξ ∈ T R = T M ⊕ N M 5o73. 7 (tz{xX5). `Y R eX M otz (u, v) → h(u, v) 5{xu!, v ⊥

p

3

⎛

m

p

n

n

p

2

M 2 = {(u, v, h(u, v))|(u, v) ∈ R2 }.

edj 6 _W M 5. ⎝_`fewh 2

⊥

(Fuu ) , (Fvv )

⊥

huu hvv − (huv )2 − (Fuv ) = 1 + h2u + h2v ⊥ 2

⎞ (−hu , −hv , 1) ⎠ N= √ . 1 + h2u + h2v

|yX5). c\nX M = {(u, v, u + v )|(u, v) ∈ R }. _WxXfy p = (u, v, u + v ) z5. (_`fdj 6.) (i) v d(O, p) q`y p zy O f M e5wx, [Zgk 2

8(

2

2

2

2

2

r = d(O, p) = O(u2 + v 2 )

2

% !

· 46 ·

po, hk O(u + v ) q` (u + v ) kq}5{{. (ii) n r → ∞ u, y_ K| . Zgk 2

2

2

2

p

lim K(p) = 0

K 0 po? 9 (k|nX5). f M eC>

r→∞

2

= {(u, v, u2 − v 2 )|(u, v) ∈ R2 }.

vh K = h1 +hh −+ hh uu vv 2 u

uv 2 v

,

K = −4 + O(r 2 )

K < 0, hk r = d((0, 0, 0), (u, v, u − v )). 10 ( ! 5rd). `Y (M , g) Zbkz~qm Riemann X, lt{k|}. c\fZy p z5~zuyf Exp f R → M , 2

2

2

2

2

(r, θ) → Expp (rθ),

hk {(r, θ)} Z R 5|$dq`. (i) d! θ , [5 σ fr → Exp (rθ ) ZgZ345? ∂Exp . e[638 J ZgZ σ a5 Jacobi 8ltd % J(0) = (ii) f J(r) = ∂θ 0, J (0) = 1 J (r), σ (r) = 0? (iii) d Rauch ! C> J(r) r. (iv) f S (r) = {q ∈ M |d(q, p) = r}, L(r) b S (r) 5{e. d (iii) 5ghy_ 2

0

θ0

p

0

p

θ0

(r,θ0 ) θ0

2

p

p

L(r) =

5t}.

2π 0

J(r, θ)dθ

! 5rd). f (M , g) bZb|~qm5 Riemann X, t{k|}. `Y S (p ) = {q ∈ M |d(q, p ) = r}, L(r) b S (p ) 5{e, d Gromov-Bishop ! ~hZ b L(r) 5a}. 2

11 (Bishop r

0

2

0

r

0

,

.

σ [0, +∞) → M (M , g) σ (t) ≡ 1,

§3.1

n

n

d(σ(0), σ(t)) = |t|

(3.1)

?

Æ. (M , g) = S (1) ⊂ R , S (1) = {(x, y, z) ∈ R | x + y + z = 1}. σ(t) = (cos t, sin t, 0). σ (t) ∈ N M , ∇ σ = [σ (t)] = 0, σ . σ (t) ≡ 1, σ(2π) = σ(0), 2

2

3

2

3

2

σ

T

σ(t)

2

2

2

d(σ(2π), σ(0)) = 0 < 2π.

Æ (3.1) t . , Exp B (O) → M , Gauss 1.23 d(σ(t), σ(0)) = |t| t . , t d(σ(t ), σ(0)) < t . a > 0, p

n

ε

0

0

0

d(σ(t0 + a), σ(0)) d(σ(t0 + a), σ(t0 )) + d(σ(t0 ), σ(0)) < t0 + a.

Ev = {t | d(σ(t), σ(0)) = t, σ (0) = v, v = 1}

[0, +∞) [0, t ] t . E σ(0) "#. "# # . 0

0

v

= [0, t0 ],

! σ(t ) 0

$! %& '!

· 48 ·

σ [a, b] → M Riemann , σ(a) = p, σ(b) = q, " {J(t)} σ #($) Jacobi * J(a) = J(b) = 0, b > a. ! p q σ (! q p σ # #). %" #+#. 3.2(Jacobi ) (M , g) Riemann , σ(t) = Exp (tv), v = 1, q = Exp (t v) " q p σ # , , n

3.1

n

p

p

0

d(σ(0), σ(t0 + ε)) < L(σ|[0,t0 +ε] )

ε > 0 . $ ( &'(% ) 3.2 . &' (% & ) σ ϕ '( )-# p q, σ = ϕ = 1() ! 9). *"* ' ϕ(t − ε) σ(t + ε) . ψ. #&

ϕ σ , Æ ϕ (t ) = σ (t ). ,

0

0

0

0

$9 L(ψ) < d(ϕ(t0 − ε), q) + d(q, σ(t0 + ε)) = ε + ε = 2ε,

L(ψ) /+ ψ 01. +, 2, L(ψ

ϕ|[0,t0 −ε] ) < 2ε + (t0 − ε) = t0 + ε.

#

L(σ|[0,t0 +ε] ) = t0 + ε > L(ϕ|[0,t0 −ε]

σ|[0,t0 +ε]

'-# σ(0) σ(t

0

+ ε)

ψ),

.3. -%,

d(σ(t0 + ε), σ(0)) < t0 + ε.

§3.1

%4$567.,

· 49 ·

/0 3.2 ( , &1' 8 {β (·)} s

−1s1

(i) β0 = σ|[0,t0 +ε] ,

s ,

(ii) βs (0) ≡ σ(0) βs (t0 + ε) = σ(t0 + ε), ∂βs (t) , σ (t) ≡ 0, (iii) t , ∂s s=0 d2 L(βs ) (iv) c = < 0, ds2 s=0 n βs [0, t0 + ε] → M , L(βs ) βs .

9/ 01 ) # β = σ , : ∂L(β ∂s (iv) , s

0

s=0

= 0,

(

c L(βs ) = L(σ) + s2 + o(s3 ) < L(σ|[0,t0 +ε] ), 2

c < 0 (iv) 0, |s| s = 0. /( (iv), & ;:<+2. 3 α [a, b] × [−1, 1] × [−1, 1] → M

n

,

(t, v, w) → α(t, v, w) ∂α ∂α 45)-, T = ∂α W = , V = . 6, 01 720 ∂t ∂v ∂w b

L(v, w) = a

*

∂α (t, v, w) dt = ∂t

∂L (v, w) = ∂v

a

b

b

a

T dt.

∇T V, T

dt, T

b ∂ ∂ 2 L(v, w) ∇T V, T

= dt ∂w∂v ∂w a T b ∇W T, T

∇W ∇T V, T + ∇T V, ∇W T

= − ∇T V, T

dt. T T 3 a

T

(0,0)

≡ 1, ∇T T

(0,0)

≡ 0 ≡ T, W

(0,0)

≡ T, V

(0,0)

.

(3.2)

$! %& '!

· 50 ·

+ (3.2) =8, *, 2 ∂ 2 L = ∂v∂w (0,0)

b

{ −R(W, T )V, T + ∇T ∇W V, T

a

+ ∇T V, ∇T W − ∇T V, T ∇T W, T } dt

b

{ ∇T V, ∇T W − R(W, T )V, T +T ∇W V, T −T V, T T W, T } dt b b { ∇T V, ∇T W − R(W, T )V, T } dt. = ∇W V, T + =

a

a

6, <>

a

b

{ ∇T V, ∇T W − R(W, T )V, T } dt

I(V, W ) = a

V W !. &'( , +"/',.45 ϕ| /, -

',".45+. a = t < t < · · · < t = b V , W 45, *+

[0,t0 −ε]

0

1

k+1

ψ.

9

[ti ,ti+1 ]

∇T V, ∇T W = T ∇T V, W − ∇T ∇T V, W

.=8?

b

I(V, W ) = a

=

k

{ ∇T V, ∇T W − R(W, T )V, T } dt ∆ti (∇T V ), W −

i=1

(3.3)

b a

∇T ∇T V + R(T, V )T, W dt,

∆ti (∇T V ) = lim ∇T V − lim ∇T V.

&@, V |

[ti ,ti+1 ]

tti

Jacobi *, I(V, W ) =

tti

∆ti (∇T V ), W .

(3.4)

i

# ,

b b ∂ 2 L = ∇W V, T + { ∇T V, ∇T W − R(T, V )T, W } dt ∂v∂w (0,0) a a b = ∇W V, T + I(V, W ). a

(3.5)

§3.1

%4$567.,

· 51 ·

+ (3.3)∼(3.4), * 3.2 . 3.2 # σ(0) σ(t ) σ , ($) Jacobi * J J(0) = 0 = J(t ) " J(t), σ (t) ≡ 0. ',.45 Jacobi * ⎧ ⎨ J(t), 0 t t , ˆ = J(t) ⎩ t t . 0, σ #.* $/0:* {E(t)} 0

0

0

0

t0

E(t0 ) = −∇σ J t .

(3.6)

0

1* 45 η [0, +∞) → R ⎧ ⎨ 0, t=0t=t η(t) = ⎩ 1, t = t . ; 1' σ #',.450:*

0

+ε

,

0

ˆ + λη(t)E(t). V (t) = W (t) = J(t)

(3.7)

β(t, s) = Expσ(t) {sV (t)}.

(3.8)

3 023 ( (i) β(·, 0) = σ, (ii) β(0, s) ≡ σ(0) (iii)

β(t

+ ε, s) ≡ σ(t0 + ε), L(s) = L(β(·, s) ), 0

[0,t0 +ε]

dL = 0. ds s=0

3, * λ

d2 L < 0. ds2 s=0

< t, β(t, ·) , ∇V W = ∇V V ≡ 0.

(3.9)

$! %& '!

· 52 ·

23 V W (3.2). (3.5) (* a = 0 b = t

0

+ ε)

d2 L = I(V, V ). ds2 s=0

14+ A I(·, ·) 1(= (3.3)∼(3.4) I(V, V ) = I(Jˆ + ληE, Jˆ + ληE) ˆ + 2λI(Jˆ, ηE) + λ2 I(ηE, ηE) = I(Jˆ, J) (3.4)

ˆ ηE(t0 ) > +O(λ2 ) = = = = = 0 + 2λ < ∆t0 J, = −2λ∇σ J2 (t0 ) + O(λ2 ).

2J (t ) * 0 < λ < |I(ηE, , > ηE)|

0

2

d2 L = I(V, V ) < 0, ds2 s=0

(3.9) , 3.2 . B. 230 #?45. 3.3 q = Exp v, σ(t) = Exp (tv). p q σ "@ (Exp ) | !. p q σ , J ($) Jacobi 0:*, J(0) = 0∂β= J(1). # {J(t)} ($), J (0) = 0. 3 β(t, s) = Exp [t(v +sJ (0))], J(t) = ∂s (t, 0), " p

p

p ∗ v

"5?

p

0 = J(1) = (Expp )∗ v J (0).

ker(Exp ) = 0 (Exp ) !. C, ξ = 0 (Exp ) ξ = 0, 3 J (0) = 0,

p ∗ v

p ∗ v

p ∗ v

β(t, s) = Expp [t(v + sξ)].

(t, 0) σ # Jacobi *, J(t) = ∂β ∂t

J(0) = 0, J(1) = (Expp )∗ v ξ = 0.

# ξ = 0, J (0) = ξ = 0, {J(t)} ($). # 6* p q σ . B.

§3.2

Ricci

#$

67 Myers '!

3.4

v ∈ S

M

n

(M p0 M

n

n

· 53 ·

Riemann

, p ∈ M . t > 0 M , (Exp ) ! 0 < t t ,

, g)

⊂ Tp0

n

0

n

p0 ∗ tv v

0

v

0

sup {d(q, p0 )} t0 .

* q ∈ M , *"* ' p ? q .3 σ [0, l] → . σ(0) = p , σ(l) = q, σ (0) = v, v = 1 Exp (tv) = σ(t). q∈M n

n

0

0

p0

d(p0 , q) = l.

(3.10)

l t . l = d(p , q) > t , σ| #Æ p # ( # t , t t < l, (Exp )| !. 23 3.3 σ(t ) σ(0) = p ). 4+ 3.2, ∀ ε > 0, 0

v

v

0

0

0

0

[0,l]

tv v

p0

v

0

d(σ(tv + ε), σ(0)) < tv + ε.

&@

d(p0 , q) = d(σ(0), σ(l)) < tv + ε + (l − tv − ε)

7 (3.10) 8D.

= l,

d(p0 , q) l t0 .

B.

9:A Myers 8BEC D + Gromov-Bishop FE#D 3.4 3 7(. 3.5(Myers) (M , g) Riemann , Ric (n−1), , g) 9F §3.2 Ricci n

(M

n

(M n ,g)

Diam(M n , g) = sup{d(p, q) | p, q ∈ M n } π.

3.4, % &5 v ∈ T M , v = 1, 3 σ(t) = Exp (tv), σ| &; p #. /3 #, p GG ' A {(r, Θ)} Exp R → M , p

p

[0,π]

p

n

n

(r, Θ) → Expp (rΘ),

n

$! %& '!

· 54 ·

Θ ∈ S

n−1

(1).

3 {e , · · · , e 1

n−1 }

T

Θ (S

n−1

(1))

:HI.

gij (t) = gij (t, v) = (Expp )∗ tv (tei ), (Expp )∗ tv (tej ) ,

ϕ(t) = ϕ(t, v) =

# Ric(M

det(gij (t, v)),

dvol(M n ,g) = ϕ(t, Θ)dΘdt.

n

, g) (n − 1),

Gromov-Bishop FE 2.9 ? ϕ(t, Θ) (sin t)n−1

HJ. sin π = 0, * t ∈ [0, π] ϕ(t , v) = 0, 23 3.3 3.4 ; (Exp ) ( !. d(p, q) = l, v

v

p ∗ tv v

d(p, q) = l tv π.

B. ;<< ( I8BECJKK== 1975 >>??+&@ALML/ 7M(, + GromovBishop FEN ) B3@(. 3.6(>??) (M , g) O Riemann Ric (n−1), Diam(M , g) = π. (M , g) Æ S (1) → (R ,g ) . * M &# p , q §3.3

n

n

(M n ,g)

n

n

n

0

n+1

0

d(p0 , q0 ) = Diam(M n , g) = π.

3 # Ric

Br (p) = {q ∈ M n d(q, p) < r}, ˆ π (0) = {(x1 , · · · , xn+1 ) ∈ S n (1) xn+1 > 0}. B 2 (M n ,g)

(n − 1),

Myers ( 3.5) Diam(M n , g) π.

#

Bπ (p0 ) = Bπ (q0 ) = M n ,

0

§3.4

Calabi-Yau

E# Ric

APCDÆQ

· 55 ·

vol(M n , g) = vol[Bπ (q0 )] = vol[Bπ (p0 )].

M

(n − 1)

Gromov-Bishop FE, vol(B π2 (p0 )) vol(Bπ (p0 )) , ˆ π (0)) vol(S n (1)) vol(B 2

ˆ π (0)) vol(B vol(B π2 (p0 )) 1 2 = . n n vol(M ) vol(S (1)) 2

@ B (p ) S (1). (M +)F, -* π

"5 #

d(p0 , q0 ) = π,

n

0

n

, g)

S

n

(1).

vol(B π2 (q0 )) 1 . vol(M n ) 2

B π2 (p0 ) ∩ B π2 (q0 ) = ∅. , vol(M n , g) vol(B π2 (p0 )) + vol(B π2 (q0 )) 1 1 vol(M n ) + vol(M n ) 2 2 = vol(M n , g).

ˆ π (0)) vol(B vol(B π2 (p0 )) 1 2 = = . n vol(M ) vol(S n (1)) 2

1>4+ Gromov-Bishop , * M

n

= Bπ (0)

S

n

(1).

B.

§3.4 Calabi-Yau BRGHST

D U(N Ricci CD8LM.

Æ. * O8 M = S (1) × R. M #"*PIO81: g = g ⊕ g , g S (1) #V 1 A + 1:. 6 (M , g) U (N Ricci . 3 B (p ) QF r p G, p = (q , 0), q ∈S (1). E3F B (p ) ⊂ S (1) × (−r, r), B (p ) D8 R ( J0 n

1

0

1

0

n

n−1 r

n−1

n−1

r

n

0

0

0

n−1

vol(Br (p0 )) Cn−1 r.

0

r

0

0

(3.11)

$! %& '!

· 56 ·

G#5 Æ HS, Calabi IJSK/ 7D8LM. 3.7(Calabi-Yau) (M , g) (O U(N Ricci Riemann . 3 B (x) QF r x GC, B (x) D8 & ;(J0 n

r

r

vol(Br (x)) c(x)r.

(3.12)

W? Gromov 0 3.7 . KL §2.5 GromovBishop T,. 5; L . , 3.8(Gromov) f , fˆ &:, " ffˆ U X 0 < r < R

r

R

f (t)dt 0

r

fˆ(t)dt

f (t)dt r

r

(t) 3 q(t) = ffˆ(t) U X. YV?

0

R

r

fˆ(t)dt =

f (t)dt

R

fˆ(t)dt q(r) 0 r r R q(t)fˆ(t)dt fˆ(t)dt

r

0 r

fˆ(t)dt

= 0

r

r

fˆ(t)dt

0

n

, g)

fˆ(t)dt r

f (t)dt,

0 r

(M

0

R

R

R

f (t)dt

3.9

fˆ(t)dt

r

. B.

r

fˆ(t)q(t)dt

#,

(3.13)

fˆ(t)dt

0

r

#$

.

R

f (t)dt rR fˆ (t)dt r

U(N Ricci , 3 B (x) r

vol(BR (x)) − vol(Br (x)) vol(Br (x)) , Cn rn Cn [Rn − rn ] vol(Br (x))

rn [vol(BR (x)) − vol(Br (x))]. − rn

Rn

§3.4

Calabi-Yau

APCDÆQ

· 57 ·

/ 3.9, * 3.7 . Calabi-Yau

-./0 (

3.7)

x SQ9 σ [0, ∞) → M

n

# (M , g) (O , σ(0) = x " n

d(σ(t1 ), σ(t2 )) = |t1 − t2 |.

3x

k

= σ(k).

(3.14)

Z[ §2.5 Gromov-Bishop D8FE, vol(Bk−1 (xk ))

(k − 1)n vol(Bk+1 (xk )). (k + 1)n

Z[ 3.9, + (* R = k + 1, r = k − 1) vol(Bk−1 (xk ))

(k − 1)n [vol(Bk+1 (xk )) − vol(Bk−1 (xk ))]. (k + 1)n − (k − 1)n

(3.15)

(3.15) , (8)! 10)

$ 10 B1 (x) = B1 (x0 ) ⊂ [Bk+1 (xk ) − Bk−1 (xk )]

(3.16)

B2k (x) = B2k (x0 ) ⊃ Bk−1 (xk ).

(3.17)

$! %& '!

· 58 ·

;

( (3.16)∼(3.18))

,

vol(B2k (x0 )) vol(Bk−1 (xk )) (3.18)

(k − 1)n [vol(Bk+1 (xk )) − vol(Bk−1 (xk ))] n n (3.16) (k + 1) − (k − 1) (k − 1)n vol(B1 (x0 )) n n (3.17) (k + 1) − (k − 1)

(3.18)

2c(x0 )k = c(x0 )2k,

c(x ) M\ x W. 13 r = 2k, (3.19) 0

0

vol(Br (x)) c(x)r.

B.

(3.19)

3.8 LS/, @%" σ # Riccati X . Gromov F Calabi +M N. IJN) [Y3]. Z[I

O 1 ]L, O [Y3] @^PK 2 QR1LMS#, PT(_ 3, ^P/Q? *BY2, +QX`aQ?R3. Calabi

4

1(

5

STÆZ%). UUVVbcW6 [\STÆ z2 x2 + y 2 + = 1. a2 b2

Wd

⎧ ⎪ x = a sin ϕ sin θ, ⎪ ⎪ ⎪ ⎨ y = a sin ϕ cos θ, ⎪ ⎪ ⎪ ⎪ ⎩ z = b cos ϕ,

XSTÆZ Riemann ]eY {(ϕ, θ)} 7^Z[f\_

g = h21 (ϕ)dϕ2 + h22 (ϕ)dθ2 .

]` h (ϕ) h (ϕ). (ii) 8 6g t = [b cos u + a sin u]

(i)

1

ϕ

2

2

2

2

0

2

1 2

du,

^X

g = dt2 + f 2 (t)dθ2 ,

Y f (0)=0 f (t ) = 0, 0 t t . 0

0

_Z!

· 59 ·

`_Z59ZZ 1, σ (t) = (t, ε) [\abC. ]^Xc θ = ε = 0 ^, 6C σ ˆ dϕ → (0, a cos ϕ, b sin ϕ) Y8he_6g t = b cos u + a sin udu i[a\abC. ˆ ˆ (iv) J(ϕ) = (a sin ϕ, 0, 0) [ σ ˆ `Z Jacobi f. Qa J(0) Jˆ(π). (v) d p = (−b, 0, 0) q = (b, 0, 0). b p q [g ? (1) = {x ∈ R ||x| = 1}. [cj p ∈ S 2(hcbÆ`Z Jacobi f). bc S cakhcid] v ∈ T (S (1)). (i) bdσ(t) = (cos t)p + (sin t)v [ga\abC? (ii) d u ⊥ v u = 1, u ∈ T (S (1)). bc (iii)

ε

ϕ

2

2

2

2

0

0

n−1

p

n

n−1

(1),

n−1

p

b

n−1

β(t, ε) = (cos t)p + (sin t)[(cos ε)v + (sin ε)u], (t, 0) [g[ σ `Zak Jacobi f? b J(t) = ∂β ∂ε (iii) Qa J(0) J(t). (iv) ]` p Z %. 3(jd67le`Z %). mefk Riemann le (M , g) ngjdo67 K 0, Yme σ d[0, +∞) → M [a\nghcd]ZabC. [ σ `Z Jacobi ff: J(0) = 0, J (0), σ (0) = 0 J (0) = 1. (i) me {J(t)} J(t) hlp'!]`g t Z[q. (ii) bih% p = σ(0) [gg %? 4(iiÆ`Z %). d M = {(x, y, z) ∈ R | z = x + y } [ R 9 ZiiÆ. (i) ]`mj%`nZjgabC. (ii) me σ d[0, +∞) → M [a\mj%`nZnghcd]ZabC, b[gg n

Mn

n

t0

2

3

2

2

3

2

d(σ(s), σ(t)) = |s − t|?

bj% O = (0, 0, 0) Y M 9[gg%? 5(kklrsZ ). d S (1) = {x ∈ R | |x| = 1} [ R (i) bc[ ml F dS (1) → S (1), 2

(iii)

n

n+1

n

n+1

9ZhcbÆ

.

n

x → −x,

b F [g[otml? ]` F = F ◦ F . (ii) kklrs[ S (1) Zgrs RP = S (1)/Z , nu Z [` F m_Z ; n. ]` RP Z . 6(pklrsZ ). bc Hopf oÆq 2

n

n

n

2

n

π

S 1 → S 2m+1 (1) → CP m

2

$! %& '!

· 60 ·

a
d

Fθ Cm+1 → Cm+1 , (z1 , · · · , zm+1 ) → (eiθ z1 , · · · , eiθ zm+1 ), √ ,zj = xj + −1yj ,θ ∈ [0, 2π].

nu i = √−1,z ∈ C [p_ (1) `Zakotml? (i) b F [g[ S (ii) ov CP = S (1)/S ,S p {F } m_Z ; n, cp CP 9 cjw% p = [z , · · · , z ] q = [ˆ z , · · · , zˆ ], b[gfq]r S (1) 9 Zw% P ∈ π (p) π Q ∈ π (q) qs d(P, Q) , nu π dS (1) → CP [gml? 2 (iii) rs[ CP Z ? 7(xtn ). e Γ = π (M ) [le M Zxtn. rme M ngak Ricci 6 7 (n − 1) Z]e g. ˜ p M Zsguvrs. tpuv ˜ → M ple M Zsguvml, M (i) d π dM ml π [ywu7tu, jqUUbc M˜ `Zvw]e j

2m+1

θ

m

1

2m+1

m+1

1

1

1

m

θ Cθ2π

2m+1

m+1

−1

2m+1

−1

m

m

n

1

n

n

n

n

n

n

n

n

X, Y g˜ = π∗ X, π∗ Y g .

bov Ric c [gvz Ric c? ˜ , g˜) |Γ | = |π (M )| n! (n − 1), ]` Diam(M , g) 4Diam(M (ii) ov Ric wxsZ{Z. nu Diam(M , g) xx Riemann leZ . 8(Milnor '!). me (M , g) [k|Z Riemann leY Ric 0. (i) ov B (p) = {q ∈ M | d(p, q) < r}, h Gromov-Bishop lp'!]` vol(B (p)) Z `q. ˜ , g˜) p (M , g) Zsguvrs, yg g Zvw]e g˜. ^Xxtn Γ = π (M ) (ii) d (M zng]_p n Zy{7, nu n = dimM . 9(Heisenberg le Milnor '!Zzh). bc Heisenberg n (M n ,g)

˜ n ,ˆ (M g)

n

(M n ,g)

n

1

n

n

n

r

n

(M n ,g)

n

r

n

1

n

n

yZ;n

⎧⎛ ⎪ ⎪ ⎨⎜ 1 3 ˜ M = ⎜ ⎝ 0 ⎪ ⎪ ⎩ 0

⎧⎛ ⎪ ⎪ ⎨⎜ 1 Γ = ⎜ ⎝ 0 ⎪ ⎪ ⎩ 0

⎫ ⎪ ⎪ ⎬ ⎟ 3 ⎟ y ⎠ (x, y, z) ∈ R ⎪ ⎪ ⎭ 1

x

z

1 0

k 1 0

⎞

⎞

⎟ ⎟ n ⎠ k, m, n 1

m

^X}zn Γ ng{Æz|,y{7 (r ). (ii) h Milnor '!|X!Æ Heisenberg le M 67Z]e. (i)

p{_⎪⎪ .

4

3

⎫ ⎪ ⎪ ⎬

˜ 3 /Γ =M

⎭

|~}}cjngj} Ricci

_Z! ov (M , g) [k| Riemann le~Yng}oÆ67 K −1. d (M˜ ZsguvrsY~kvw]e g˜, B (˜p) = {˜q ∈ M˜ | d(˜p, q˜) < r}. p)) g = _y{7. (i) hlp'!^X, vol(B (˜ (ii) bxtn Γ = π (M ) [g~g = _y{7? 10.

Mn

· 61 · n

Mn

r

r

1

n

n

n

, g˜)

[

Æ

§4.1

Cheeger

. Cheeger

,

.

(M n , g)

4.1

, p ∈ M n , v ∈ Tp M n ,

Riemann

v = 1.

cv = sup{t | d(p, Expp (tv)) = t}.

(1) p

C (p) = {cv v | v ∈ Tp M n , v = 1, cv < +∞}. (2) p

Mn

ˆ C(p) = Expp C (p).

Æ !" ,

.

(M n , g) = S n (1) → (Rn+1 , g0 )

(1)

,p

,

p

Mn

.

n

(M , g)

(2)

S

n−1

Riemann

n

p

n

p ∈ H , v ∈ Tp H ,

C (p)

, ˆ C(p)

n

cv = +∞,

ˆ q ∈ C(p).

4.2

(1)

(2)

Tp M n

C (p)

.

(M n , g) = (Hn , g−1 )

(3)

, p ∈ M n.

σ(t0 )

.

q = σ(t0 ) σ

p

p

σ

,

;

σ

ϕ

p

L σ|[0,t0 ] = L(ϕ) = d(p, q).

q

K ≡ −1. .

#Æ!"$ #

§4.1

q = σ(t0 )

σ(t0 + εi )

σ|[0,t0 ]

ϕi (

,

· 63 ·

εi → 0

%!" p

11)

L(ϕi ) < t0 + εi .

q

p σ , $ t

%

0

= d(p, q)

11

d(p, σ(t0 + εi )) < t0 + εi .

(4.1)

, d(p, σ(t0 + εi )) d(p, σ(t0 )) − εi

! p σ(t

0

+ εi )

(4.2)

= t0 − ε i .

# &

σvi [0, li ] → M n , t → Expp (tvi ),

&

vi = 1, σvi (li ) = σ(t0 + εi ),

(4.1)∼(4.3)

'

d(p, σ(t0 + εi )) = li .

(4.3)

t 0 − ε i l i < t0 + ε i .

(4.4)

, lim li = t0 .

, ' v

i→+∞

S n−1

→ vˆ.

σˆ(t) = σ (t) = Exp (tˆv), (4.4) ij

v ˆ

p

L σvˆ [0,t0 ] = L(σ).

(4.5)

( !"$Æ)(

· 64 ·

)!*" ) #$ ) ' Æ ! *+ .

1. σ ˆ = σ.

4.2(2) (Expp )∗

2. σ ˆ = σ.

.

t0 σ (0)

(Expp )∗ |t0 σ (0)

+ Æ ! , .

"#$

Gauss

-, &. ' / "$ -' .

0 % & () & * #$ Æ$/0 . $'' -,& ( ! # ( &

#$ 1 )# & *+ %

d(p, σ(t0 + ε)) = t0 + ε,

ε>0

,

.

σ(t0 ) ˆ C(p)

,

n

4.3

(M , g)

σ(0)

σ

.

p

.

.

, p ∈ M n,

Riemann

Mn

p

Injp (M n ) = inf{cv | v ∈ S n−1 ⊂ Tp M n }

= sup{t | Expp Bt (O) → M n

4.2,

.

n

4.4

(M , g)

p

}.

, p ∈ M n, q

Riemann

Mn

p

,

(1)

p

q

σ,

p

q

σ

;

ψ [0, 2l] → M n

(2)

ψ(0) = ψ(2l) = p, ψ(l) = q

d(p, q) =

1 L(ψ). 2

σ, ϕ [0, l] → M n

4.2,

L(ϕ) = L(σ) = d(p, q).

2 ,4

ϕ (l) = −σ (l).

ϕ (l) = −σ (l),

134 &.

w ∈ Tq M n , w = 0

.

(4.6)

+, $ ϕ (l) = −σ (l) ,

⎧ ⎨ w, −σ (l) > 0,

(4.7)

⎩ w, −ϕ (l) > 0.

α(s) = Exp (sw), p q σ ϕ , (Exp ) | (Exp ) | 5 Æ!-2, 23,, 6 {σ } {ϕ }, σ = σ, ϕ = ϕ, & σ ϕ ! p α(s) ( 12). 4.7 8- (4.7) 4 p ∗ lσ (0)

q

s

0

s

s

p ∗ lϕ (0)

s

0

#Æ!"$

§4.1

· 65 ·

⎧ d ⎪ ⎨ < 0, L(σs ) ds s=0 ⎪ ⎩ d L(ϕ ) < 0, s ds s=0

(4.8)

max{L(σs ), L(ϕs )} < l

(4.9)

9/ -, 1 . s>0

.

12

L(ϕs ) L(σs ),

% + , σs

p

qˆs ,

9/ (M

d(p, qˆs ) max{L(σs ), L(ϕs )} < l.

-, . & . / %

s>0

n

, g)

,

q = α(0)

Riemann

,

p

M

.

.

n

Inj(M n ) = Inj(M n , g) = inf n {Injp (M n )}. p∈M

& ." $

$5&

l(M n , g) = inf{L(ψ) | ψ S 1 → M n , ψ (0) = 0, ∇ψ ψ ≡ 0}.

Klingenberg

4.4

(M n , g)

4.5 (Klingenberg)

. Riemann

n

(M , g)

n

Inj(M , g) min

0 Mn

,

&.6$'+ &

1 π n , l(M , g) . |K0 | 2

p0 ∈ M n

Injp0 (M n ) = Inj(M n ).

&

, Æ K

c SM n → R

{+∞},

Mn

K0 ,

( !"$Æ)(

· 66 ·

SM !13 ,, & SM n

cp,v = sup{t | d(p, Expp (tv)) = t}.

(p, v) → cp,v n

= {(p, v) | p ∈ M n , v ∈ Tp M n , v = 1}, ˆ 0) cp,v , q0 ∈ C(p

!1- 0

d(p0 , q0 ) = Injp0 (M n ).

2."

4 ( %# (78 9:: & {p0 , q0 }

4.4

(i)

p0

q0

(ii)

.

p0

q0

ψ S1 → M n,

$ ) 43 43 (i)

π

|K0 |

,

1 L(ψ) = d(p0 , q0 ). 2

Rauch

;; (

,

'

+

0 t<

π d(p0 , q0 ) . |K0 |

,

n

Inj(M ) d(p0 , q0 ) min

/

KM n K0 )

1 π , L(ψ) . |K0 | 2

.

< =< /04Æ. =4 5> :>?6? ' @ 12 $: @ & §4.2 Cheeger

.

Riemann

,

.

Fermi

.

σ S1 → M n,

9:: Mn

. Mn

4.5

t → σ(t)

0A σ(t)

Nσ(t) M n = {ξ ∈ Tσ(t) M n | ξ, σ (t) = 0},

B!"$CA )2A7BD §4.2

'

· 67 ·

Cheeger

Fermi

Nσ M n =

&

F Nσ M n → M n , (σ(t), ξ) → Expσ(t) ξ.

$ 9/ ξ

Nσ(t) M n .

t∈S 1

-'$'&*

,

, F∗ |(σ(t),ξ)

Æ !-2.

Dε (σ) = (σ(t), ξ) ∈ Nσ M n | t ∈ S 1 , ξ < ε ,

() & *, E σ Fermi @8. Fermi @8FD /3 $' +. 4.6 Σ Riemann (M , g) 99: , q ∈ Σ , ϕ &[0, b] → M !" Σ q #,

& "C

F Dε (σ) → M n

k

k

n

n

w ∈ Tϕ(0) Σk

k

ϕ (0), w ≡ 0

-,

(4.10)

.

&

β [0, b] × (−ε, ε) → M n , (t, s) → β(t, s)

Æ !" # 2.7 8 -

β(0, s) ∈ Σk , β(b, s) ≡ q, β(t, 0) = ϕ(t), V (t) =

,

Σ

k

q

∂β (t, 0), ∂s

,

b ∂ 1 d [L(β(·, s)] V (t), ϕ 0= = (t) − V, ∇ϕ ϕ dt ds s=0 0 ϕ (0) ∂t =

1 ϕ (0)

=−

[ V (b), ϕ (b) − V (0), ϕ (0) ]

V (0), ϕ (0)

, ϕ (0)

ϕ (0), V (0) ≡ 0

V (0) ∈ Tσ(0) Σk

-, / .

.

ϕ

( !"$Æ)( &*

· 68 ·

#$ :4 4.6

,

Æ & *. . Fermi

(M

Riemann

& &* ;; $' :>? Æ

G4H ,

.

n

(M , g)

Riemann

n

KM n −1. Diam(M , g) d0

: 5

, g), Fermi

F Nσ M n → M n

4.7 (Cheeger, Heintze-Karcher) n

n

vol(M , g) C0 ,

M

,

n

σ

& .#$ 4

L(σ)

C0 , an (sin hd0 )n−1

an = voln−2 (S n−2 (1)). Dd0 (σ) = {(σ(t), ξ) | t ∈ S 1 , ξ d0 }.

4.6

Fermi

&*

4

(ii)

J (0) = 0, J(0) = 1

ξ = 1, J

J(0) = 0, J (0) = 1

Jacobi

, K

Mn

−1,

(

4

J(t) sinht.

4

s ∈ S 1 , tξ ∈ Bdn−1 (O) = {x ∈ Rn−1 | |x| < d0 }, 0 n

C0 vol(M , g)

σ(s)∈σ

L(σ)

0

d0

t=0 d0

.

2.6)

(4.12)

(4.13)

ξ∈S n−2 (1)

det(F∗ )dtdsdξ

an (sinht)n−2 coshtdt

L(σ)an (sinhd0 )n−1 .

.

2.8

det(F∗ )|(σ(s),tξ) (cosht) · (sinht)n−2 ,

&

/

ϕ

KM n −1,

,

J(t) cosht.

(4.12)∼(4.13)

2

ϕξ (t) = Expσ(s) (tξ)

1

$

Diam(M n , g) d0 ,

& A7 Æ I 7K;;

Mn

$

F Dd0 (σ) → M n

Æ & *. K −1, Jacobi I $. J s ∈ S , ξ (i)

(4.11)

;Æ5<=BE6

§4.3

· 69 ·

2 F >": & 4.8 (Cheeger) (M n

Diam(M , g) d0 , vol(M

n

9: Riemann , Æ |K , g) C , (M , g) Æ n

, g)

0

Inj(M ) min π, n

&

Æ Inj

∈ Mn

n

) = Inj(M n ).

q

%# ' + n

0

p0 (M

Injp0 (M ). (1)

p0

1,

C0 2an (sinhd0 )n−1

an = voln−2 (S n−2 (1)).

p

Mn |

n

q0

,

,

0

ˆ 0) ∈ C(p

(4.14)

d(p , q ) =

KM n 1,

0

0

Rauch

;;

2.5,

d(p0 , q0 ) π.

(2)

: 7

σ

' 4.7

(

p0

q0

1 L(σ) = d(p0 , q0 ). 2 K −1) L(σ)

/

C0 . an (sinhd0 )n−1

.

!?7@A
§4.3

94 $+C $ H1 C) D7! : I L,

*M 6 ; - - ' <

$ Æ Cheeger

,

.

Γ

,

|Γ | =

4.9

G(k0 ) = {Γ | Γ

Γ

.

, |Γ | k0 },

Riemann

4.10

(1)

,

n

(M , g)

{p1 , · · · , pk } ⊂ M

G(k0 )

Riemann

.

.

, δ > 0,

n

d(pi , pj ) > δ,

i = j,

(4.15)

( !"$Æ)(

· 70 ·

E < < . NJ < 5O

E < 6 < 8 Æ$'KP <

+ Mn

{p1 , · · · , pk }

δ

{p1 , · · · , pk }

(2)

M

{p1 , · · · , pk }, 4.11

&

δ

M

(M n , g)

n

δ

δ

.

,

.

Riemann

, {p1 , · · · , pk }

Mn

vol(M n , g) W0 . k

W0 , bn δ n

L5> n = dimM Q ,. {p , · · · , p } δ <, $ i = j , n

bn

1

k

d(pi , pj ) > δ,

B δ (pi )

2

B δ (pj ) = ∅, 2

W0 vol(M n )

{p1 , · · · , pk }

,

{p1 , · · · , pk }

, δ < Inj(M n )

δ

.

n

Croke

M (

k

vol B δ (pi ) .

[Cr3]),

C 4 * ,

vol[Br (p)] ˆbn rn ,

& *5> Q, ' ˆbn

n

.

kˆbn

(4.16)

2

i=1

(4.16)∼(4.17)

r

1 Inj(M n ) 2

* (4.17)

34

n k δ vol B δ (pi ) W0 . 2 2 i=1

/ $ < M E )2 L

. R)! 3-'

6

< -

.

Inj(M n ) > δ, {p1 , · · · , pk }

(M n , g)

δ

.

Γ = Γ(M n ,g)

(1) Γ (2)

{p1 , · · · , pk };

d(pi , pj ) < 2δ, i = j,

Riemann

n

(M , g)

ˆn

(M , gˆ)

M=

pi

pj

(M , g)

δ

k

i=1

. n ˆ (M , gˆ), {p1 , · · · , pk }

n

Bδ (pi )

.

ˆn = M

k

i=1

Bδ (ˆ pi ).

{ˆ p1 , · · · , pˆk }

;Æ5<=BE6

§4.3

Inj(M n ) > δ

Æ f

· 71 ·

ME NSF & & $T = &

Inj(M n ) > δ,

fi Bδ (pi ) → Bδ (ˆ pi )

i

= Exppˆi ◦ Exp−1 pi ,

G. NSFME H>9:&* & ?I@$ J$ &$ K

6OU PA "C4LVB Q Æ M !" #

E CN . CN Æ {fi }

ˆ n. f Mn → M m p∈ Bδ (pij )

{fi1 (p), · · · , fim (p)}

,

j=1

ˆn M

.

4.12

,

n

(M , g)

Ω⊂M

(1)

δpq ⊂ Ω,

δpq

(2)

Riemann

n

.

.

p, q ∈ Ω

(M n , g)

Ω

{q1 , · · · , qm } ⊂ Ω, Ω

p

.

n

(M , g)

m

q

y∈Ω

,

Exp−1 y qi = 0,

E 4L 4L & #Æ

4L M i=1

y

{q1 , · · · , qm }

M

n

.

.

n σ [0, l] → M n (M , g) l t, q0 = σ(0), q1 = σ(l), y=σ {q0 , q1 ) 2

(i)

13

⎧ l l −1 ⎪ ⎪ ⎨ Expy q0 = − 2 σ 2 , ⎪ l l ⎪ ⎩ Exp−1 . y q1 = σ 2 2

(

13).

d(σ(t), σ(0)) =

· 72 ·

( !"$Æ)( DE4 ORFR L 'ORF 4L M

{q , q , q } R L, y {q , q , q } (ii)

1

2

3

1

2

(R2 , g0 )

,y

(

3

14).

14

$" 6?4L.

C N 5M 4L ,

.

n

4.13 (Whitehead) 1 r < min{π, Inj(M n )}. 2 (1) Br (p) Mn

(M , g)

Riemann

;

{q1 , · · · , qm } ⊂ Br (p),

(2)

m

F M

Æ K

y ∈ Br (p)

M n 1,

Exp−1 y qi = 0.

i=1

F -'!B3, & (1)

[ChE]p103.

(2),

m

E(q) =

1 [d(q, qi )]2 . 2 i=1

@ GT ∇E(q) = −

KM n 1, d(q, qi ) < 2r π

$

m

Exp−1 q qi .

i=1

q ∈ Br (p)

-,, ;;#, B (p) D q r

i

,

q → [d(q, qi )]2

B (p) N3 ,. E(q) O q ∈ B (p) N3 ,. ' E B (p) M

S5 / r

y.

r

.

r

§4.4

Cheeger

)(

· 73 ·

PQWT X46$"I X §4.4 Cheeger

1970

H Cheeger 4.14

.

n, d0 , C0 > 0,

M(n, d0 , C0 ) = {(M n , g) | |KM n | 1, Diam(M n ) d0 , vol(M n ) C0 } .

; *F &@YU $JK RM S $ 7 >V

;; M(n, d0 , C0 )

.

,

[Pe].

n

(M , g) ∈ M(n, d0 , C0 ).

.

.

.

Ric(M n ) −(n − 1),

K −1,

Gromov-Bishop

vol(M n , g) vol(Bd0 (p)) d0 a ˆn (sinhs)n−1 ds = W0 . 0

7K .

.

Inj(M ) min π,

C0 2an (sinhd0 )n−1

n

7F < < .

Mn

δ

δ

k k0 =

Γ

Mn

= δn .

.

(M n , g)

{p1 , · · · , pk }

)2<

TR)!

, 0 < δ < δn .

W0 . bn δ n

.

pi

-'

pj

⇔

⎧ ⎨ (i) i = j, ⎩

(ii) d(pi , pj ) < 2δ.

G(k0 ) = {Γ | |Γ | k0 } .

7L NSFME 2 0 .

.

,

(4.18)

δ < δn ,

*ME &*

M(n, d0 , C0 ) → G(k0 ), M n → ΓM n .

(4.19)

" 4.11

(4.20)

· 74 ·

M

M(n, d0 , C0 )

.

( !"$Æ)(

ME NSF &

Riemann

|ΓM n | = |ΓMˆ n | = k k0 ,

&

(M n , g)

ˆ n , gˆ) (M

ΓM n

ΓMˆ n

{fi }ki=1

fi Bδ (pi ) → Bδ (ˆ pi ), q → (Exppˆi ◦ Exp−1 pi )(q).

7S H>FME $ "

%# .

j=1

#

.

1 δ < min{π, Jn } , 4 m Bδ (pij ), m k0 , {fi }

4.13

B2δ (x)

C N. p ∈

{fi1 (p), · · · , fim (p)} ⊂ B2δ (fi1 (p)).

C N " # M 4L 2=4 %H>9:&* & & 4L 4 UVF & ! #

H6 Æ WM)Z[W/\ 2 %H > 9:& * & & +H>F &2Æ/F$& B2δ (fi1 (p))

,

4.13

{fi1 (p), · · · , fim (p)}

,

y0 .

,

ˆ n, f Mn →M

p → {fi1 (p), · · · , fim (p)}

Pxy

σxy

Peters

1988

,

,

d(x, y) < δ, σxy

Ppi pj

Ppˆi pˆj

.

x

y

. S.

SO(n)

,

ˆn f Mn → M

.

/

#

.

X] L^F Cheeger

Grove, Petersen

YZ[:X$& $

={(M n , g) | KM n −1, Diam(M n , g) d0 , vol(M n , g) C0 }, d0 , C0 )

.

ˆ M(n, d0 , C0 ) ˆ M(n, n5

\N

· 75 ·

!"$OBPYCA SB!"$T&bcU bcUÆ bcU _V[ef

$% ZQ

R_]`B Riemann 5<, aZQ ^ Santalo dU, Berger-Kazdan W, bcU

, Croke 1980). (M n , g) inf n Inj(M n ,g) (x) δ0 . Croke

1(

x∈M

Schwarz

,

0 r δ0

voln−1 (∂Br (p)) [voln (Br (p))]

Cn∗

n−1 n

g\ `] Rab^cÆX BhX di B_e_ `] f di B_e_ )( ZQ R_` 5< Ya Z f = E6 di B_[_ ZQ Æ gZQ Æ h\B E6 ^;Bjkli_\]cbj^ e Cn∗ n = dimM n d , [voln (Br (p))] dr 0 r δ0 , voln (Br (p))

,

(i)

(ii)

Inj(M n , g) δ0

0 r δ0 . .

n

1989).

2(Yamaguchi

(i)

.

(M , g)

Riemann

vol(M n ) v0 , {p1 , · · · , pk }

.

(M n , g)

δ0

,

k

.

Inj(Min , gi ) δ0

(ii)

vol(Min , gi ) v0 ,

(M1n , g1 )

Γ1 ∼ = Γ2 .

(iii)

ϕ M1n → M2n .

_

¯ v¯) = {(M n , g) | Inj(M n , g) δ, ¯ vol(M n , g) v¯}. M(n, δ,

^ Bcami ZQ e (ii)

3.

f

¯ v¯) M(n, δ,

(M1n , g1 )

→

a_\]cbd R_kn\` _j^ B .

(M2n , g2 )

L(f ) = maxn

)le_

δ0 8

(M2n , g2 )

Riemann

v∈T M1 v=0

5
,

f∗ vg2 vg1

f

Lipschitz

hXf

.

d(M1n , g1 ), (M2n , g2 ) = inf {log[L(f )] + log[L(f −1 )]}. (i)

_T

f Æha bif

f

2

1

= S ×S

1

2

,

M(T 2 , A0 ) = {(T 2 , g) | Area(T 2 , g) A0 , Kg ≡ 0}.

jkcV R_o`Bbif ldm_bif M(T 2 , A0 )

.

(ii)

N(T 2 , D0 ) = {(T 2 , g) | Diam(T 2 , g) D0 , Kg ≡ 0},

jkcVbif o` )( meld )nB opf Bia ld N(T 2 , D0 )

.

).

4(Mumford

k

Cheeger

)(BohfgBh< ZQ

,

M Σ2k , l0 =

Σ2k , g | Kg ≡ −1, l(Σ2k , g) l0 ,

,

Σ2k

R `n

( !"$Æ)(

· 76 ·

`]

qj ='rstoBuv Ya Æ di B!"$B_e_ ppee_ 5
.

l(M n , g) l0

(i)

k

(Σ2k , g)

Riemann

(ii)

(Σ2k , g)

M

.

R

v0

D0 ,

,

∂Ωk ,

Ωk

pi =

Cheeger

2

.)

→ −∞

Cheeger

.

2

2

(x, y) ∈ R | x + y 1, (x, y) ∈

k

B( 1 )i (pi ) 16

p Ω =B_)w Ω T& ˆ , (i) = ∂Ω xy Z Σ \` c ∂ Ω ˆ n
Æ

i=1

ˆ 2k × [0, 1], ˆk = Σ Ω Σ2k

,

, 3 Ωk . Σ2k = 2, 1 , 0 ∈ R2 , Dε (p) = (x, y) ∈ R2 | d((x, y), p) < ε . 2i

ˆ 2k = Σ

Kg ≡ −1,

k

K(Min ,gi )

Diam(Min , gi )

3

.

?(

).

vol(Min , gi )

.

3

(Σ2k , l)

5(

p

(M n , g)

Kg 0,

k

k

2 k

k

k

k

k

k

Σ2 k

∂Ωk

2 k

2 k

2 k

2 j

3

2 k

2 k

2 j

√

k

k

2 i

Σ2 i

k1

0

j −1 k 2π

e

ϕλ S 3 (1) → S 3 (1),

Rcgj^. (ii) (iii) (iv) (v)

(z1 , z2 ) → (λz1 , λz2 )

di5< cV wr `(}Æ5<

B~}| Rxg\ RxÆ

Mk3 = S 3 (1)/Zk π Diam(Mk3 ) . 2 lim vol(Mk3 ) = 0 k→+∞ 3 Mk k1

.

?

Cheeger

)(~~

?

Æ

Diam Σ2i → +∞,

Æ Riemann

20

.

,

,

.

.

,

,

.

,

¦ Riemann Æ

Æ Æ

,

Riemann

. Riemann

30

,

Alexandroﬀ

.

Riemann

Riemann

, Morse

.

,

Anosov

Birkhoﬀ

.

.

§5.1

(M n , g)

.

, p ∈ M

mann

σ(0) = p

n

v ∈ Tp M

σ (0) = v φt

,

n

Rie-

σ(t) = Expp (tv)

.

TM → TM d (p, v) → Expp (tv), [Expp (tv)] . dt

{φt }t∈R , ⎧ ⎨ φ (p, v) = (φ ◦ φ )(p, v), s+t s t ⎩ φ−1 = φ−s .

{φt }t∈R

! s

d (σ (t)2 ) = 2σ (t), ∇σ σ ≡ 0, dt

,

SM n = {(p, v) ∈ T M | v = 1, p ∈ M },

{φt }t∈R

SM n

,

φt SM n → SM n . T Mn

(M n , g) .

SM n

Riemann

{φt }t∈R

G.

,

(T M n , G)

.

,

(T M n , G)

T Mn

!!#$! " %

" % "

! " Æ ## $##

#

" &'

$ " $ " ! % $ !% $%" % ! & # $ &! ! ## $ ## ! (" "

· 80 ·

Riemann

5.1.1

Mn

n

Mn

,

n

(M , g)

T Mn

Riemann

,

TM

2n

.

n

g

G.

G

,

π

T M n → M n,

(p, v) → p.

π∗ T (T M n) → T M n

,

T (T M n)

,

(p, v) ∈ T M n, kerπ∗ |(p,v)

.

n

−dim[Tp M ]} = 2n − n = n. ,

kerπ∗

Tp M

{w1 , · · · , wn }

{dim[T(p,

n

n

.

n

T(p,v) (T M )

(M , g)

,

σi ui (0) = v d ξi = (σi (t), ui (t))|t=0 = (wi , ui (0)). dt {ξ1 , · · · , ξn } . T (T M n)

kerπ∗

wv

T(p,v) (T M n ) = kerπ∗

kerπ∗

(p,v)

w

(p,v)

Tp M n ∼ = Rn .

(p,v)

wH

)]

Riemann

.

(p, v)

⊕ H(p,v) .

T(p,v) (T M n )

,

n

σi (t) = Expp (twi )

, {u1 (t), · · · , u1 (t)}

H(p,v)

v) (T M

H(p,v) .

,

Tp M n

T Mn

w,

H(p,v)

w

.

g

G

& # "%

"%' ' ! # #(&

u, wG = uv , wv g + π∗ u, π∗ wg . T(p,v) (T M n ) = kerπ∗ ⊕ H(p,v)

Mn

(p,v)

{(x1 , · · · , xn )}, {Γkij }

p

{(x1 , · · · , xn )}

Christoﬀ

,

(p, v) ∈ T M n

n ∂ ∂ w= + Vi Yi . ∂xi ∂vi i=1

n

T(p,v) (T M )

wH

⎧ n ⎨

⎫ ⎞ ⎛ n n ∂ ⎬ ∂ Yi = −⎝ Γijk vj Yk ⎠ ⎩ ∂xi ∂vi ⎭ i=1 j=1 k=1

(M n , g) ∂ v = vi , ∂xi i=1

Riemann

n

$!$#'#)%)

§5.1

· 81 ·

£

wv = w − wH .

( % )* " & +* +, ' % ##- ## ! ' ' & % ## .(" / , '.( " /('0 '0"" * ,) Æ , ! $ "%'

% ! "% * + "% ' Æ-, ! ' ! "% ' 5.1.2

$ Λ (M 1

1

n

Mn

)

n

π ˆ Λ (M ) → M k

Λ1 (M n )

,

.

n

n

∗

π ˆ .

1

β ∈ Λ (M )

n

ξj ∈ T (Λ (M )),

(ˆ π ∗ ◦ β)(ξ1 , · · · , ξk ) = β(ˆ π∗ ξ1 , · · · , π ˆ∗ ξk ).

5.1

π ˆ Λ1 (M n ) → M n

(1)

ηπˆ

,

Λ1 (M n ) → Λ1 (Λ1 (M n )), α→π ˆ∗ ◦ α

Λ1 (M n )

(contact form).

(2)

(

)ηπˆ

Ω = dηπˆ .

Λ1 (M n )

Ω ∈ Λ2 (Λ1 (M n )).

.

Ω = dηπˆ

p

Λ1 (M n )

.

α=

Mn

{(x1 , · · · , xn )}

.

{(x1 , · · · , xn )}

α

n

yj dxi .

j=1

{(x1 , · · · , xn , y1 , · · · , yn )}

Λ1 (M n )

Λ1 (M n )

,

Ω

Ω = dηπˆ = d(y1 dx1 + · · · + yn dxn )

*

' Æ, ! ))+ * ! ! . !£"¡# /

( . Æ , ! ) ' ) *&! " &! = dy1 ∧ dx1 + · · · + dyn ∧ dxn . Mn

Ω

,

Λ1 (M n )

.

5.1.3

T Mn

TM

n

1

n

Λ (M )

.

.

Mn

Riemann

"

· 82 ·

!!#$!

!1 # *

* &+

, ! , '0 " (, " , ! '0 " $%&¥ ª'()* ,

+! ,* + , 2 - 0, )3 " 4 " & +

- " 5 ©, , '1 ." -/ .."% / "%* *

"% ! "' -/

#

Riemann

g

T M n → Λ1 (M n ), (p, v) → v, ·g ,

Fg

p

X ∈ Tp (M n )

Fg (p, v)(X) = v, X

Ωg =

Fg∗

(

◦Ω

TM

TM

n

n

, ηg =

Liouville

.

Fg∗

Fg

◦ ηπˆ

TM

).

T Mn

Ωg

T Mn

.

n

ηg

,

.

T Mn

5.1.4

(M n , g)

Riemann

T M n → R, 1 (p, v) → v2g . 2 p

Eg

, Ωg

T Mn

,

T Mn

T Mn

,

Zg

Ωg (Zg , ·) = −dEg .

(M n , g)

5.2

Zg

Riemann

{φt }t∈R

,

.

dφt = Zg . dt

(5.1)

(5.1)

,

(5.1)

.

{(x1 , · · · , xn )}

M

n

Z=

n j=1

∂ 2 Eg = gij , ∂vi ∂vj

∂ ∂ +Vj Y ∂xj ∂vj j

∂Eg ∂xi

=

(x,v)

.

n 1 ∂gkj vj vk 2 ∂xi j,k=1

n

∂gik ∂ 2 Eg = vk , ∂xi ∂vj ∂xj k=1

T Mn

,

Eg =

1 gjk vj vk . 2 k,j

(5.2)

$!$#'#)%)

§5.1

&

· 83 ·

,

Ωg (Zg , ·) = −dEg

(5.3)

Ωg = Fg∗ (dv1 ∧ dx1 + · · · + dvn ∧ dxn ),

(5.4)

⎧ ∂ ∂Eg ⎪ ⎪ ⎨ Ωg Zg , ∂x = − ∂x , i i ⎪ ∂ ∂E g ⎪ ⎩ Ωg Z g , . =− ∂vj ∂vj

(5.5)

% 0-0 6 (g ij )

(gij )

. (5.5) ⎧ ⎪ Y j (Zg ) = vj , ⎪ ⎪ ⎨

(5.2)

n 1 ki ∂gjl gil ⎪ k ⎪ g −2 vj vl . ⎪ ⎩ V (Zg ) = 2 ∂xi ∂xj

- '

2

(5.6)

j,i,l=1

Christoﬀ

n

Γijk

1 il = g 2

l=1

(5.6)

(5.7)

∂glk ∂gjl ∂gjk + − ∂xj ∂xk ∂xl

⎧ ⎪ ⎪ Y j (Zg ) = vj , ⎪ ⎨ ⎪ ⎪ V i (Zg ) = − ⎪ ⎩

..& Zg =

Zg

⎧ n ⎨

∂ vi − ⎩ ∂xi i=1

n

.

(5.7)

Γijk vj vk .

j, k=1

n

j, k=1

⎫ ∂ ⎬ , Γijk vj vk ∂vi ⎭

T Mn (p(t), v(t)) ⎧ dxi ⎪ ⎪ = vi , ⎪ ⎨ dt n n d2 xi dxi dxk dvi ⎪ i ⎪ . Γjk vj vk = − Γijk ⎪ ⎩ dt2 = dt = − dt dt

ÆÆ 6 , 7 j,k=1

.

dφt = Zg . dt

.

j,k=1

(5.8)

"

· 84 ·

!!#$!

$%& /0+ / & + , ! % 8, !'0" 5.1.5

Riemann

Liouville

{φt }

5.3 (

)

.

{φt }, Ωg

T Mn

ηg

.

⎧ ⎨ φ∗t Ωg = Ωg , ⎩ φ∗ ηg = η g t n

©,*% " 1 * SM

Riemann

t

(M n , g)

! (5.9)

SM n

.

LZg

Zg

.

⎧ ⎨ LZg Ω = 0, ⎩ LZ ηg = 0. g n

1 (

(5.10)

SM

,

LZg Ωg = d[Ωg (Zg , ·)] + dΩg (Zg , ·, ·) = d[−dEg ] + 0

# (& , !

3 Æ 91 0 21 = −d2 Eg = 0.

Ωg = Fg∗ Ω

(1)

(

),

dΩg = Fg∗ (dΩ) = Fg∗ (ddηπˆ ) =

d(dFg∗ ηπˆ ) = 0;

(2) d2 = 0.

X ∈ T(p,v) (T M n ), ηg (X) = v, π∗ (X)g .

,

(p,v)

,

LZg ηg

7

= {d ◦ i + i ◦ d}η n Z Z g g g SM n SM = dv, π∗ (Zg ) n + Ωg (Zg , ·) n SM SM + dEg = d v2 n n SM

= 0. .

SM

:$3 + , ! '0" &" ! # ; * © ,.. '1 / .."% ! / * ! % # #(( ; 3) 14* #(( 7 12

+ 2 +*+, <4 3 2 5 =0 3 #135 §5.2

· 85 ·

(M n , g)

5.4

TM

n

Ωng

Riemann

.

G

TM

n

T Mn

, Ωg

g

.

(T M , G)

n

= a n Ωg ∧ · · · ∧ Ωg .

, ηg

n

bn ηg ∧ Ωn−1 , g

, (SM , G)

an

n

bn

n

.

dvolG = Ωng .

(5.11)

(5.11)

p ∈ M n,

.

Expp Rn → M n

{(x1 , · · · , xn )},

{Γkij (x)}

Christoﬀ

Γkij (O) = 0,

p = Expp 0.

Ωg

= dv1 ∧ dx1 + · · · + dvn ∧ dxn

(p,v)

dvolG

(p,v)

= dv1 ∧ dx1 ∧ · · · ∧ dvn ∧ dxn ,

(n!)dvolG

dvolG

bn =

(p,v)

= Ωg ∧ · · · ∧ Ω g . n

SM n

= bn ηg ∧ Ωn−1 g

1 . (n − 1)! 5.5 (

,

.

an =

.

)

{φt }t∈R

.

(M n , g)

Riemann

(T M, G)

,

, {φt }

5.3

5.5

Liouville

.

§5.2

.

Synge

.

T Mn

(SM n , G|SM n )

.

.

1 n!

!!#$! +

2 5 - $ 4 3 ,-. ) Æ /0 6 + 2 + -$4 1 6 3 ©, % 1 6 # ( % > % 1 6 3 2&&73 7 3> " 2

.83 4 , 2 4 2 1 93 " 3 0 2 (" 5 3 1 6 3 32 &73 28+ 7 ' 6562 3 4 4 62 6 5 62 4 4 ) 5 6 62 5"

7% 833 + 7 > 3 =0 ? 6 & 7+ 7 > 8* '58

: 7 7 8 . 5 7 8 ..- 7 * $ + 7 9 37 $ > 8' & 3 7 5 > + ) 3 +* 626 # &7 & 7% 3" &7 & +/ 3> &7 , 7 3& #

" "

· 86 ·

Riemann

(M n , g)

5.6 (Lusternik, 1951)

Riemann

,

(M n , g)

.

5.6.

.

n

5.7

(M , g)

(M n , g)

π1 (Mn ) = 0,

.

},

lo = inf{L(σ)|σ

1

σi S → M

1 (i) L(σi ) l0 + , i (ii) σi

L(σ)

Mn

{σij }+∞ j=1

,

1

σ S →M .

[ϕ].

,

σ

,

Min-Max

Birkhoﬀ

Birkhoﬀ

Birkhoﬀ

.

,

1

σ S →M

n

L(σ).

(M n , g)

Inj(M n , g),

L(σ) = l0 .

.

.

Inj(M n , g)

, {σi }

.

(M n , g)

.

Arzela-Ascoli

n

,σ

σ

(= q0 ).

,

N

L(σ) . N

N

qi

qi+1

.

τi

β(σ),

, Birkhoﬀ

15

σ

N

σ [0, 1] → M n

qi+1

p0 , p1 , · · · , pN −1

pi ,

, L(σ) = L(β(σ))

L(σ)/N <

q0 , q1 , · · · , qN

qi

,

.

.

β(σ)

.

L(σ)/N < Inj(M n , g),

, S 1 = [0, 1]/{0, 1}.

β(σ)

{σs }0s1

σ0 = σ, σ1 = β(σ).

s1

L(σs1 ) L(σs2 ).

s2

σ [0, 1] → M n

=σ

.

[ϕ].

.

σ

σ

n

,

(iii) σi

τi

Mn

Riemann

i , N

.

i = 1, 2, · · · , N.

σ

σ 12 1 s ∈ 0, , σs 2

.

,

σ 12

i N

:$3

§5.2

· 87 ·

:

15

⎧ s ⎪ ⎪ ⎨ τi (t), i +t = σs ⎪ i N ⎪ ⎩ σ +t , N i i 2s + σ σ N N N

0 t 2sN , 2sN t N1 ,

# 8

.5 7 $+ 3> .. % 73 8' $+6 ;7 3> 3 ) < &" ! 8 &7 3 & 6 + 1 6 9 6 6 83(+1696 83 5 . &7 . 6 Æ & 7 ." * + ) 3 &4 ' Æ /" 78 8 & % -$ 4 τis 2s 0, N

= σ 12

.

i 1 + ; N 2N

,

σ1

σ1

.

,

τis

t∈ i 1 + σ1 N 2N i 1 i+1 1 + , + N 2N N 2N σ 12 σ1 . ,

Λcg (M n ) = {σ S 1 → M n |Lg (σ) c},

5.8 (Birkhoﬀ

)

β Λcg (M n ) → Λcg (M n ) (i) β

Λcg (M n ), →

Birkhoﬀ

Λcg (M n )

.

L(β(σ))

id Λcg (M n ) → Λcg (M n )

(ii) β

c < Inj(M n , g), N

N

.

σ

,σ

;

β(σ)

;

σ ∈ Λcg (M n ),

(iii)

L(β(σ)) L(σ),

σ

.

5.9 .

5.8

Birkhoﬀ

g

S2

Min-Max

Riemann

5.6

,

(S 2 , g)

.

!!#$! ©, % # ; 3==0 ! 9;

=& Æ + > 2 5 1 9 8& *&+ 9 $4 3* 9 5 = $99 . 6 ! "

· 88 ·

δ0 = min

(S 2 , g)

,

σ

Inj(M n ) π ,√ , 2 K0 Bδ0 (p)

K0 = max2 {Kg (p)}. p∈S

.

δ0

σ S1 → S2

,

β(σ), β 2 (σ), · · · , β k (σ), · · ·

L(σ) < δ0 ,

,

Riemann

.

S 2 (1) S 2 (1) = {(x, y, z)x2 + y 2 + z 2 = 1}. S 2 (1)z = 0, x 0} , F

:

S2

F S 2 (1) → S 2 (

16).

{(x, y, z) ∈

[−1, 1]

16

F ([−1, 1]/{1, −1})→ (Λg (S 2 ), Λ0g (S 2 ))

y → F (·, y, ·).

? & $ : 7 / 164*

4 & &7 < ; * &+ ,5 * $

< 82 & c = max {L(F (·, y, ·))}.

N ,

−1y1

c < lnj(S 2 , g). N

lk = max L(β k ◦ F (·, y, ·)). −1y1

, β k ◦F

k, lk δ0 .

,

, βk ◦ F

5.8

.

F , deg(F ) = deg(β k ◦ F ) = 0,

deg(β k ◦F ) = 0. F

,

l k δ0 > 0

k

.

k,

{lk }k1

,

(5.12)

, lim lk = eˆ δ0 > 0. k→+∞

yk ∈ [−1, 1]

L(β k+1 ◦ F (·, yk+1 , ·)) = lk+1 .

:$3 % -/

, .83 2 & 0 24 5

7 : =

;= + ©, 4 28+ % %>< 9&7 9& > 35 . * 5 / , 28+ * $ + ,2

& % & ( -$4 % &?8 &+.6-% % ! &

4 9 = @A$ & $ * ? ==0 % §5.2

· 89 ·

σk = β k F (·, yk+1 , ·).

{σk }

, {σk }

{σkj }

lim σkj =

j→∞

σ ˆ.

L(β(ˆ σ )) = L(β( lim σkj )) = L( lim β(σkj )) j→∞

j→∞

σ ). = lim lkj +1 = ˆl = L(ˆ j→∞

5.8, σ ˆ

.

5.9

.

,

5.6

5.6

Mn

5.7,

n

πk (M )

M

n

n

, Hk (M , Z)

k

Hi (M ) ∼ = πi (M ) = 0 n

Hk (M n ) .

M

n

Hurewicz

n

.

πi (M ) = 0

1 i k−1

Mn

.

n

k

.

1 i k−1

,

,

n

i = k

πk (M ) =

Hn (M n , Z) = Z = 0.

,

k0 = inf{k 2|Hk (M n , Z) = 0} 2.

Hurewicz

kˆ0 = inf{k 2|πk (M n , Z) = 0} 2. ˆ

F S k0 (1) → M n

B

ˆ k−1

R

ˆ k−1

(x0 , · · · , xkˆ ) ∈ R

B

ˆ k+1

ˆ k−1

πkˆ0 (M n )

F

= {p ∈ R

|x = 1, x0 0, x1 = 0}.

ˆ k−1

Σ

.

|p 1}.

ˆ k−1

Σ

B

ˆ k−1

ˆ k−1

= {x =

.

F

ˆ

ˆ

F (β k−1 , ∂B k−1 ) → (Λg (M n ), Λ0g (M n )).

ˆ ˆ ˆ ˆ p ∈ B k−1 ∼ = Σk−1 ⊂ S k (1) ⊂ Rk+1 ,

{x1 = 0}

p

.

N

,

max{L(F (p, ·))} p

N

(M n , g)

< Inj(M n , g).

δ0 > 0.

j

β ◦ F (p, ·)

, ( F

-> lj = max L(Fj (p, ·)). ˆ p∈Σk−1

πkˆ (M n )

,

l j δ0

ˆ

p ∈ Σk−1 ,

Fj (p, ·) =

"

· 90 ·

!!#$!

$ 4 4

Riemann

* : 3&)=

$4 <; & .83 4 4192

4 & # .

< 8 2/ (& &

j

ˆ

5.8

ˆ

B j ◦ F (B k−1 , ∂B k−1 ) → (Λg (M n ), M n )

.

F

,

5.9

.

,

σk ∈ Image(β k ◦ F )

.

L(β ◦ σj ) = max {L(β j+1 ◦ F (p))}, ˆ p∈Σk−1

{σj }

,

L(ˆ σ) = L

lim σji

i→∞

Ascoli

5.8

= lim L(σji ) lim lji = ˆl, i→∞

.

σji → σ ˆ.

,

{lj }

i→+∞

,

L(σji ) L(σji +1 ) = L(β ◦ σji ),

1

:." & .. ,2

0 24 ? : 5 3 $ + > 7 @@;AA B ;. B < * 1 CB 0 2= . B < * . 6 - 0, (< * , C. .B<* B<* + - > lim L(σji ) lim lji = ˆl.

i→+∞

i→+∞

,

L(ˆ σ ) = lim L(σji ) = ˆl. i→+∞

,

L(β ◦ σ ˆ ) = lim L(β ◦ σji +1 ) = lim lji = ˆl = L(ˆ σ ). i→∞

5.8(iii),

i→∞

L(ˆ σ ) = ˆl δ0 > 0.

σ ˆ

.

§5.3

Hopf

,

Hopf

3.3,

.

.

n

5.10

(M , g)

F = Exp

Riemann

.

T M n → M n × M n,

(p, v) → (p, Expp v)

(

)

(

(M n , g)

detF∗ |(p,v) = 0

(M , g)

),

.

.

n

(p, v) ∈ T M n

.

{J(t)}

(M n , g)

Riemann

σ

Jacobi

DC=D#!!$ D " $+.! B<* *& + + - 5.B<* C ":& % $ + . B<* >$ *5 $E ?E 2& !84 & . B < * E 4* & &+ ":& C

2 +

2 6F - +.B<* ." * + )

$E ©, ,)"% * ! ) $ + ) .=/ §5.3

· 91 ·

Hopf

,

J (0) = 1,

J(0) = 0

σ

σ(0)

, J(t) t.

Rauch

.

.

n

5.11

(M , g)

,

(M n , g)

(T n , g)

,

Riemann

.

1948

, E. Hopf

5.12

g

n

(T , g)

.

R. Gulliver

mann

Tn

n

(M 2 , g)

g

p1 , p2 ∈ M

2

n=2

M2

1975 ,

g

Kg (p1 )Kg (p2 ) < 0.

, Hopf

,

Gulliver

5.12

M

2

.

.

5.13 (E.Hopf, 1949, n = 2, L.Green, 1958) .

Rie-

(M n , g)

(M n , g)

,

Mn

Scalg (p)dvol 0

(M n , g)

.

SM n

II

SM n → ⊗2 (M n ),

(p, v) → II(p,v) .

˜ n, ξ = (p, v) ∈ S M

σ(t) = Expp (tv)

Bξ =

+∞

Bt (σ(t)),

# @A* F ."FG $+ t0

˜ n , g˜) (M

Mn

g

,

=* #

!

><

t 1 < t2

Æ; &?%

Bt1 (σ(t1 )) ⊂ Bt2 (σ(t2 )).

w ∈ Tp M n , w ⊥ v,

,

g˜.

(5.13)

(5.13)

ht (w) = II∂Bt (σ(t)) (w, w)

ht (w) = −∇w w, σ (0), t > 0.

η−ε (w) = II∂Bε (σ(−ε)) (w, w) = −∇w w, σ (0),

!!#$! , ><= < "

· 92 ·

3) )"

, ht (w) < η−ε (w).

Hξ

Riemann

ht (w)

.

∂Bξ =

6 # ,-0

!$%"+ -0 = " = $B 3) G."8& &

& IIξ (w, w) = II∂Bξ (w, w) = lim II∂Bt (σ(t)) (w, w). t→∞

SM n

{φt }

Riccati

,

Gromov-Bishop

,

IIφt (ξ)

,

II φt (ξ) + II2φt (ξ) + R(t) = 0,

IIφt (ξ) (·, ·)

{Ei (t)}

(5.14)

(n − 1) × (n − 1)

.

En (t) = σ (t),

σ(t) = φt (ξ) = Expp (tv)

Rij (t) = R(Ei , σ )Ej , σ = R(σ , Ei )σ , Ej

R(t) = (Rij (t)).

R(t)

tr(R(t)) = Ric(σ (t), σ (t)).

Ricci

sphere)∂Bξ

IIξ

∂Bξ

,

(Horo-

.

(n − 1)(λ21 + λ22 + · · · + λ2n−1 ) (λ1 + · · · + λn−1 )2 ,

(trIIφt (ξ) ) +

§5.1

1

SM n

0

1 (trIIφt (ξ) )2 + Ric(σ (t), σ (t)) 0. n−1

Liouville

,

(trIIφt (ξ) ) dvolSM n dt =

SM

n

= SM n

..&

{trIIφt (ξ) − trIIξ }dvolSM n (trIIφt (ξ) )dvolSM n − (trIIξ )dvolSM n SM n

= 0. (5.16)

Mn

(5.15)

Scalg dvolM n = an = an

SM n 1 0

1 − n−1

Ric(ξ, ξ)dvolSM n

SM n

0

1

Ric(φt (ξ), φt (ξ))dvolSM n

SM n

(trIIφt (ξ) )2 dvolSM n dt 0,

(5.17)

HI #." " &** +) an

· 93 ·

.

(5.17)

(5.15)

⎧ ⎨ (trIIφt (ξ) )2 ≡ 0, a.e., ⎩

(5.18)

* 02 6 ( >##

% ( 83 ( 5 >@G 7 HHJ A, *

83 & . !8IC .> C.B<* 2& 83 ))

??K@C8#LI@ A B I C?JBJM K ND C LO@ KE AA CMC?JBQMK NDM B CLFP# L I AI B CMC C?JLFP#RN !!$A B D C=D LCDDE BQM KNDM GABS TLUCF?HJRN#LFPL #DCMG E? JDC = D#N IM C = D# !! HI F # J D $ L G F@DVOC?JPRN##$3 !! ABL F#HEJD OQ GW?#A$B3XF$ LSJJQCMGHLFPDMAAHPK F IL AADCB=D LL $ C CMFJKEJ@QYGRI J IH J K # ÆJ M AA B $ XK SNX # K D IL LNZRLU[I G:F I C #LTUDVJMA S IIφt (ξ) ≡ 0,

IIφt (ξ) = λ(t)I,

a.e.,

Riccati

Rφt (ξ) = (Rij (t)) = 0

.

(M n , g)

.

Rφt (ξ) (·, ·)

,

ξ

Rφt(ξ) (·, ·)

,

.

ξ

, E. Hopf

1948

.

40

,

IIξ (·, ·)

,

,

Ballmann, Brin

,

1994

, Hopf

, IIξ

5.12

Burago

(

Expp

.

Sn

(ii)

(n 2).

p

p0

q

(M , g)

Ω

˜ Ω

[BI].

x

˜ n , g˜) (M

,

),

Sn

M

p

q,

?

Riemann

R

˜n M

.

n

n

Mn

,

?

3.1(

{p, q} ∈ M ,

n

§3.1)

˜ h 3. (i) h ˜ ˜ h (0) h (0) h(x). (ii)

,

det [F∗ (x)] = 0

¯ n , gˆ) (M

.

)

(

Riemann

n

2. (i) ˜ n , g) (M

.

˜n F R →M

(M , g˜) n ˜ Tp M → M˜n

(iii)

,

n

˜n

(ii)

Ivanov.

Gulliver

)

˜n F Rn → M

1. (i) ˜ Mn

Burns

(M , g)

˜

˜ h(0) = R(0), h (0) = h (0) = 0,

?(

Taylor

2

.

h(x)

.)

˜ Ω ⊂ Ω.

∂Ω

˜ ∂Ω

(

.

k∂Ω (p0 ) k∂ Ω˜ (p0 ),

p0 , ψ [−ε, ε] → ∂Ω k∂Ω (p0 ) = ψ |p0 N p0 = (0, −1)). ( ,N

∂Ω

p0 ψ(0) = p0 , N

"

· 94 ·

Riemann

!!#$!

: H? J MINWOZ? J MI A B CSMOOTX #?J@ HYUDTR ZWOZ?J@ A B I C?V$3 CP #?BQKP\[I[ C G S TX #?J@ S Q IL C TX #O?@MMA OU TX #P@S LN$!#UO@DPSVW S #RCS #CEH $ S\DNS VK GDEF TRL : $3$WPYQ#F AB S? JRN# !!$A B I C?JWPYQ S LFP MDC=Q G I SW? #O P #$3 X ]G H"?YILKVKD NWC #?JQIDSMRS RGN? $3 LY LO@ KE X^ CWPYQ #?JF B SS?JF IL !!S_VI #GYNST C]@Z[DE LY I C # #: $ 3 17

Jacobi

4(

Jacobi

,

0

y2 (t) = c(t)y1 (t)

). (i)

y2 (t) = y1 (t)

σ R→M

(ii)

n

t

a

, {Ei (t)}

1 ds. y12 (s) σ

Rij (t) = R(σ , Ei )σ , Ej , A(t) = (aij (t))

0).

y +ky =

y1 (t)

∇σ E i =

(

Jacobi

A (t) + R(t)A(t) = 0

.

detA(t) = 0

,

Bc (t) = A(t)

Jacobi

(5.19)

˜ = A(t) A(t)

A∗

A

,

c

t

A−1 (s)[A−1 (s)]∗ ds

Bc (c) = 0. t

0

(5.19)

(5.19) −1 −1 ∗ A (s)(A (s)) dsC1 + C2 ,

a∗ij = aji , C1

Jacobi

C2

.(

(5.20)

(5.20)

,

.)

5(

˜ n , g˜) (M

).

˜n M

˜n → ϕ M

Riemann

.

(i)

n

˜ , g˜) (M

,

˜n σp,q [0, d(p, q)] → M

p

q

,

hϕ (p) = d(p, ϕ(p)).

∇hϕ .

(ii)

p

{ϕ−1 (p), p, ϕ(p)}

hϕ

(ϕ ◦ σ)(t) = σ(t + c) ˜ n → Mn Riemann ,π M

(iii) ,

π◦σ

[ϕ]

t

,

σ

, ϕ ∈ π1 (M n ). .

σ

ϕ

,

˜n σ R→M

.

(M n , g) ϕ

HI

· 95 ·

HZK PILK GL X`QG?V #F I $ L A B CTaS #?JF^[ C LO@ K IL EG LS AB C SF^[ Q I @ IU #?F C #?JQID S D C=DQ ILLO@ QBD O@_VbWK (iv)

5.7

6 (O Sullivan

). n

˜ n. σ R→M

ϕ

ϕ

.

n

ϕ ∈ π1 (M )

(i)

).

ϕ ∈ π1 (M n ), ˜ n , g˜) (M n , g), (M

,

π1 (M )

(

ψ eπ1 (M

ϕψ = ψϕ

n)

hϕ (x) = d(x, ϕ(x)).

hϕ (ψ(x)) = hψ−1 ϕψ (x).

ϕ ∈ π1 (M n ) σ

(ii)

˜n → R , hϕ M

, x0 ∈ σ

ϕ

˜ n, k ∈ Z y∈M

?

hϕ

˜ n , g˜) (M

, x0 = σ(t0 ).

,

,

kd(x0 , ϕ(x0 )) = d(σ(t0 ), σ(kc + t0 )) k−1 d(ϕj y, ϕj+1 y) + d(σ k x, σ k y) d(x0 , y) + j=0

LSB DCD=CD=$ADQBHE #QICJD F^[C H#TB W#L%YUH`QIL DU C\D AUL F#X?JDO@?V:$3P\ OU@aT' VPK AB SMDC=D IL c $UHND CIL #X?D O@ JbNT[ 3O]E #FP\ « W%Y G Db#ILFXYÆVW#EH K@C #LI G S TFH C #?JQGN I YL CM@ S?JSQG !! L CMZ D@DZ^ZVbNcJ #:$ 3 LY VbNcJ # : $ 3 K@C #LI G C?J[_ #:[H Z@ `Q[\ L #$ 3!CM`Z@_CMCNd]VTN CCMC@HE?J F# bY#^J C 2d(x0 , y) + kd(y, ϕ(y)).

˜ n , g˜) (M ˜ n , g˜) (M

(iii)

(iv)

hϕ

x0

n

ϕ ∈ π1 (M )

hϕ

(

.

)

?

hϕ (x) ≡

(i)∼(iii)

n

.

C

,

(M , g)

.

S 1 → M n → M n /S 1 .

(v) (Hopf

(M n , g)

).

, π1 (M n ) = Z ⊕ · · · ⊕ Z = Zn .

Mn

n

. (

˜ n , g˜) (M

(v)

7(

).

p

n

(

Hopf S

3

)

.)

.

3

?

, g

S

3

(S 3 , g)

.

(M n , g)

Riemann

(M n , g)

,

?

8(

{φt }

).

Σ2

(ergodic)? ( ?

, (Σ2 , g)

2

SΣ2

{φt }

2

vol (Ω)[vol(SΣ − Ω)] = 0?)

.

Ω(

SΣ2

φt Ω ⊂ Ω)

I¦ J¢K¨LM - 9* & &

C W 4ea\\`b\` * 7 8=* =4c"c" 2 NOPQ+ 8+ - * +

88+ + -

@A=X Æ ] "&+

] "* & + ©, % % # *

@A=> 4 * 30

.

.

§6.1

,

.

.

6.1.1

Rauch

.

n

˜ , g˜) (M

6.1 (Cartan-Hadamard)

Riemann

,

˜n → M ˜n Expp Tp M

,

˜ n , Expp Rn = T p M

, M˜n

.

{(t, Θ)}

Rn

,

Θ ∈ S n−1 (1).

˜ n, F = Expp Rn → M

(t, Θ) → Expp (tΘ)

.

∂ ∂θi

S n−1 (1)

.

Gauss

∂ ∂ F∗ ≡0 , F∗ ∂t ∂θi ∂ ∂ F∗ = = 1, ∂t ∂t

F∗ ∂ ∂ = 1. ∂θi ∂θi

(6.1)

$3f`Q[\$d[\ ! =

$ " 7 , 02 1 & * , * ,

8+ 1 5 @A=X ] "* &+ 7 , 8+ - +*, 1202- e a

d g > b-g> + Y ©, b-g!f* # -f* &

@A=>

) ..2 f* #(( §6.1

· 97 ·

∂ σΘ (t) = Expp (tΘ). J(t) = F∗ |(t,θ) t σ ∂θi ∂ ∂ J(0) = 0, J (0) = ⊥ σ (0) J (0) = ∂θi ≡ 1. ∂θi , ∂ = J(t) t, t F∗ ∂θi

p

Jacobi

.

K 0,

Rauch

(6.1) ,

v(t, θ) ∈ Rn

F∗ |(t,θ) v v

.

F

.

.

6.1,

, abc

Cartan-Hadamard

Cartan-Hadamard

6.2 ( ˜n M

Cosine

(M , g˜)

a

b,

c

c2 a2 + b2 − 2ab cos θ.

abc

x

.

Cartan-Hadamard

abc

.

θ,

Cosine

˜n

)

˜n M

.

˜ n , g˜) (M

6.1,

Rn

.

y.

p

(6.2)

abc

,

6.1

˜ n, Rn → M

F

v → Expp v

x∗ = F −1 (x)

.

R2 √ a2 + b2 − 2ab cos θ.

y ∗ = F −1 (y).

{O, x∗ , y ∗ }

c¯ = d(x∗ , y ∗ ) =

c = d(F (x∗ ), F (y ∗ )) c¯ =

a2 + b2 − 2ab cos θ.

7 c * 5&& ] $E "&? d 12 d % d5;( . d 5& ] $E " .

c2 = a2 + b2 − 2ab cos θ

˜n M

2

.

Cartan-Hadamard

6.3 (

,

)

abc 2

.

abc

π.

.

abc

,

Cartan-Hadamard π

, abc

e Z@`Q[\#!! © , g > 7 d g> E d 9 % $f^$& ,* "

· 98 ·

abc

R

2

{a, b, c},

{a, b, c},

,

R

)

{θ1 , θ2 , θ3 }.

{θ1∗ , θ2∗ , θ3∗ }.

2

θ1∗

+

θ2∗

+

θ3∗

(

18

= π.

: > g d g a& & ,." ** 3) ." * 0 ge 7 RST(UV¤W§XY 7 - @A* 1=) .- +

+ -

$5 " I )

+! ! " ©, ' #(( 5 - 18

θi θi∗ .

c

θ3

.

(6.3)

Cosine

a2 + b2 − 2ab cos θ3 c = a2 + b2 − 2ab cos θ3∗ ,

− cos θ3 − cos θ3∗ .

θ2∗

θ2 θ3∗ ,

, θ3 θ3∗

(6.3)

i = 1, 2, 3

,

.

θ1

.

.

.

6.1.2

.

6.4

n

(i)

σ

(M , g)

Jacobi

, {J(t)}

Riemann

,

d2 J(t) 0. dt2

(ii)

2

,

(6.4)

(M n , g)

d J(t) 0, dt2

σ

n

(M , g)

J, J d J(t) = dt J

.

σ

Jacobi

$3f`Q[\$d[\

§6.1

d2 J = dt2

· 99 ·

[J, J J + J , J J] −

J, J 2 J

J2 −R(σ , J)σ , JJ2 − J, J 2 + J 2 J2 = . J3

- I ) "* ! * " #(& (i)

, −R(σ , J)σ , J 0,

d2 J 0. dt2

,

(ii)

(6.4)

Jacobi

p = σ(0), E ⊥ σ (0),

.

K(σ , E) = R(σ , E)σ , E 0.

{J(t)} J(0) = E, J (0) = 0. d2 J −R(σ , E)σ , E · 1 − 0 + 0 0 = dt2 t=0 1 = −R(σ , E)σ , E.

σ

Jacobi

712 @A* =

# @A* 5 ©=* , *! ." % 8 7 7 &" % 7 9 4 .

6.5 (

˜ M

n

,

)

˜ n , g˜) (M

Cartan-Hadamard

, σi R →

i = 0, 1.

h(t) = hσ0 ,σ1 (t) = d(σ0 (t), σ1 (t))

.

h

˜n ηa [0, 1] → M σ0 (b)

(

σ1 (b)

σ0 (a)

,

a < b, a+b 1 [h(a) + h(b)]. 2 2 σ1 (a)

. ˜ F (s, ·) [a, b] → M

19).

:

19

(6.5)

,

ηa (s)

˜n ηb [0, 1] → M ηb (s)

"

· 100 ·

F

e Z@`Q[\#!!

˜, [0, 1] × [a, b] → M

2 "

$ 7 " 2 &

=*

(s, t) → F (s, t). ∂F ˜n (s0 , t) F (s0 , ·) [a, b] → M ∂s t∈[a,b] ∂F (s0 , t) (6.4) t→ t , ∂s ∂F ∂F 1 ∂F a+b ∂s s, 2 2 ∂s (s, a) + ∂s (s, b) (6.6) dηb 1 dηa = + 2 ds ds

0 , Jacobi . 0

* > ..& .

L(η) η , 1 ∂F a+b a+b a+b ds h L F ·, = s, ∂s 2 2 2 0 1 1 dηa dηb 1 ds + ds ds 2 0 ds 0

7

=

1 [h(a) + h(b)]. 2

@A* = ' 12!* 4 * & 3=& .*##

@A8 _ ©, !* ,

3& 1 ] _ 4 * & 8 I7 - * * -%# # - 7 ! * & # d 9 $ ] _ ( . g a& .

Cartan-Hadamard

.

˜n M

6.6

Ω Cartan-Hadamard n ˜ , x∈M PΩ (x) ∈ Ω

(i)

,

d(x, PΩ (x)) = d(x, Ω);

(ii)

PΩ

˜ n → Ω, M

x → PΩ (x)

.

˜ n, x ∈ M

(i)

d(x, yx ) = d(x, Ω). yx ,

yx∗

yx

,z

σyy∗

l = d(x, z).

zxyx

α + α∗ = π.

Ω

yx∗

∈ Ω, .

σyy∗

zxyx∗ ,

yx ∈ Ω

,

η

d(x, yx∗ )

= d(x, Ω). ˜ η [0, l] → M n z σyy∗

z

π . 2

x

z,

α

π α . 2

σyy∗

α∗ (

20).

Cosine

§6.1

$3f`Q[\$d[\

· 101 ·

:

20

l2 + a2 − 2al cos α > l = d(z, x).

h- A . *

A* . *<; : * ;,

* ." * * E ) ("& 6A . * # da

5(. # d(x, yx )

d(yx , z) = a = 0.

2a = 0, Ω

yx

Ω

x

x

.

, d(yx , yx∗ ) =

.

(ii)

d(x, y) d(PΩ (x), PΩ (y)).

{x, y} ⊂ Ω

. 1)

x ∈ Ω,

y

,

y ∈ Ω.

xyPΩ (y)

Ω

Ω

(6.7)

. 2)

{x, y}

xyPΩ (y) .

PΩ (y)

).

(6.7)

Cosine

θ

π ( 2

PΩ (y)

,

d2 (x, y) d2 (x, PΩ (y)) + d2 (y, PΩ (y)) − 2d(x, PΩ (y))d(yPΩ (y)) cos θ

;06* ..;06 d2 (x, PΩ (y)),

(6.7)

. 3)

:

21

˜ n \Ω. {x, y} ⊂ M

% σ1

PΩ (x)

e Z@`Q[\#!! # d 3 ) 9 ..%2 <= "

· 102 ·

7 7 3) % 0 2 (" - (" & h` =* * 7 Z2 - * * 8 & 2 = 8 & =& i .' & @A8_ Z[\]+ &.<=&) $ & h @A @A* !-& - #- 3> , $%b^ +) - ] & ,- &$% ] & =& + $Ef *

-$% $ + 4 ]-&$% .@a- & 5& ] $Ef - $% ] & 4] .@a- & ©, $ Æ/0 * ] &5 3 =& 0 22

=* E 0 2 < * ˜ n. , σ1 [0, 1] → M

x

.

xPΩ (x)PΩ (y)

θ1

PΩ (·)

,

, h (0) 0.

h

PΩ (y)

y

PΩ (x) , θ2 ( 21). π π θ1 θ2 . h(t) = 2 2 h (0) = − cos θ1 − cos θ2 0. 6.5

,

d(σ1 (t), σ2 (t)).

˜n σ2 [0, 1] → M

,

h (t) 0

.

t 0, h [0, 1] → R

d(PΩ (x), PΩ (y)) = h(0) h(1) = d(x, y).

.

KM n 0

˜ n , g˜), (M

2

2

6.6

6.6

.

2

{(x, y, z)|x + y + z = 1}

Ω

. Ω

,

2

PΩ (q)

,

2

PΩ S (1) → Ω

S (1) =

q = (0, 0, 1)

,

.

6.1.3

.

˜ n , g˜) (M

.

A

B,

Haus-

dorﬀ

Hd(A, B) = inf{r > 0|Ur (A) ⊃ B, Ur (B) ⊃ A},

˜ n |d(x, A) < r} Ur (A) = {x ∈ M

A

r

.

{σ1 , σ2 }

+∞.

Y1

Y2

Hd(σ1 (R), σ2 (R)) <

Hd(Y1 , Y2 ) < +∞,

(

).

6.7 (

(1)

σ1

˜ n , g˜) (M

)

Cartan-Hadamard

˜ n , g˜) (M

σ2

.

.

a = Hd(σ1 , σ2 ),

˜n ϕ R×[0, a] → M

ϕ(R×{0}) = σ1 (R)

σ2 (R).

ϕ(R×{a}) =

.

(2) (Eberlein, 1982)

Y1

Y2

, a = Hd(Y1 , Y2 ).

˜n ϕ Y1 × [0, a] → M

ϕ(Y1 × {0}) = Y1

ϕ(Y1 ×

{a}) = Y2 .

,(1)

(2)

.

.

6.5,

h1 (t) = d(σ1 (t), Y2 )

σi R → Yi

.

(2).

.

Hd(Y1 , Y2 ) = a

,

$3f`Q[\$d[\

< =* 5 ** * 3 ) # ,

3=& 1 2 &##

@A8 _ )* 5. f*! 6? f* d 5(. a §6.1

· 103 ·

h1 R → R

. h1

,

h2 (t) = a,

Yi

.

˜ n → Yi PYi M

6.6

(PY1 ◦ PY2 )

(PY1 ◦ PY2 )(x1 ) = x1 . x1

a

h2 (t) = d(σ2 (t), Y1 ).

,

.

h1 (t) = a

θ

.

x1 ∈ Y1 ,

{x1 , x1 , PY2 (x1 )} π . Cosine , 2

.

Y1

x1 =

h : _ E ." * < ; , &" ,

@A= &."& .@ .. # ( > ! # ,

) * =& ) f* b&g"bcf* g 9d5( -, . dÆd 2 & d5;( . & cf* d + ) & ] - + & - ÆÆ ?

'1 ] . @ a- 7 a2 + d(x1 , x1 )2 − 2ad(x1 , x1 ) cos θ > a.

a = d(x1 , PY2 (x1 ))

x1 = x1 ,

,

PY1 ◦ PY2

.

PY2 ◦ PY1

PY1

= idY1 ,

,

= idY2 .

Y2

PY2

Y1

PY1 |Y2

,

.

˜ n, Y1 × [0, a] → M

ϕ

(y1 , t) → σy1 ,PY2 (y1 ) (t),

˜ σx,y [0, l] → M

y2 = PY2 (y1 )

x

y2 = PY2 (y1 ),

{y1 , y1 , y2 , y2 }

,

Σ

y1

y1 ,

Y1

Y2

PY1 ◦ PY2 |Y1 = π . 2

22),

Σ

( 6.3) π .Σ 2π 2 ϕ(σy1 ,y1 × [0, a])

.

.

{y1 , y1 } ⊂ Y1 . Σ(

PY2 ◦ PY1 |Y2 = idY2 .

idY1

y

2π.

Σ

.

˜n ϕ Y1 × [0, a] → M

.

: . & - > < ! .i5c 22

,

.

e Z@`Q[\#!! [jd e ekgij ! - F F G 5&+f ^ . "&7 7 &+3 7 - Æ/ ^_`abVc /de+

- $ 4 6 & !

* 2 + .@* -4 * & ©, j= < 5 * j 3Æ , 3 < 5 2& ) * -, 1 ]" .; *=*? 3

=* ' .;I* .;* 4 & % 8 7#6 97 *% ac 1 <; .;* "

· 104 ·

§6.2

Preissmann

Cartan-Hadamard

Mn

,

Mn

,

Mn

,

n

π1 (M )

.

M

K(π, 1)

n

.

n

π1 (M )

.

6.2.1

Cartan

(M n , g)

, ϕ ∈ π1 (M n )

Riemann k

k, ϕ = 1.

Cartan

ϕ

ϕ

ϕ = 1.

,

Z.

.

6.8 (Cartan

˜ n , g˜) (M p

,

˜ n , g˜) ) (M ˜ n ). G ⊂ Iso(M ˜n M G

G{p}

A.

Cartan-Hadamard ˜n M p

, G

G

.

A

A

.

˜ n → R, M

rA

x → max{(d(x, a))}. a∈A

˜ n , g˜) (M

x→∞

Cartan-Hadamard

rA (x) → ∞.

,

rA

,

,

rA

x0 .

.

x0

rA

.

rA

σ

x0 x∗ 0

d(ˆ y , a) <

π . 2

, d(·, a)

,

x0

x∗0

˜n x∗0 ∈ M

,

˜n x∈M

yˆ,

max{d(x0 , a), d(x∗0 , a)}

rA (ˆ y ) < rA (x0 ) = rA (x∗0 ),

rA (x∗0 ) = rA (x0 ) = inf rA (x).

23

.

rA

:

23

θa +

Cosine

θa∗

= π, max{θa , θa∗ } x0 .

TaSf $fKhPK , 2

.; * .; * * 7 12

-

.i ! - $ 4 5& ! 6 ©, % F FG $4 % Ag.@? G < *&; 5 5.G * +&E! - $47 .* < fg¬`! +

. @\ % .@ % §6.2

· 105 ·

Preissmann

G(A) = A.

G{x0 } = {x0 },

x0

G

.

.

˜ M

π1 (M n )

M

ϕ

.

.

G

,

G{p} ˜n M G ⊂ π1 (M )

. G

n

Z.

,

6.2.2

g

ϕ ∈ π1 (M n ), ϕ = 1

x0 .

, G = ϕ

n

.

G = ϕ

,

Z.

˜ M

, g˜

˜ n , g˜) (M

˜n M

π1 (M n )

ϕ

n

,

Mn

,

ϕ

n

rA

.

(M n , g)

6.9

.

rA (G(x)) = rA (x). G(x0 )

,

.

Preissmann

˜ n , g˜) (M

Cartan-Hadamard

,

ϕ

˜ n , g˜) (M

,

˜ n |d(x, φ(x)) = inf d(y, φ(y))}. Min(φ) = {x ∈ M ˜n y∈M

G

˜ n , g˜) (M

,

Min(G) =

!

Min(φ).

& 2 - + !\ &!

2 -=&+.@ *

, $%` - + .@ &!©, ! 02 $` 5 9 k< * +) l& ! ,

. @ * ! 6 * =* φ∈G

.

6.10

Z.

ϕ

ψ

π1 (M n )

(M , g)

(1) Min(φ)

,

˜ , g˜) (M

(2) ψ[Min(φ)] = Min(φ), D×R . (1)

x

ψ, ϕ

.

Ω × R; ϕ

.

2

Z⊕

,

n

Min(φ) R

"

2

Min(ψ)

,

Min(φ)

"

R

Min(ψ)

.

x ∈ Min(φ),

d(x, ϕ(x)) = c = inf {d(y, ϕ(y))}. ˜n y∈M −1

{x, ϕ

δϕ (y) = d(y, ϕ(y))

σx R → Min(ϕ).

σx

σx (0) = x

{x, y} ⊂ Min(φ),

{σx , σy }.

(x), ϕ(x)}

ϕ(σx (t)) = σx (t + c). ϕ

,

h(kc) = d(σx (kc), σy (kc)) = d(ϕk (σx (0))),

h(t) = d(σx (t), σy (t))

ϕk (σy (0))) = d(x, y) = b

Ω×R

k

.

h(t)

t

,

e Z@`Q[\#!! ! & - 6 &

= .@a- < 02$Ef &4] & $% c ..& ,

* 5 =& k &

$Ef 2 . @ , !\ -/

"

· 106 ·

{k1 , k2 }

t,

k1 < t < k2 .

h

,

h(t) max{h(k1 ), h(k2 )} = b,

Hd(σx (R), σy (R)) = a b ˜n η R× [0, a] → M

.

6.7

η(R× {0}) = σx (R) η(R× {b}) = σy (R). n ˜ s ∈ [0, a], η(·, s) R → M σx . ϕ

,

η(R × {s})

, η(R × {s}) ⊂ Min(ϕ).

δϕ (x) = d(x, ϕ(x))

= δϕ−1 ({c})

η(R × [0, a]) ⊂ Min(ϕ).

c = inf {δϕ (y1 )}

,

˜n y∈M

, Min(ϕ) = δϕ−1 ([0, c])

x0 ∈ Min(ϕ),

.

Ω = {y ∈ Min(ϕ)|Exp−1 x0 y, σx0 (0) = 0}.

Ω × R.

Min(ϕ)

(2)

ψ

ϕ

,

& =& % @A8_ ! , -/

δϕ (ψ(x)) = d(ψ(x), ϕψ(x)) = d(ψ(x), ψϕ(x)) = d(x, ϕ(x)) = δϕ (x),

ψ(Min(ϕ)) = Min(ϕ). (1)

, Min(ϕ)

Pϕ

˜ n → Min(ϕ) Pϕ M

.

˜ n, x∈M

,

##c 2 & ,

6.6

ψMin(ϕ) = Min(ϕ),

Pϕ (ψ(x)) = ψPϕ (x).

(6.8)

δψ (Pϕ (x)) = d(Pϕ (x), ψPϕ (x)) = d(Pϕ (x), Pϕ (ψ(x)))

(6.9)

& , & -& & - ! *

4 & $ +

$% $E f ]& ] f ^ $&

f ^ $& # % $Ef 5 $% $&] &.@ # d(x, ψ(x)) = δψ (x).

(6.6) (

Min(ϕ)

"

Min(ψ) = ∅,

Min(ψ)

x ∈ Min(ϕ)

σxϕ (0) = x

, "

Pϕ (Min(ψ)(

Min(ψ),

σxψ (0) = x,

,

x ∈Min(ϕ)

"

{ψ ◦

ψ◦

"

σxϕ

σxϕ

σxϕ

, ψ σxϕ

σxψ

.

σxϕ (R)}k∈Z .

Y1 = R2x ,

R2x1 × [0, a],

σxψ ,

.

.

R2x ⊂ Min(ϕ)

R2x

Min(ψ)

)). ,

Min(ψ),

x ∈ R2x , ψ, ϕ

,

(1)

ϕ

ψ(Min(ϕ)) ⊂ Min(ϕ), k

Pϕ (Min(ψ)) = Min(ϕ)

, R2x1

a = Hd(R2x1 , R2x2 ).

"

Min(ψ), R2x2

TaSf $fKhPK * ..% 0 2 & . @ 7 12

2 © 7, , !!\* 5& ! 6 &

.i - $ 4 !\ ! - & 5 : ] f ^ < ; 1 # 7 5 hi¬`!jklmn+ *

- 44 m] a h n" l, - i < 8 2

e i

!\ ,91

!\

o> §6.2

· 107 ·

Preissmann

x0 ∈ Min(ϕ)

"

Min(ψ).

D = {y ∈ Min(ϕ)

,

Min(ϕ)

6.11 (Preissmann

"

!

2 Min(ψ)|Exp−1 x0 y, Rx0 = 0}.

D × R2x0 .

Min(ψ)

.

(M n , g)

)

Riemann

n

,

Z.

π1 (M ) ,

n

H

, π1 (M n )

Cartan

π1 (M )

,

{ϕ, ψ} ⊂ H, {ϕ, ψ}

H

˜ n , g˜) (M

Z,

H = ϕ0 ∼ = Z,

.

R2 ,

lϕ0 = inf {lϕ }

.

lϕ = inf {δϕ (y)}.

.

˜n y∈M

ϕ∈H ϕ=1

6.2.3

.

.

.

G

G = Gk Gk−1 Gk−2

· · · G1 G0 = 1 (i) Gi

Gi+1

,

(ii)

Gi+1 /Gi

G

.

,

,

,

⎧⎛ ⎪ ⎪ ⎨⎜ 1 G= ⎜ ⎝ 0 ⎪ ⎪ ⎩ 0

.

⎞

⎟ m ⎟ ⎠ 1

k,

l

1

⎫ ⎪ ⎪ ⎬ k, l, m ∈ Z ⎪ ⎪ ⎭

o> $ !\ ^ o> // 2 - mn : & ;= C

2 - +

5 $E G5+ ":&5bFG ; &! + C + j j j j & jj

,

0

.

G

.

?

,

6.13.

1968

, J.Wolf ,

M

,

n

(M n , g)

Preissmann

π1 (M n )

(M n , g)

,

n

n

n

Z

, π1 (M )

π1 (M n )

Bieberbach

n

|π1 (M )/Z |

.

1971

, Lawson-

,

n

.

Gromoll-Wolf

(§6.3)

,

[Y1],

e Z@`Q[\#!! & - 12 $E:& 2 - ©,&!$ +) Æ; ] " ]_- _- $E:&"$E:& +!

, ] _ - I9 ] a- , &!4 &! 1& - + . @& ! # , $` F FG &c ! *

] "$E:& 7 j jd + j j

2 - + - +. @& ! *

2$E 5 . G Æ ©, G + Æ & ! &

& ! - , = +

> , !\ 2 $* Ælp '?: -=& +

= .@ - * ?: * ? :

> * "

· 108 ·

[LY]

[GW].

6.10(2)

6.12 (

(M n , g)

)

Zk

Mn

k

f T

k

→ M

.

n

T

f∗ π1 (T k ) → π1 (M n )

˜n f˜ Rk → M

(

π1 (M n )

,

α1 , · · · , αk ,

k

k

f

,

, f

, π1 (M n )

).

π1 (M n )

,

,

Zk .

Zk ∼ = Zα1 ⊕ Zα2 ⊕ · · · ⊕ Zαk = k " Min(G) = Min(αi ) , Min(G)

G

6.10(2)

i=1

D × Rk ,

˜n → M π M

.

π({x0 } × Rk ) = Σk

x0 ∈ D,

.

Rk

G

k

.

.

.

[Y1]

6.13 (

G

M

n

)

n

(M , g)

Riemann

n

π1 (M )

,

,

Min(G)

D × Rk

(1) G({d} × Rk ) = {d} × Rk ,

d∈D

(2) {d} × Rk /G

,

{d} × Rk

(3) Φ = {ϕ ∈ G|Φ ,G

}

G

G

m

m = 1,

Z

.

|G/H| < ∞.

Min(ϕ)

ϕ

G = Gm Gm−1 · · · G0 = {1}

,

G

.

k

H

˜ n , g) → (M ˜ n , g) ϕ (M

.

,

,

.

G

,

6.12(2)

6.12,

(1)∼(3)

.

(6.13.A)

X

Cartan-Hadamard

G

,Ω

X

,

(1)∼(3)

(6.13.A)

m−1

A

X

, G(Ω) = Ω.

.

m=1

.

(6.13.A)

.

G = Gm Gm−1 = A Gm−2 · · · G0 = {1},

Min(G)

"

,

Ω

$ K@PK e 02-/ !\ e, 1 2 + , i ! .@ - & .. 2 +? : * 7 m`ij & ! - k ! & 2 5" - /f

+ 2 - + ( - $ 4 ? . @

©, + '4 4 / d

& % & + =Æ

2 + - G + . - $ 4 ?© ,! I' I4 2 i 2 & , 2 4 & -/

§6.3

Gromoll-Wolf

· 109 ·

Lawson-Yau

Min(A) = DA × RkA

G

,

−1

ϕ∈G

,

ϕ

G/A

.

A

Aϕ = A.

δψ (ϕ(x)) = d(ϕ(x), ψϕ(x)) = d(x, ϕ−1 ψϕ(x)) = δϕ−1 ψϕ (x),

ϕMin(ψ) =Min(ϕ−1 ψϕ).

ϕMin(A) = Min(ϕ−1 Aϕ) = Min(A)

G(Min(A)) = Min(A).

G = G/A

,

DA

G (DA ) = DA .

Min(G ) = D × RkG ,

(6.13.A),

D×R

kG

×R

kA

=D×R

k

(1)∼(3)

§6.3 Gromoll-Wolf

,

.

Min(G) =

.

Lawson-Yau Mn

K(π, 1)

n

π1 (M )

.

.

.

n

6.14 (Gromoll-Wolf, Lawson-Yau)

(M , g)

π1 (M n ) = Γ 1 × Γ2

,

(M1 , g1 ) × (M2 , g2 )

(M n , g)

.

π(Mi ) = Γi

6.14

(i = 1, 2).

,

.

.

A

Cartan-Hadamard

X

X

,

Con(A)

A

.

6.15

˜ n , g˜) (M

˜ n /(Γ1 × Γ2 ) M ,

.

˜ ,Con((Γ1 {x0 })/Γ1 ) x0 ∈ M Mn

,

.

{xi } ⊂ Con(Γ1 {x0 }),

{ϕi } ⊂ Γ1

Mn =

Γ2

n

.

Γ1 {x0 }) → ∞.

, π1 (M n ) = Γ1 × Γ2

Cartan-Hadamard

{ψi } ⊂ Γ2

d.

d(ψi (x0 ), x0 ) = d(ϕi ψi (x0 ), ϕi (x0 )) d(xi , ϕi (x0 )) − d(ϕi ψi (x0 ), xi ) d(xi , Γ1 {x0 }) − d → ∞,

d(xi ,

d(ϕi ψi (x0 ), xi )

e Z@`Q[\#!! g % , !\ c k "

· 110 ·

{ψi } ˆ ˆ {ψ1 , · · · , ψm }

i = j

, ψi = ψj .

Γ2

Γ2

.

sup

y∈Con(Γ1 {x0 })

Γ1

.

d(y, ψˆj (y)) rj = d(x0 , ψˆj (x0 )),

j,

δψ-1 ψˆj ψi (x0 ) = d(ψˆj ψi (x0 ), ψi (x0 )) i d(ψˆj ψi (x0 ), ψˆj ϕ−1 (xi )) + d(ψˆj ϕ−1 (xi ), ϕ−1 (xi )) i

i

i

+d(ϕ−1 i (xi ), ψi (x0 )) = d(ϕi ψi (x0 ), xi ) + δψˆj (ϕ−1 i (xi )) + d(xi , ϕi ψi (x0 ))

, 2 ? a -$4 ? <!\ ! 1 ; 7 + 4

] + .@&! ©, %

=&

2 .; ! .;& ! % .@&! # .@ "%% d + rj + d.

˜ n /(Γ1 × Γ2 ) Mn = M

,

i, k

ψi−1 ψˆj ψi = ψk−1 ψˆj ψk ,

ψi−1 ψk

ψˆj

{ψˆ1 , · · · , ψˆm }

.

,

.

Γ2

˜ n , Con(Γ1 {x0 }) M ˜ 1 , g˜1 ) × (M ˜ 2 , g˜2 ). (M

6.15

Riemann

,

. n ˜ (M , g˜)

F = {Ω|Ω ⊂ Con(Γ1 {x0 }), Γ1 Ω = Ω, Ω = ∅, Ω

Con((Γ1 {x0 })/Γ1 )

,F

[0, ly ],

$%,

Ny

x0 ∈

}.

N0

% Con N0 Ny

Ny = Con(Γ1 {y}).

ly = d(y, N0 ),

Γ1

x ∈ N0 ,

N0 .

Con(Γ1 {x}) = N0 . y ∈ N0 ,

Γ2

.

(M n , g), π1 (M n ) = Γ1 × Γ2

6.16

,

N0 ×

N0 .

&

& * ## / b* & b g d 16b g5

] $E- $ =& .; 5 N0

Γ1 .

Σ0 = {x ∈ N0 |d(x, Ny ) = d(N0 , Ny )}.

Γ →N

Ny

˜n Py M

1

,

Σ0

Γ1

z ∈ Σ0 , ϕ = 1 ∈

.

,

y

d(z, Py (z)) = d(ϕ(z), Py (ϕ(z))) = d(N0 , Ny ) = ly ,

{z, Py (z), ϕ(z), Py (ϕ(z))} .

Σ0

Γ1

,

N0

π . 2

,

N 0 = Σ0 .

QnKQÆSK@PK & . , . @ &

.g I .g * %

= 9 $% $+& * E # 02!8 $Ef * @ * ! .g -f*7# -d<; , + 2+ - .@

+ .-$ 4 ? , $ 4 , A g ©, I -$4 , !\ 4 & - $ 4+

] " < , = 1 5 >* 91 5 54 . @

# 5 $ , 4 2 c -$4 ? <; , % ## , , , A g I 4 $ 4 * &

4 4 2 a 4 * & 5 g+ g f <+k + Ag 7 * E 5 , 4 4 '2 \obcodm`ij .? E

2- + -$4? ^

:.@ # $E §6.4

· 111 ·

Eberlein

% Con N0 Ny

N0

x

,

Ny

,

x ∈ ∂N0

y ∈ N0

y = σ(t0 + ε),

xp ∈ Int(N0 ).

{x0 } × [0, l].

z

z

x0

{x0 , z, y}

,

.

σ

t0 = d(x0 , x). , % ∼ Con N0 Ny = N0 × [0, ly ].

N0 × [0, ly ],

,

Ny

x0

, N0 = Con(Γ1 {x0 })

.

.

6.17 ˜ n , g˜) (M

Γ2 .

N0 × Y .

.

N0

N0

˜ n , g˜) , (M

N0 × [0, ly ].

(M n , g)

Riemann ˜ 1 , g˜1 ) × (M ˜ 2 , g˜2 ), Γ1 (M ˜ 1) = M ˜1 (M

, π1 (M n ) = Γ1 × Γ2

,

(1) Γ1

˜2 M

,

(2) Γ2

˜2 M

.

(1)

Γ1 , ϕ = 1

ϕ

˜2 M

Γ1 ˜2 M

Γ1

.

Γ2

δϕ |M˜ 2

.

˜ 2 , g2 ) , (M

.

˜ n /Γ Mn = M

k 1. ˜2 M

δϕ ˜2 ∼ ˜ , M = Rk × M 2

Γ2

,

ϕ ∈

,

,

Γ1

.

.

˜n → M ˜i Pi M

(2)

,

ρ2 (Γ )

βn → 1.

.

6.16

.

βn = βˆ

βn → 1,

.

˜2 M

Γ1

βn = ρ2 (rn ) rn∗

.

∗ ξn = rn+1 (rn∗ )−1

.

βn+1 βn−1 ≡ 1

˜ 1 /Γ1 M ˜ 1) Iso(M

[ρ1 (rn∗ )]−1 [ρ1 (rn )]

,

.

˜2 M

6.15

rn∗ = (ρ1 (rn∗ ), βn )

˜n M i

, ρi

˜ 1. x1 ∈ M ˜ 1 )× {1}, → 1, ξn ∈Iso(M ˜2 M βn ≡ 1, ,Γ

.

6.14

6.15∼

6.17

.

§6.4 Eberlein

.

π1 (M n )

(M2 , g2 )?

(M1 , g1 )

,

, dim(Mi ) 1.

(M n , g)

(M n , g)

(M1 , g1 ) ×

e Z@`Q[\#!! & - o " ! 8$E g . B'=h% pqrs

- 4-$4 &k % % #

$E ":& $ . @

,$4 l4 $ 4 &0 )4 (% $4 ! "

· 112 ·

g

,

.

Lawson-Yau

6.4.1

Wolf

Calabi

π1 (M2n−k )

Riemann

→ Rk .

g 0 = g10 ⊕ g20 .

Mn = T k ×

˜ n , g˜) (M

Zk × π1 (M2n−k ). ˜ n−k M (i) Zk 2 (ii)

.

Calabi, Lawson-Yau

(M2n−k , g20 )

ρ

Calabi

π1 (M2n−k )

M2n−k ,

(T k , g10 )

k

,

˜ n , g˜) = (Rk , g˜0 ) × (M ˜ 2 , g˜0 ). M n (M 1 2 n−k k n ˜ M Γ = Z × π1 (M2 ) (

6.17),

Rk

,

,

.

π1 (M2n−k )

R

π1 (M2n−k ),

k

(ϕ, ψ) ∈ Zk ×

.

˜ n , g˜) ρt Zk × π1 (M2n−k ) → Iso(M

, i 3 - 2 $ + * &+ E

i o #

- $ 4 &k $

2 ? E & %

#

? E > "& > < mh` (% !-$4 &k 4 &k

-$4 &k ll ti+ & f ^ , ^ f^, d hn n ) -.@\ # ? < f^1$& " -/ e!\ <Æ

ρt (ϕ, ψ)(x, q) = (ϕ(x) + tρ(ψ), ψ(q)),

˜ 2. (x, q) ∈ Rk × M

˜ n , g˜), ρt (Γ ) ⊂ Iso(M ˜ n /ρt (Γ ) Mtn = M

,

˜ n /ρt (Γ ) M

˜ n /Γ , M

Mtn = T k × M2n−k

g t = g˜/ρt (Γ )

,

t > 0, ρ

.

M2n−k

H1 (M2n−k , Z) = Zk ,

,

H1 (M2n−k , Z)

=

M2n−k

k = 2m, m

G

η π1 (M2n ) →

ρˆ Zk → Rk ,

.

.

.

H1 (M2n−k , Z) = Zk . Rk

M2n−k

π1 (M2n−k )/[π1 (M2n−k ), π2 (M2n−k )].

aba−1 b−1

, [G, G]

6.4.2

H1 (M2n−k , Z)

,

ρ = ρˆ ◦ η.

π1 (M n )

[LY].

Eberlein

˜ n , g˜) (M ˜ n , g˜) Cartan-Hadamard (M

.

(Rk , g0 )

π1 (M n )

?

Klein

A(x, y) = (x, y+1) −1

−1

AB = A

.

R2

A, B R2 → R2 ,

B(x, y) = (x+1, 1−y).

B2

π

B

.

B 2 ,

2

B , A

π

A

π = A, B.

B

1.

2.

,

2.

QnKQÆSK@PK : ;= 78 j j

2 - $+ 4 &. e!\?+< G F FG .@ - o 4 + GFG .? & * &+ # ©, 02 &! g ,! e ! & &! , e & &k -/

+) * ! !\ % * $eÆ; ?# , ? ÆÆ % ? ! ,

e $ ?? 1 e !\ ,

. !\ 1 & G % & f ^ h; G 7 6 /A9& ,

G §6.4

· 113 ·

Eberlein

,

.

6.18(Eberlein,

)

(M , g)

Riemann

n

Γ1

,

n

π1 (M )

n

(1) π1 (M )

k 1.

,

Γ1

C(Γ1 )

|π1 (M n )/C(Γ1 )|

C(Γ1 )

;

˜ n , g˜) (M

(2)

(Rk , g0 )×

Riemann

˜ n−k , g˜2 ); (M 2

(3) Mn ˆ ) = Zk ⊕ π1 (M ˆ 2 ), π1 (M

ˆ M

ˆ 2) π1 (M

n

Γ1

π1 (M )

Zk ,

Γ1 = α1 , α2 , · · · , αk .

n

ϕ ∈ π1 (M ),

,

C(Γ1 ) =

.

Γ1

(1)

ˆ 2, Tk × M

ˆ M

Γ1

Γ1 → Γ1 ,

Adϕ

α → ϕ−1 αϕ.

Γ1 = Zk

Aut(Γ1 )

Γ1

.

,

π1 (M n ) → Aut(Zk ),

Ad

ϕ → Adϕ .

ϕ−1 αϕ = α

Adϕ = 1

α ∈ Γ1 = Zk

,

Γ1

ϕ

.

& C(Γ1 ) = ker(Ad) = ϕ ∈ π1 (M n ) | ϕαϕ−1 = α

, Γ1

C(Γ1 )

.

'

α ∈ Γ1

C(Γ1 ) = ker(Ad),

.

π1 (M n )

C(Γ1 )

.

Γ1 .

C(Γ1 )

n

ϕ ∈ π1 (M ),

−1

ϕ

hϕ ∈ C(Γ1 )(

C(Γ1 )

,

,

C(Γ1 )

k

Z

L

,

Γ = π1 (M )

|π1 (M )/C(Γ1 )|

,

Adϕ

π1 (M n ) Ek

.

.

Γ = α1 , · · · , αk =

{β1 , · · · , βm }

Γ1

{β1 , · · · , βm }

m!,

.

).

H = Γ1 .

L = max{αi | i = 1, · · · , k}, " {β1 , · · · , βm } = BL (0) Γ1 . Adϕ

Ad(Γ )

m!,

H

h ∈ H,

.

π1 (M )

n

,

C(Γ1 ) n

C(Γ1 )

ϕ−1 hϕ ∈ H,

Γ1 ⊂ H

H

.

|Γ /Γ1 | = |Ad(Γ )|

* (2)

e Z@`Q[\#!! .@ ,

"

· 114 ·

Min(Γ1 ) = Min(Zk ) = Rk × D

.

,

,2

1 , G G FG 2 , =Æ 5 6 1 & =Æ * , - 0 2 j j - -$4 # 2 ? (& ,$ ?4 4 #% ## $ , e ( . @ 4 $4 '

4 &k& (

A g , 2 i 1

2 % I9 &+ 7 .@

e+!\ 2 + fF ^FG, + ) e! \©, &! e !\ &! 02= & GFG 5 &+

+ 4 e I9

f ^ , GFG *&+ % F FG qc(

: 3 ! α

i

˜ n. Min(Γ1 ) = Min(Zk ) = M

∈ Γ1 , ϕ ∈ C(Γ1 ),

δαi (ϕ(x)) = d(ϕx, αi ϕ(x)) = d(ϕx, ϕαi (x)) = δαi (x),

Min(αi )

C(Γ1 )

,

˜ n, M ˜n Min(αi ) = M ˜ 2 , g˜2 ). (Rk , g0 ) × (M

αi ∈ Γ1

Γ1

⊂ C(Γ1 )Min(αi ) ⊂ Min(αi ), ˜n ˜ n , g˜) = Min(Γ1 ) = Min(Zk ) = M (M

,

, C(Γ1 )/Γ1

C(Γ1 )

).

˜ 1 = Rk , ρ i M ˜ →M ˜i M ϕ ∈ Γ(

,

ρ2 (Γ1 )

Γ

,

ϕ = (ρ1 (ϕ), ρ2 (ϕ)). ˜2 M ρ2 (C(Γ1 )))

, ρ2 (Γ )( ˜ 2 /C(Γ1 ) M ,

˜ n /C(Γ1 ) = M ˆn M

Γ = π1 (M n )

Γ1

˜ n = Rk × M ˜2 M

ϕ ∈ C(Γ1 )

˜ M

˜2 M

.

ϕ ∈ C(Γ1 ))

ρi Γ → Iso(Mi ).

.

˜2 Rk × M

Γ1

6.17

.

C(Γ1 ){x0 }

.

,

(

Γ = π1 (M )

C(Γ1 )

˜ n /C(Γ1 ) M Mn ˜ n = C(Γ1 ){x0 } M

˜ n , g˜) (M

(3)

x0 ∈ Min(αi ).

.

ˆ 2. Tk × M

ˆ2 = M ˜ 2 /C(Γ1 ), M

.

.

˜ , g˜) = (Rk , g˜0 ) × (M ˜ 2 , g˜2 ) (M

,

6.17

π1 (M n )

Zk .

6.19(Eberlein ˜ , Mn Mn ˜ 2 , g˜2 ) (Rk , g0 ) × (M

(M n , g)

)

, g˜

g

.

Riemann n ˜ (M , g˜) deRham

k

R (k 1)

π1 (M n )

Zk .

Γ1

π1 (M n )

˜ n , g˜) (M

ˆ M

,

Rk (k 1), ˜2 → Mn Rk × M Mn

Γ1

˜ n , g˜0 ) = (Rk , g0 ) × (M ˜ 2 , g˜0 ), (M ˆ 2. Tk × M

˜ n , g˜) (M

deRham

Mn

.

.

Zk

,

6.18

M

˜ 2 , g˜2 ) (Rk , g0 ) × (M ˆ ˆ 2. M T k ×M

˜ = π M

π(Rk × {y})

QnKQÆSK@PK % o:& 7 c , op .,

#

! 7 m . ##:&

3& & *

# ,- f ^ , !8 G FG i . ]"$%" G FG !8 ,

! .@ ^, -f^,-f^ , ! f ^,f #

. @ eq.@ -r*]"$%" E n( \ / r r * ] "$%" , ? * ]"$%" ! 8=

3 )% $` ! .@ ! % ! 8

<& % G e % <& G& , G eFG 2 . @\ ! 1 . @\ k ! *

7 # .@*

!\ % n" §6.4

· 115 ·

Eberlein

(M 2 , g) = R2 /Z ⊕ Z = S 1 (1) × S 1 (1)

.

,

R2

2π.

R2 = {Rv1 } × {Rv2 },

v1

v2

(T 2 , g0 )

π({Rvi } × {z})

, vi = (ai , bi ),

deRham

˜ 2 , g˜2 ) (Rk , g0 ) × (M

ˆ M

˜n

k

˜ , g˜) = (M

.

Mn

.

.

n

,

˜ 2 , g˜2 ) (M

ˆn M

ai bi

,

ˆ M

,

k

.

.

˜ 2 , g˜2 ) (M , g˜) = (R , g0 ) × (M ˜n → M ˜n ϕ M n

ϕ ∈ Γ = π1 (M ) ˜ n−k , g˜2 ) , β(ϕ) (M 2

deRham

˜ n , g˜) (M

,

,

.

ϕ = (α(ϕ), β(ϕ)),

α(ϕ)

R

k

.

R2

,

,

.

(M n , g)

,

Klein k

.

ˆ M

n

6.18(1)

.

A = Rk

Rk

ϕ ∈ Γ,

.

ˆ ϕ A → A, Ad

ψ → [α(ϕ)]−1 · ψ · [α(ϕ)].

{α1 , · · · , αk }, L = sup αk . A = α1 , · · · , αk = Zα1 ⊕

A

1ik

ˆ ˆ Zα2 ⊕ · · · ⊕ Zαk . A BL BL , Ad(Γ ) Ad(Γ ) " ˆ ˆ CΓ (A) = ker(Ad) Γ , Γ /CΓ (A) ⊂ Ad(Γ ) , n ˆ =M ˜ /CΓ (A), ˆ M M CΓ (A) Γ .

.

.

M

n

. .

Riemann n

ˆ , gˆ) (M

ϕˆ

ˆn M

x0 , ϕˆ

,

ˆn M

}

ˆ n , gˆ) (M

.

,

x0

ˆ M

ϕˆ

n

.

ϕˆ∗ |x0

ˆ n → T x0 M ˆn ϕˆ∗ |x0 Tx0 M

H = {ϕˆ ∈ G | ϕ(x ˆ 0 ) = x0 },

ˆ n, H →G→M

,

,

x0

ϕˆ

ˆ → G = {ϕˆ | ϕˆ M

e Z@`Q[\#!! #

! 2 ,

21 &#( 1 -/

, $` \ p j r * ] "$%" " 1

.@

7

"

&

, $` 1 .5Æ; @ ":& 1Æ;-, 1 , 21 1 ! " $% ! j ]

Æ; , j Æ;&k ? , e,

$`e 7 oAAupqfA q g ! 4 hv.; / "*% .; "

· 116 ·

H ⊂ O(n)

.

,

G . ˆ n , gˆ) = (M ˜ n , g˜)/CΓ (A) (M

H

ˆ n , gˆ) G = Iso(M

,

.

A

{v1 , · · · , vk }. v˜i

,

,

T k.

k

,

k

ˆ n) π1 (T ) → π1 (M ˜ n , gˆ) (M Rk Γ

vi

R

k

,

A.

ˆ , Tk x0 ∈ M

CΓ (A)

.

G

,G

k

,

k

Z .

k

. Z ⊂ A,

Zk

,

A

x0 ˆ , π1 (M n ) Z

,

G

Tk

k

,

, n ˜ (M , gˆ)

{ϕˆi (t)}t∈R

{ϕˆi (t)} ϕ˜i (t) n k ˜ , g˜) = (R , g0 ) × (M ˜ 2 , g˜2 ) (M

{ϕ˜i (t)} ⊂ A

.

k

vi ˜ (M , g˜)

.

Rk

CΓ (A)

Γ

T

k

CΓ (A)

.

.

§6.5 Gromov

20

80

, Gromov

,

(Minimal Volume).

Mn

6.20

n

.

Mn

MinVol(M n ) = inf {Voln (M n , g) |Kg | 1, (M n , g)

\r7 .;

Mn

,

}.

Riemann

|Kg | 1

.

.

2

,

k

|Kg | 1.

2

.

Σ2k

g

Gauss-Bonnet

( ) Area Σ2k , g

.

Σ2k

Riemann

,

|Kg |dA Kg dA = |2π(2 − 2k)| , Σ2 Σ2k

k

Riemann

§6.5

Gromov

:!!!$H`Qj!!

· 117 ·

( ) MinArea Σ2k = |2π(2 − 2k)|.

, "8& ?E .k * " , 2 & r o - o, :&, !":& .; > &" r 2 s n"

n" &+ ":& ; 0 6 .; + , 2

n" n"! >

e Æ; ? n" p j - i 0 2 5 &" o $ ! 8 X2 s 4 9 39 " &

" - s g "o % $ s & got " g< g?E " d 6 -- $ + &+ t ":& 9 +l& - m k 1 / !\ , & ]" o # Σ2k

,

S2

2

,

k = 0.

,

MinArea(S 2 ) = 4π.

, S2

.

,

n

M =Σ

n−k

Mn

.

k

×T .

T = S1 × ) ( n−k , MinVol Σ × Tk

.

· · ·×S 1 = Rk /(εZ)k ,

Tk

,

k

.

= 0.

T k → M n → Σn−k ,

.

k

.

, Mn

,

T k → M n → Σn−k

.

Zk ∼ = π1 (T k )

π1 (M n )

(

n

M ,

Eberlein-Yau

,M

,

M

π1 (M n )

,

n

).

g, n n−k k ˆ M =Σ ×T .

n

,

n

MinVol(M ) = 0.

n=3

, Gromov

,

,

.

.

Vα =

M

{V1 , · · · , Vm }.

m

Σ2α × S 1 ,

,

Σ2α

Σ2α

2

g

.

,

) Σ22 ,

M

Σ21

Σ22

.

3

α=1

.

Vα

3

3

Σ2α

2

Σ2α

,

=

m %

Vα

Σ2α

,

3

,

∂Σ2α × [0, ε]

1

= S × [0, ε].

)%( 1 ( S × m = 2, M 3 = Σ21 × S 1

,

[T 2 − D2 ] = Σ21 = Σ22 ,

2

24.

,

(

)

S

1

,

V1 = Σ21 × S 1 V2 = S 1 × Σ22 * + * 1 + % M 3 = Σ2 × S 1 S × Σ2 f

.

f

( 2) ) ( ∂Σ × S 1 → S 1 × ∂Σ2 , (t, s) → (s, t)

,

"

· 118 ·

&!+

e Z@`Q[\#!!

f ◦ f = Id.

: 9 on" 2 s & 9 Æ/

2 - $

# e G" FG:& & o -& !& -& :&, , &!\ # :&$+ :& Z , s 9 " - 9 Æk p Æk

9 l "9% xs n( / = d . ; > "9 * I9 ^ $ + p j i& :n 8

s 9u r ;q& ! + rm 8 1 r ;q

2.g - 4 +" * & 5 s 9 2 7 > 1#s 24

,

.

.

k=1

Riemann

Mn =

,

Uk

(1)

T

(M n , g)

6.21

m %

αk

×Σ

Vk → Uk

Uk

n−αk

,

T

(2)

αk

Uk

T αk

,

Vk =

αk 1.

αk

Uj

Vk

,

T βj,k ,

Vj

T αj

Vk " Vj Vk

T βj,k

.

Mn

,

.

M 3,

3

Schroeder

,

Gromov

,

[Bu 1], [Bu 2]

[Sc].

3

,

3

MinVol(M ) = 0

3

M

n

,

MinVol(M ) = 0

,

6.22(

−1 KM n 0

Vol(M n , g) εn

3

.

M

Mn

.

6.22

Buyalo

n

?

Cheeger

Cheeger

Riemann

)

(M , g)

.

,

εn ,

, Mn

[CCR1]

.

n

,

.

MinVol(M n ) = 0. ,

.

$!#uOPK$RtuOPKno voijsuvoijpq w 0 2 - i&

+f ^ FG 3 "&7 > && 2 + - 35+ & &7 . 6 Æk

& - 2 .g

& $ + d "&7 > - / /r&+ & s > < w

+

- 2.g & - & + " + E * &+ ! 8- 2.g & &+ E * &+ *i&

& * i

+ : * "3 * ! 35 CB

&7 " 8 *&+ " 8 " % C

. &+ " 8 & &+ " 8 &/

- *t . l9 3 s t% 3 % * &+ 3) kC

)s3 &7(> < ^

&+(&7 -;06 )w ) 0 6 +

3 + &7 . 6 & 2 i - .@&! .&t*, Æ *&+ §6.6

· 119 ·

[CCR1].

§6.6 ,

M

.

n

R

n

6.1,

.

M

n

.

.

(M1n , g1 )

6.23

,

(M2n , g2 )

M1n

,

Riemann

M2n

.

A.Borel

,

,

M1

M2

M1

M2

M1n

M2n

?

, Farrell

Jones

,

.

6.24 (Farrell-Jones[FJ1∼2] ) ,M1n

,

(M1n , g1 )

(M2n , g2 )

M2n

n 6,

.

[FJ2]

, Farrell

M2n ,

M1n

Jones

,

M1n

,

.

,

n = 3, 4

,

.

M

3

M

4

M1n

.

3

.4

6.24

.

n=3

Poincar´e

3

3

M

Poincar´e

4

M

.

4

6.24

,

Pontryajin

.

.

MinVol(M n )

,

.

n

MinVol(M ) = 0

.

Gallot

,

Besson-Courtois-

.

6.25([BCG])

(M n , g)

,

.

Novikov

Hn

4

,3

,

Mn

Hn /Γ (

(M n , g) ∼ = Hn /Γ ,

MinVol(M n ) = MinVol(Hn /Γ ).

).

,M n

Hn /Γ .

e Z@`Q[\#!! #

* kc " &7E E Æ s9 ) > &7

u;q . + rm 8 1u;q s 9 + " :&, "

2.g pj- +&7&7.6.6 5 s 9 \4 - > 1 1"* ", . w

j j . s # d v

- " , w v"qg> w y w4 &Æk -/

,/ < 5 u zp

" 35 , : xA Au 4 xx{ g(.g 2 u 7 ( u # ) v u > u

2.g - i& >u* # >

&7 3

3

i& 2. g + u > 7 > u u uw "

· 120 ·

MinVol(Hn /Γ ) = vol(Hn /Γ , g−1 ),

Gromov-Thurston K ≡ −1

Riemann

6.25

6.24

Hn /Γ ,

g−1

.

Mn

(n 6)

,

MinVol(M n ) > MinVol(Hn /Γ ).

MinVol(M n ) = 0

0

Mn

,

.

, MinVol(M n ) =

, Cheeger

(

[CGR2])). 6.26(

Cheeger

M1n

)

n − 2, M2n M1n ,

g2

,

Riemann

M2n

MinVol(M2n ) = 0.

,

M1n ∼ M2n

6.26

F M1n → M2n ,

.

n

H /Γ

,

1950∼2000

50

.Mostow,Gromov,Ballmann,Berger,Pansu, Burns-Spatzier, Katok, Eberlein,Besson-

Courtois-Gallot,Corllett,Mok-Siu-Yeung,Jost-

,

.

M1n

.

(

6.24

21

1950

M2n

6.26)

,

.

.

,M.Kac

?(

Can you hear the shape of drum?)

(

)Riemann

(M n , g)

Laplace

.

.

6.27 (

)

(M n , g)

,

l(M n ,g) π1 (M n ) → R,

σ → l(M n ,g) (σ),

Mn

l(M n ,g) (σ) = inf{Lg (ψ)|ψ dψ ds . Lg (ψ) = S 1 ds g

(M n , g)

Riemann

Riemann

M.Kac

ψ S1 →

}, Lg (ψ)

σ

,

g,

(M n , g)

Laplace

.

HIe |8 > uw 2.g i + & & > ( B < :. @

9 +

- i& 2.g "9 + B < 7r . @& ! 6 v .- ??K@C8#LI} `Q G [ H #[ \ A B CJ [ _ S # SDR^ P [ [ H I ? J #w w x Jw w xK w y#`Q G P z NI L F#XJww! yvyq{vP zKsNISyvÆwYL Lr #NIGwwxK #XJzDx^xbQG HI #?JzD yvÆw zD#O@ww!yw$ xS Lr #yw$ b?PC yvPt`Q GNI G x#[\S YIL_V{t VK I LN|?JyzS #`uzS ]vH{#zD} Jy S[HVK #[_ YJIGXJzD x| { zP H #W ~b WK b WK A B I C{ H D M A $ YIL YJIHAB JKCJSMs SRI#QG|w AAB CVTNS #:[ 3 I CLULTUDV J C S #V^SxDQ YIL · 121 ·

6.28 (

(M1n , g1 )

)

(M2n , g2 )

,

(

),

?

n=2

,

Croke-Otal

.

n=3

,

Croke

.

(M13 , g1 )

6.29 (Croke)

3

,

(M23 , g2 )

,

,

.

n3

6.28

.

(

1(

)

Σ2k

).

k

τ.

.

gτ ,

,

.

,

gτ

τ

τ

v

Σ2k

,

aτ (v).

τ

aτ (v)

v,

2π,

v

Kτ (v) = aτ (v) − 2π.

Gauss

Kτ (v1 ) + · · · + Kτ (vm ) = 2π(1 − k),

(Σ2k , τ )

{v1 , v2 , · · · , vm } ε

,

Gauss-Bonnet

).

2 ( l

0

f (s)ds = 0

Fourier

(ii)

.(

.)

(i) (Wirtinger

l

[f (s)]2 ds

f [0, l] → R

).

2π l

2

l

0

[f (s)]2 ds.

.)

Ω ⊂ R2

∂Ω σ

.

σ = ∂Ω

[0, l] → ∂Ω,

s → (x(s), y(s)).

σ

vj

f (0) = f (l),

0

(

Σ2k

,k

[x (s)]2 + [y (s)]2 ≡ 1. ∂Ω = σ , l l 2Area(Ω) = [xdy − ydx] {[x (s)]2 + [y (s)]2 }ds 2π 0 Ω ,

=

l2 . 2π

l

"

· 122 ·

H $ #%Y}W LO@ F#@Iw KE #LII D AB !!K@CÆw_V W~bWKCMKE (iii)

(i)

(ii)

A(Ω)

R2

Ω

mann

.

Gromov

3(

[l(∂Ω)]2 4π

).

(M n , g)

CJ

.

Voln (B n (1))

e Z@`Q[\#!!

LFP#Z@`Q|H[\#RN

Rie-

n

LNC C{z|}#LU { ^ F M QY @ K@C #LIIfKh # ? J D yvyzzILY CJSDR#Z@` Q [\# !! MAB TaS @ JN)K #ÆS X {? T$~}PzUH ~Gyv~\}WLI `Q|H[\# !! #]@Z[|} A{??B JT$#CJ{SDRZ@ JN)K # ÆS O? ÆSP?zÆPHÆS|QAB #TaSCJSDRZ@CM{?? `Q|H[\# !!aS B #?#J]@ÆSZ[Y|}IL {?? J bY#T${zPH LN CT ??JN)K ÆS YJIÆwWP~} I {S~~ CJS !! {#$3C|U# OUQGJJ`\}#D $ LNJF~DE IVJLU#'#P~H D ` \} H W b WK $ ) }? ~~ Voln (Ω)

n

[Voln−1 (S n−1 (1))] n−1

[Voln−1 (Ω)] n−1 ,

S n−1 (1) = ∂B n (1), B n (1)

n 4 (n = 3

,

B.Kleiner

).

(M n , g)

).

4(

Riemann

Mn

,

π1 (M n )

Z⊕Z

(M n , g)

,

.

n

(M , g)

Riemann

g˜)

Mn

.

, (M n , g)

˜ n, (M

π1 (M n )

Z⊕Z

?

Z-

5(

,

). (M n , g) ˜ n , g˜) ˆ (M Z

Z⊕Z

n

(M , g) n

π1 (M )

2

R ,

n

.

Z⊕Z

π1 (M )

Riemann ˆ Z

.(

R2 /Z = S 1 × R → M n ,

ψ

(s, t) → ψ(s, t).

(M n , g)

Riemann

,

,

t1

∂ψ ∂ψ n ∂t (s, t1 ) − ∂t (s, t2 ) n < ε < Inj(M , g), SM

· SM n

,|t1 − t2 |

1

1

ϕ S ×S →M

n

.

ψ|S 1 ×[t1 ,t2 ]

ϕ∗ (Z ⊕ Z) ∼ = Z ⊕ Z ⊂ π1 (M ).) n

t2

Æ

Riemann

.

,

70

.

20

, Cheeger-Gromoll

,

.

. Cheeger-Gromoll .

, . Calabi Æ . . , . . G , G σ, σ G . , L G → G ! L G → G, §7.1

σ

σ

h → σh.

", ! R G → G R (h) = hσ. (L ) L , G g ! gˆ G . gˆ . !" G Harr µ G 1, µ(G) = 1. # e # X Y , σ

X, Y gˆ |e =

σ

σ ∗

(Lσ )∗ (Rσ )∗ X, (Lσ )∗ (Rσ )∗ Y dµ σdµ σ ,

σ

$ d (σ) σ µ !. ", T (G) ! µ

X, Y gˆ |σ =

h∈G

G

G

σ

! ·, ·

σ ∈G

.

(Lh )∗ (Rσ )∗ X, (Lh )∗ (Rσ )∗ Y dµ (h)dµ (σ ).

%" &#"#$' ($

· 124 ·

{X, Y } G ) !, !% ˆ X Y = 1 [X, Y ], ∇ 2

(7.1)

$ ∇ˆ gˆ %. "& (7.1) !&. !$' "&. G * ' Aut G → G, h

(# ,

z → h−1 zh.

G

(7.2)

# e #++ G g, "& (7.2) z (

Adh Te (G) → Te (G),

(7.3)

X → d(Auth )|e X.

G , Ad g → g "Æ'. % X, Y Z G ) !, h = exp(tZ), &-$#, t % h

) &7.4'

Adexp(tZ ) X, Adexp(tZ ) Y = X, Y .

(7.4)

d (Adexp(tZ ) X)|t=0 = [Z, X]. dt

(7.5)

[Z, X], Y + X, [Z, Y ] = 0

(7.6)

,

t

t = 0 (*&7.5',

!(., $ , Lafontaine * ) 68 + (* (7.6))

.

'

Gallot, Hulin

ˆ X Y, Z = XY, Z + Y Z, X − ZX, Y 2∇ +[X, Y ], Z − [X, Z], Y − [Y, Z], X = 0 + 0 − 0 + [X, Y ], Z − 0 = [X, Y ], Z

(7.7)

!(.. 7.1 % G . G ) ! X, Y Z, % ˆ X Y = 1 [X, Y ], ∇ 2

§7.1

&#"#$'($ /

+

· 125 ·

, R(X, Y )Z =

*(

,

1 [[X, Y ], Z], 4

R(X, Y )X, Y =

(7.8)

1 [X, Y ]2 . 4

0)+ [X, Y ]f = XY f − Y Xf *+, f (.. -,!. Bianchi "& [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0.

(7.9)

-1"& (7.9) (7.1). 2 R(X, Y )Z

* (7.6)

= −∇X ∇Y Z + ∇Y ∇X Z + ∇[X,Y ] Z 1 1 1 = − [X, [Y, Z]] + [Y, [X, Z]] + [[X, Y ], Z] 4 4 2 1 1 = [Z, [X, Y ]] + [[X, Y ], Z] 4 2 1 = [[X, Y ], Z]. 4

1 [[X, Y ], X], Y 4 1 = − [X, [X, Y ]], Y 4 1 = [X, Y ], [X, Y ]. 4

R(X, Y )X, Y =

-. -13 7.1 !, !! G = G ×R , 4 G !./ . ,- G , + . G/H . ( , $ H G . //. G/H . ( 0% ! /5 O’Neill 1&. O’Neill ! ( #+006 1

1

H → G → G/H

k

%" &#"#$' ($

· 126 ·

,1 201. 17 201 Riemann Riemann 89 (Riemannian submersion) : ;3. "# 7.2 (Riemann 89) % (M , g) (N , h) ) Æ Riemann , %2 C *+ ψ M → N < $ % (1) ψ < ( C *+89) ψ(M ) = N , M 3 x, ψ ψ | < & =4; (2) N 3 q ∈ N , C *+ < & 20 ψ (q) = F . X p 4. F , ( n

1

k

n

k

1

n

k

n

∗ x

k

k

1

−1

q

q

ψ∗ |p Xh = Xg

X ⊥ F (.. 5 ψ Riemann 89. % ψ M → N C *+ Riemann 89, N ! ξ ∈ T N , p ∈ M , %2 ξ p 23 ξ˜ ξ˜ ⊥ F = ψ (q), $ q = ψ(p), ξ˜ = ξ ψ | ξ˜ = ξ. 25#, M p # T M .! q

n

k

k

2

k

n

q

n

∗ p

−1

n

p

Tp M n = Hp ⊕ Tp Fq ,

(7.10)

$ H 20 F p .6, N p 4>3. O’Neill Riemann 89 5! . "' 7.3 % ψ M → N C *+ Riemann 89 N )#.! ξ η % p

k

q

n

k

2

1 ˜ υ η˜ = (∇ ∇ ˜] , ξ η) + [ξ, η ξ 2 Y = Yh Yυ

$ Y (7.10) .! $, 3 υ

Mn

V 6#

Nk

Fq ,

k

→R

q

(7.11)

6# 20 F !. &- q

˜ η˜]υ 2 = KN k (ξ, η), ˜ η˜) + 3 [ξ, KM n (ξ, 4

$ K K ( . !25# f N

k

(7.12)

*+,,

˜ η˜](f ◦ ψ) − [ξ, [ξ, η](f ◦ ψ) = 0.

(7.13)

˜ V ](f ◦ ψ) = 0. [ξ,

(7.14)

&#"#$'($ /

§7.1

· 127 ·

$'? 1& 2∇X Y, Z = XY, Z + Y Z, X − ZX, Y

(7.15)

+[X, Y ], Z − [X, Z], Y − [Y, Z], X,

(7.13)∼(7.15) !

⎧ ˜ ˜η˜, Z ˜ = ∇ξ η, Z, ⎪ ∇ ⎪ ξ ⎪ ⎪ ⎨ 1 ˜ ˜ Y ], V , ∇X˜ Y˜ , V = [X, ⎪ 2 ⎪ ⎪ ⎪ ⎩ ˜ Y˜ = −[X, ˜ Y˜ ], V . ∇V X,

(7.16)

6"&0 X Y .#7, ˜ Y˜ ]v 2 . ˜ Y˜ ) + 3 [X, KN k (X, Y ) = KM n (X, 4

,, 7.3 (.. -. 68, 77! Riemann 4 5 879(: 8 )9 ([Y2] p670). 1956 ,Milnor 98::'@ #8!;!'@ #8 . 2 7 068 Σ , + 8 0 ; ;;. (# 7 0< ( 8 8 0<98 ;;, < Σ A;!'@ S , 7 [Mi]. 7 07, ++ 28 '@ S , 4 15 68 ' 2011 7

7

7

7

B ; 7 Grove

.

S 3 → Σ7j → S 4 .

15 Milnor 68%=: , 9 0689 , 7 Hitchin

Ziller[GZ]

*0=> Cheeger.1973 > CP #CP ( (7 [Che1] ). C',30 D9

98. E?, Gromov 2/ C(n), < 0 n ( n 0 M % (

[Hi]).

n

n

n j=0

bj (M n ) C(n),

n

%" &#"#$' ($

· 128 ·

$ b M ) j Betti , 7 [Gr1]. Cheeger ?@F*#( = 17 CP #CP # · · · #CP .

> §7.2 A??B@@AABC 1965 , 8;! CB6=% M ( Riemann , + π (M ) ./ .> D (7 [Y2] p671). C# Preissmann 9. §6.2 D6 Preissmann % M ! ( M π (M ) ./ . Z '. !DKShankar 1998 ?# (7 [Sh]). ! (#. SO(3) ' Z $ Z = {0, 1} ∼= Z/2Z. ! Z , Z Z >D. , -=(., <@&6 . "' 7.4 ([Sh]) 2 7 0 ( M . Z ) = N SO(3) M E . * M /(Z Riemann ,N ' Z Z . ( ¯ , U (3) 3 0EGE C E.. A 3 × 3 E=4, A A F ),A A * A. n

j

n

n

n

k

n

n

1

n

1

n

n

2

2

2

2

2

7

7

7

7

2

2

7

2

2

3

T

U (3) = {A|AA¯T = Id},

$ Id G"=4.dim (U (3)) = 9. R

,

⎧⎛ ⎪ ⎪ ⎨⎜ z Z = ⎜ ⎝ 0 ⎪ ⎪ ⎩ 0

0 z 0

⎫ ⎪ ⎪ ⎬ ⎟ ⎟ 0 ⎠ |z| = 1 ⎪ ⎪ ⎭ z 0

⎞

U1,1

⎧⎛ ⎪ ⎪ ⎨⎜ z = ⎜ ⎝ 0 ⎪ ⎪ ⎩ 0

"Æ'#B , 0 z 0

⎫ ⎪ ⎪ ⎬ ⎟ ⎟ 0 ⎠ |z| = 1 . ⎪ ⎪ ⎭ 1 0

⎞

- U (2)×Z U (3) E , U ;. U /U (3)/Z , ;8 (cosets) {U

1,1

1,1

⊂ U (2).

1,1 vZ

}v∈U(3) ,

§7.3

Cheeger-Gromoll

HIHJ($

· 129 ·

. M , 6201 7

(U (2) × Z )/(U1,1 × Z ) → M 7 → CP 2 .

=%)F, SO(3) = SU (2) = (U (2) × Z )/(U "Æ , 6201"I

1,1

× Z ),

* SO(3) M 7

SO(3) → M 7 → CP 2

Riemann 89. Eschenburg CK M ( U (3) D) . ( ([Es2]). -. ; Æ M Ricci ;, Ric (n − 1)c > 0, M .LE " πc , 87 ?G. M ( M . MF M = {(x, y, z) ∈ R |x + y = z} ( H+ 9 G Æ;. ( . Cheeger-Gromoll . 7

n

2

Mn

n

n

n

2

3

2

2

§7.3 Cheeger-Gromoll NOBP

! Q Æ . Cheeger-Gromoll " 17J HR S6, 4 ! 98 ".; !'@ G @1. "# 7.5 (GI8totally convex set) Ω Æ M 8. % I3T Ω FJ σ [0, 1] → M %GT Ω , σ([0, 1]) ⊂ Ω, Ω M GI8. 7.5 9 (FJ σ KFJ. !, GI8 .I8. DI8GI8. + " M <(8 S (1). , Ω 38 Ω = {p }, $ p 2 0 #8 U4. ! Ω I8. H"KU4 LF σ, σ MK Ω , Ω GI8. Ω M GI8, " f (x) = d(x, Ω). * Morse !, M 'V"I Ω. Ω MK M ::17GJ. M Æ ( , KL?M; GI S. 5 M; M (soul). n

n

n

n

2

0

n

0

n

n

n

n

%" &#"#$' ($

· 130 ·

!$ , NF ' M S. !NO?O; GI 8 WX# M Busemann ,. "# 7.6 (1) % σ [0, +∞) → M PF<$ d(σ(t ), σ(t )) = |t − t |, $ t (i = 1, 2) < σ P (4E) F (ray). (2) σ [0, +∞) → M P4EF, , n

n

n

1

2

1

2

i

n

ησ (x) = lim {t − d(x, σ(t))} t→+∞

σ Busemann ,. !2# F%4EF. + - M = S × R = {(cost , sint , z )|t ∈ R, z ∈ R} ⊂ R . ! GY. - M + , 4 S - G ,

M S × {z } P M LF. LFA4EF. ZHF σ(t) = (cos(αt ), sin(αt ), βt ), $ α + β = 1, αβ > 0. -ZHF S R QI[F ZHF σ . - M = S × R F. 0 α = 0 7 5 F σ 4EF. < L.F σ(t) = (x , y , t) M = S × R 4EF. 1 Cheeger-Gromoll \ $ R . 7.7 % M Æ ( % σ [a, +∞] → M P4EF, 6* Busemann , η M → R I,. &- $ η ((−∞, c]) GI8. % ϕ[a, b] → M P#S F ϕ )I3[ T η ([−∞, c]) . sˆ ∈ [a, b], η (ϕ(ˆs)) c. , 2

1

2

3

2

1

1

2

0

2

2

1

2

1

0

2

0

1

n

n

σ

n

−1 σ

n

−1 σ

σ

δt (ˆ s) = d(σ(t), ϕ(ˆ s)).

Q< lim [t − δt (ˆ s)] c.

<8 Q< (a, b) 8 U = {s| η (ϕ(s)) > c}. U 8, 3 7.7 (.. T" U MK sˆ ∈ U =U (s , s ) sˆ ∈ (s , s ) ⊂ U, ! η (ϕ(s )) = c = η (ϕ(s ))( = % ). % ψ X ϕ(ˆs) σ(t) P δ F, t→∞

σ

1

1

2

σ

1

σ

2

t

αt = ϕ(ˆs) (σ(t), ϕ(s1 )), βt = ϕ(ˆs) (σ(t), ϕ(s2 )).

2

t

§7.3

Cheeger-Gromoll

HIHJ($

· 131 ·

- α + β = π, JV α β E " π2 . -( QL d(σ(t), ϕ(s )) d(σ(t), ϕ(s )) JV <$ δ + (s − s ) , t

t

t

t

1

2 t

2

δt min{

2

2

δt2 + (s1 − s2 )2 .

min{d(σ(t), ϕ(s1 )), d(σ(t), ϕ(s2 ))}

"I

1

[d(σ(t), ϕ(s1 ))]2 − (s1 − s2 )2 , [d(σ(t), ϕ(s2 ))]2 − (s1 − s2 )2 }.

0 t → ∞, "4K] ησ (ϕ(ˆ s)) = lim [t − δt (ˆ s)] t→∞

max{ησ (ϕ(s1 )), ησ (ϕ(s2 ))} = c.

,3(.. -. GI8. @ 1 . .' 7.8 % M Æ Riemann , {q , q , · · ·} M; MW06. % σ [0, l ] → M X q # q P KF #S . 06 {σ (0)} X^ # v ∈ T (M ) σ (t) = Exp (tv ) σ [0, +∞) → M P q 9 F. !25# q ##8 S (1) ⊂ T (M ) 8.

MW06 {σ (0)}. X^ 06 {σ (0)}. EY (# {q } M;0 6, d(q , q ) = l → +∞. M r > 0, σ | → σ | LX^. - σ | % KFJ, r > 0, σ | %P K F* σ [0, +∞) → M P q 9 F. -.

66J. "' 7.9 % M Æ ( , {σ }

q 9 4EF 8N. Ω = η ((−∞, 0]) MK q GI 8. $ η σ Busemann , η (x) = lim [t − d(x, σ (t))]. 3 7.7 , G η ((−∞, 0]) GI8. ! q ∈ η ({0}). R Ω = η ((−∞, 0]) . ;8. 5, Ω . M;0 6 q → ∞, *1 7.8, ?#P4EF σˆ , η (q ) → +∞,

Ω 6ON, _S. -,,Ω . ;8. ! Ω L8 .8, L8. M Æ, ;L8.8. GI8 .8 GI8. * Ω GI8. -. n

j

j

ji

∞

q0

0

n

0

j

+∞ i=1

∞

n

∞

n−1 q0

j

j

n

q0

n

q0

ji

j

0

j

ji [0,r0 ]

ji [0,r0 ]

0

n

∞

∞

0

0

0

1

∞ [0,r0 ]

∞ [0,r0 ]

0

n

λ λ∈λ

0

λ∈Λ

σλ

λ

σλ

−1 σλ

λ∈Λ

j

−1 σλ

0

t→∞

0

−1 σλ

∞

n

λ

σ ˆ∞

j

−1 σλ

%" &#"#$' ($

· 132 ·

-1;!17 )Z, I8O; . % Ω I8+ 0 k, Ω k 0 *, (k − 1) 0 6;;, Cheeger Gromoll & -$ *"Æ8N Ω(−t) = {x ∈ Ω|d(x, ∂Ω) t}.

"' GI 8

% M Æ ( Æ, % GI8, 46;; ∂Ω. Ω = {x ∈ Ω| d(x, ∂Ω) t} . n

7.10

Ω

(−t)

?@ 7.10, ?@C# Calabi IM .

9.

"#

9P

% f M

7.11

n

→R

XN,, - f q 3 X ? 0

Hess(f )|q0 (X, X) C,

([ MK q 8 *+ , h U → R 0

1) h(q0 ) = f (q0 ); 2)

U f (x) h(x);

3)Hess(h)|q0 (X, X) C,

$9P Hessian, Hess(h) Hess(h)(X, Y )

@$' ? . {Σ } /*+"Æ , % T (X, Y ) 6 #? ∂t∂ )9&. / )9 & T < $ ? T + T + R(t) = 0, $ R(t) = R % , R (X, Y ) = R ∂t∂ , Z ∂t∂ , Y . (# Σ = ∂Ω ∂Ω I8, 6 *@ ∂t∂ , T (X, X) 0. 0( R 0 7 T = −T − R 0. )0 t OP7T PDP. 8N Ω *@ )9 &0Q0 Ω I8. = XY h − (∇X Y )h = ∇X ∇h, Y .

Riccati ∂ = ∇X , Y Σt ∂t { t} Riccati

t

t

2 t

t

(t)

t

t

(−t)

(0)

t

t

t

t

(−t)

2 t

t

(−t)

§7.3

Cheeger-Gromoll

HIHJ($

· 133 ·

66*+"Æ / {Σ } A(. . Q \, "Æ / {∂Ω } *+, ]@* 7.11 D`R ab. 7.10 RS. "' 7.10 0 " q ∈ Ω t = d(q , ∂Ω) > 0, , f (q) = d(q, ∂Ω). (# f Ω → R QQXN,, "P ∂Ω # q PKFJ σ [0, t ] → Ω σ (0) ∈ ∂Ω σ (t ) = q . Ω k 0O;, ∂Ω (k − 1) 0. - σ [0, t ] → Ω P KFJ, σ | σ| .KFJ. )!1&, ∂Ω σ (ε) #"Æ k 0GE R , σ (ε)⊥T (∂Ω ) ∂ ∂t . ∂Ω 0 ε < t (.. , Σ = ∂Ω . * Riccati ? , T σ , ∂Ω )9 ∂ & (6*@ ∂ε ) . < Ω σ(ε) I . " ε → t Ω q I . -. GI8 Ω, t

(−t)

0

0

0

0

0

0

0

0

0

0

(−t)

0 [0,ε]

k

0

σ0 (ε)

−t

0

(−ε)

0

(−ε)

(−t0 )

(−ε)

0

(−ε)

t

0

[ε,t0 ]

0

(−ε)

0

0

rmax (Ω) = max{d(x, ∂Ω)|x ∈ Ω}.

0 tˆ = r (Ω) 7, , Ω = {x ∈ Ω | d(x, ∂Ω) tˆ}, dim(Ω ) < dim(Ω). E6$1, U0%SVT, JV UTV 1 0. M 0 n 0, : E n U, '#GI8 Ω , Ω M; . :!, GI8.G8. 5 M;GI W Cheeger Gromoll . , J ! J. "# 7.12 % M Æ ( Riemann , 2GI M; S( ), M ;!'@ S M @ ! N S. &-$ , X Y ) M T S ##, < $ X ⊥ S

Y ∈ T S, (−tˆ)

max

(−tˆ)

n

0

0

n

n

n

n

K(X, Y ) ≡ 0,

$ K(X, Y ) X Y ,( ( . #]<. Ω GE R I8, U (Ω) T Ω U 2 cU, ∂[U (Ω)] . C *+ X , Federer 98 (7 [Fe]). - , $QQ% Ω I ε > 0, 9 ( ∂Ω C *+. n

ε

1

1,1

ε

%" &#"#$' ($

· 134 ·

0Y R /( M 7 Ω I8 ε Z!E, ε > ε > 0, ∂[U (Ω)] '5 C *+, -,<@ R > 0, 8 N (S) = {(x, v)|x ∈ S, v⊥S, v R} ;!'@ 8 U (Ω ), $ {Ω } Cheeger Gromoll I8 /. 0 R Z!E7 ( R E S GL), ), ;Q Exp N (S) → M d^. , Bootstrap ?@ 3 $PDP= ;!'@. A LGI8, , A GL δ(A) = sup{ε | U (A) 3 x 2 A B3 }, 0 ε < δ(A), 2 U (A) # A KÆeQI. V K = max{K(x) | x ∈ U (Ω )}. GIL 8 A ⊂ Ω¯ , % n

n

0

1,1

ε

R

ε

R−ε

u

R

s

n

ε

ε

0

1

R

R

1 π √ δ(A) δ0 (R) = min InjM n (ΩR ), ,1 , 4 K0

$ δ (R) /5 GIL 8 A V". "& )W*W QL _. 6"& KÆe QI. ! [N (S) − N (S)] ! # U (Ω ) − U2 (Ω ) ;!'@. =f@. %CK N (S) # U (Ω ) ;!'@, 6J! 3 $ [N (S) − N (S)] # [U (Ω ) − U (Ω )] ;!'. ;E `[, )'@W D, * N (S) # U (Ω ) '@, $ R > R + 1. *, # G ;!'@ F N (S) → M . XN( Y , g#, . -. 0

R

ε

ε

R−ε

ε

R1 −ε

ε

R2 −ε

R−ε

R−ε

R1

R2

ε

R2

R1

R1 −ε

ε

R2 −ε

2

1

n

§7.4 Cheeger-Gromoll hYAÆXi

! . 4< n + 1 0GE R YZI X % ( . a\, M = {(x, y, z)|z = x + y } R MF, + . Gauss Qb, M R YZI X , M '@ G8 Z 8, * M .'@ R . n+1

2

2

2

3

n

n

n+1

n

n

§7.4

Cheeger-Gromoll

j[][ \k

· 135 ·

&-$, Gromoll Meyer , M Æ ( , M .'@ R . 636 9,Cheeger Gromoll =. !25#% M Æ ( , M QQ38, M '@ R . 45 7.13 (Cheeger-Gromoll =) % M Æ ( , M 3 p ( [, M .;!' @ GE R . = 1972 . B 30 D, Marenich,Walschap Strake " YZ !, ' Perelman . Perelman ?@, /5]l Sharafudinov m^\ (7 [Shv] [Yim2]). $ , @^Y I, I8 ]lJ. 7_ _]^

`n 6 2003 , 7 [CaS2], G"Æ8^ φ R × [0, l] → M , n

n

n

n

n

n

n

n

n

n

0

n

n

(s, t) → φ(s, t),

φ(R × [0, l]) M P8^>bO. ! Q Cheeger-Gromoll I8 / 6 >bO. "# 7.14 (6= >bO) % {Ω } Cheeger Gromoll GI8 /. P4>F φ(R × {t}) %MKoGI8 Ω ;; ∂Ω , >bO φ(R × [0, l]) 6= Cheeger-Gromoll GI8 / {Ω }. ! Q ?@ !J*+>bO . "' 7.15 ([CaS1]) % M Æ ( , % M R ;!'@, M 3 x, %2XN !J*+ >bO Φ = {φ } X M x. &-$, !J >bO φ %

Cheeger–Gromoll I8 / 6=. * Cheeger–Gromoll =(.. Guijarro Perelman ?@, _]^ `n 6 7.15 EY (7 [CaS2]). <_, 7.15 !J*+ >bO {φ } Cheeger-Gromoll I8 / 2. R` %, !)$ 7.15. ) E! Q , *& Cheeger-Gromoll I8 / % &HR S6. n

u

u(t)

u

n

n

n

n

i 1iN

n

i

i

%" &#"#$' ($

· 136 ·

7.4.1

Cheeger-Gromoll

789:;<=8 GI8 / aa GI8/ % Æ (

! EJ Cheeger-Gromoll

2! I8 / 6& (. 7.16 (Cheeger-Gromoll

.

Mn

)

a0 = 0 < a1 < a2 < · · · < am < am+1 = ∞

,

G

{Ωu }u0

0 u > a 7, dim(Ω ) = n; 0 u a 7, dim(Ω ) < n. (2) Ω = S M . -, S G MW*+M; . (3) 0 u > 0 7, Ω GIO; . dim[Ω ] = k , + 6; ; ∂Ω (k − 1) 0XN. (4) u ∈ [a , a ] 0 r u − a , / {∂Ω } ("Æ /, (1) M n = ∪ Ωu .

m

u0

u

m

u

n

0

u

u

u

u

u

0

j

j+1

0

j

u−r r∈[0,u0 −aj ]

Ωu0 −r = {x ∈ Ωu0 |d(x, ∂Ωu0 ) r}.

a

(5)

u > a , u − a m

m

= max{d(x, ∂Ωu )|x ∈ Ωu }.

0 0 j m − 1 7,

* dim(Ω ) < dim(Ω ). KÆeQI 2 . Ω M 8, , U (Ω) = {x ∈ M , d(x, Ω) < ε} cd Ω U 2 cU. 8 Ω abGL δ = sup{ε| 2 U (Ω) # Ω KÆeQI}. 0 Ω = {p } 387,δ = Inj (p ) " M p GL. I8 abGL c0! Cheeger-Gromoll . 7.17 % {Ω }, a = 0 < a < · · · < a 3 7.16 6, T > a , K = max{K(x)|x ∈ Ω } M Ω ( ; Ω X/I8 A, + a8GL ;. j+1

− aj = max{d(x, ∂Ωaj+1 )|x ∈ Ωaj+1 },

aj

aj+1

n

n

ε

Ω

ε

0

{p0 }

u

0

Mn

0

1

n

T +1

δA δ0 (T ) =

n

0

0

m

m

T +1

T

1 π min InjM n (ΩT ), √ , 1 , 4 K0

$ δ (T ) /5 A /. @ *WQL 3. 2 ' )3 q = q ∈A U (A) 3 p d(p, q ) = d(p, q ) = l < δ (T ). q # p KFJ σ [0, l] → M . )!1&π (σ (0), A) π2 . -,, )W `3 q q *W[E 2 . E?, ( K 0

1

2

1

δ0 (T )

n

i

pq1 q2

1

2

2

0

qi

i

i

0

§7.4

j[][ \k

Cheeger-Gromoll

· 137 ·

8 P 5 "b)W . (# l < √πK ,

"b)W )*W[E π2 . *WQL q q *W. E π2 , _S. T > a , !" [0, T ] ! u = 0 < u < · · · < u = T, {a } {u } 8, Ω ⊂ U (Ω ). KÆeQI, 6!J*+FJ. "# 7.18 (Cheeger-Gromoll !JF) % {Ω }, T > a , δ (T ) ! 0 = u < u · · · < u = T 6. % P Ω → Ω K ÆeQI, Ω 3 x, x = x, x = P (x), · · · , x = P (x ), $ j = N, N −1, · · · , 1. !JF {σ } X S x CheegerGromoll !JF, $ σ X x # x KFJ. 6$, ( {x } E 36. ∗

pq1 q2

0

pq1 q2

m

m i i=1

1

0

N j j=1

uj

2

1

N

uj−1

δ0 (T )

u

0

1

N

j−1

T

j−1

m

N

uj

N −1

j

0

uj−1

N −1

j−1

j

j

>?

7.4.2

j−1

@AB0CD0EFGH !J*+F % + # KFJ 2 j

Cheeger-Gromoll

% {σ } Cheeger-Gromoll FJ σ [0, l ] → M . x ∈ ∂Ω , j

i

i

j

n

i

,

xi−1

xi

xi−1 = xi . .

wi

wi

ui (t) = ωi − d(σi (t), ∂Ωωi ),

-1 7.16 σ (t) ∈ ∂Ω . # I d. M 8 Ω y ∈ Ω, Ω p #I i

ui (t)

n

Ty− (Ω)

d(Expy (tv), Ω) n =0 , = v ∈ Ty (M ) lim sup t t→0+

$ Exp ;. (##IF . 0 v ∈ T y

7, −v e

# I. Q<&6a . "' 7.19 % P T Cheeger-Gromoll !J*+F >cd[, σ = {σ } u (t) 6. 6 >cd[ P , Cheeger-Gromoll I8 / {Ω } # I a . 0 0 t t l 7, − y (Ω)

σ

j

i

σ

0

1

i

Pσ [Tσ−i (t) (Ωui (t0 ) )] ⊂ Tσ−i (t) (Ωui (t1 ) ).

u

%" &#"#$' ($

· 138 ·

# I a % "I 6* @ I a %. <1-@ I . "# 7.20 (1) % Ω M I8, σ [0, l] π→ M P σ(0) = p ∈ Ω 9 F, σ (0) # I T (Ω) W 2 , σ (0) Ω p e@. (2) I8 Ω M @ I n

n

− P

n

N + (Ω, M n ) = {(p, v)|p ∈ Ω, d(Expp (tv), Ω) = tv, 0 ||tv|| < δΩ }.

", !6@ I N (Ω , int(Ω )). ) % (p, v) ∈ N (Ω, int(Ω )), v = 0, σ = (3) (4E@ v Exp t X p # ∂Ω KF v Ω p 4E@. v "' 7.19 0 $', "Æ / {∂Ω }, a ua , +

u

u+ε

+

u+ε

u+ε

p

(p,v)

u

u

j−1

j

Ωu = {x ∈ Ωaj d(x, ∂Ωaj ) aj − u},

M

T.

D,Yim Q/5

T

/ C

T,

max{d(x, Ωa ) x ∈ Ωb } CT (b − a),

$ 0 a < b T (7 [Yim 1]∼[Yim 2]) -, u ∈ [a , a ), !" j−1

j

δ0 (T ) 0 = ε = εu = min (aj − u), . CT

\p KÆeQI ,7 6@I " 4E@,( I8 QQ@&6R JK % 4E@ # I _ fWE " R!b"?@& g , P EF V

P Ωu+ε → Ω,

,

∂Ωu+ε ) ε}.

Np+ (Ωu , Ωu+ε )

,

7.21

Ωu = {x ∈ Ωu+ε d(x,

v0

Ωu(t0 )

σ(t0 ) π . 2

− Tσ(t (Ωu(t1 ) ) 1)

.

,

Pσ (v0 )

v0 = σ (t0 ), ψ [u(t0 ), u(t1 )] → " v0 # , ψ(u) = Expσ(t0 ) (u − u(t0 )) p1 = v0 ∆σ(t0 )σ(t1 )p1 . K 0, .

Mn

v0

8 )W -( < )W *W .= " π, 3! QL , * ψ(u(t1 )).

σ(t1 ) (p1 , σ(t0 )) + σ(t0 ) (p1 , σ(t1 )) π − p1 (σ(t1 ), σ(t0 ))

π . 2

§7.4

Cheeger-Gromoll

$# Ω *@). ,

I %

u(t1 )

Q< .

j[][ \k

· 139 ·

p1 (σ(t1 ), σ(t0 ))

- −ψ (t ) ∂Ω

π ( 2

− βv (t) = σ(t) (Pσ (v0 ), Tσ(t) (∂Ωu(t) )). π π βv (t) t t0 . βv (t0 ) = , 2 2 ∂ + β 0 ∂t

(. -

1

u(t1 )

<@

t0

t (.. <_, <@ 0

2

β(t1 ) β(t0 ) + O(|t1 − t0 | ),

(7.17)

$ O(ε ) P 2 fce. 6"&! Fermi L cd[. Fermi L < 6 F R × [t , t ] → M , % $ 2

1

m

1

σ(t0 )

0

)

m

1

m

n

1

(x1 , · · · , xm−1 , xm ) → Expσ(xm )

m−1

xk Ek (s) .

k=1

, Fermi

Y(I G", F F ∗

F∗ (0,···,0,x

m)

= Id

(7.18)

x (.. , R = Span{σ (t ), v } σ (t ) v ,( 2 0F . V φ [0, b(t )] → M σ(t ) # p KF, , R = Span{σ (t ), φ (0)}

σ (t ) φ (0) ,( 2 0F . -1 (7.18) m

2 t0

0

n

1

0

1

0

1

0

t1

2 t1

1

t1

t1

1

σ(t1 ) (R2t1 , Pσ (Rt0 )) = O(|t1 − t0 |2 ),

$ O(l ) l 6'P e. V v(t) = P (v ) , β(t) = (v(t), T 2

2

σ

0

σ(t)

− σ(t) (Ωu(t) ))

,

βv (t1 ) σ(t1 ) (v(t1 ), p1 ) = π − [σ(t1 ) (σ (t1 ), v(t1 )) + σ(t1 ) (p1 − σ (t1 ))] + O(|t1 − t0 |2 ) = π − [σ(t0 ) (σ(t1 ), p1 ) + σ(t1 ) (p1 , σ(t0 ))] + O(|t1 − t0 |2 )

π + O(|t1 − t0 |2 ). 2

= β(t0 ) + O(|t1 − t0 |2 ).

(7.19)

%" &#"#$' ($

· 140 ·

-, ∂∂tβ 0 a,. R 7.21 (.. 0 σ (t ) = vv 7, σ PX σ(t ) # ∂Ω 4EF. )!1 &, # I T (Ω ) −σ (t ) G. - Ω I, G# I IMKoG*. <,7, R 7.21 (.. 7.19 . -. +

t0

0

0

0

0

− σ(t1 )

u(t1 )

u( t1 )

1

789:MNO"'0 ( "' 0 $'

u

7.4.3

= . 7.15 ,M S G *+ . dim[S] 1, !"P #S F c R → S c (0) = x

c (0) = 1. , {V (s)} F c >c! V (0) = σ (0), $ {σ } P Cheeger-Gromoll !J*+F. φ R × [0, l ] → M , Cheeger-Gromoll n

0

0

1

0

1

0

1

1

0

j

n

1

(s, t) → Expc0 (s) [tV1 (s)].

)Q<&6J. JK 7.22 % M , {Ω }, {p }, {x }, {σ } φ 6. 6J( . (1) 6 F c → S, >cd[ P , @ I N (S, Ω ) ; (2) φ G 8^; * ∂φ (s, t) F t → φ (s, t) >c!; (3) 4>! ∂s ∂φ ∂φ

,( ( (4) ∂s ∂t n

u

j

j

j

0

c0

1

+

ε

1

1

1

1

1

K

∂φ1 ' = ∂s ∂t

& ∂φ

1

,

& ∂φ1 ∂φ1 ' ∂φ1 ∂φ1 R , , ≡ 0. ∂t ∂s ∂t ∂s

V (0) = σ (0) 4E@, FJ σ [0, l ] → M X S

∂Ω KF, )9!1&, Cheeger-Gromoll R 7.22 (., 7 [ChG] 1.10. ?; N (S, Ω ) 4E@,(, -1 [ChG] 1.10, 6 TF c >cd[, 4E@K>cd['4E@. ,R 7.22(1) (.. (M sˆ ∈ R, P6* 4.F σ t → φ (ˆ s, t) 1

1

1

l1

+

ε

0

sˆ

1

1

n

§7.4

Cheeger-Gromoll

j[][ \k

· 141 ·

σ >c! {W (t)}, $( W (0) = c (ˆs) = ∂φ∂s (ˆs, 0). Q< sˆ

j

Wsˆ(t) ≡

1

0

j

∂φ1 (ˆ s, t), ∂s

(7.20) &

sˆ t (., b2R '7.22 "& (7.20) ._. - ∇ W ≡ 0, ( W (.) >c!. "& (7.20) &@!$-c. )$ , a 3 7.16 , u(t) = −d(σ (t), ∂Ω ) + a ! ±W (t) ∈ T (∂Ω ).

7.19 # I a , (# ±W (0) = ±c (ˆs) ∈ T (S), *. ∂φ1 ∂t

sˆ

1

sˆ

sˆ

a1

1

u(t)

0

sˆ

σsˆ(0)

±Wsˆ(t) ∈ Tσ−sˆ(t) (∂Ωu(t) ).

-1 Cheeger-Gromoll ,{Ω } %I8, <. ±W (t) ∈ T (∂Ω ). )$ (. )9$ Calabi df hq (barrier surface method). ->c! ±W (t) 6#X / {∂Ω } ! φ (R × [0, l ]) σ 6# Mr Ψ R × [0, l ] → M , u

sˆ

sˆ

1

u(t)

1,ˆ s

1

− σ(t)

u(t)

sˆ

n

1

(s, t) → Expσsˆ(t) [sWsˆ(t)].

5 Mr Σ = Ψ¯ (R × [0, l ]) + 8 Σˆ J σ G. ! "&2 2 1,ˆ s

1,ˆ s

1

2 1,ˆ s

= Ψ1,ˆs ((−ε, ε) × [0, l1 ])

%

sˆ

∂Ψ1,ˆs ∂Ψ1,ˆs ∂Ψ1,ˆs ∇ ∂Ψ1,ˆs = ∇ ∂Ψ1,ˆs = ∇ ∂Ψ1,ˆs = 0, ∂s s=0 ∂t s=0 ∂s s=0 ∂t ∂t ∂s

+)9& T !D , ,

ˆ 2 (X, Y Σ 1,ˆ s

F / {γ } e,

t t∈[0,l1 ]

)|(0,t) ≡ 0.

ˆ2 γt = Σ 1,ˆ s

(

∂Ωu(t) ,

"Æe F /. C Calabi

[Ca]

h(x) = dM n (x, c0 (R)),

9, g Æ

%" &#"#$' ($

· 142 ·

h h, (upper barrier function) ˆ h(x) = dΣˆ 2 (x, c0 (R)).

8 F / {ˆγ } "Æ F /. σ (·) ) N -1 e9P , 1,ˆ s

, ,

ˆ −1 (t), ˆ2 h γˆt = Σ 1,ˆ s ˆ h h . Calabi

t

sˆ

ˆ ˆ = Hess ˆ (h)(W λ(t) sˆ(t), Wsˆ(t)) HessM n (h)(Wsˆ(t), Wsˆ(t)) = λ(t). Σ

$', Σˆ σ (·) G, & 2 1,s

sˆ

K(t) = KM n (t) = RM n (σsˆ(t), Wsˆ(t))σsˆ(t), Ws (t) = KΣˆ 2 (t), 1,s

-,, !

(7.21)

?

Riccati ⎧ ˆ ⎪ ⎨ ˆ2 ∂ λ + K(t) ≡ 0, λ + ∂t ⎪ ⎩ λ(0) ˆ = 0,

(7.22)

ˆ ˆ 0. H {Ω } I8 /, λ(t) 0, ∂∂tλˆ = −[λ(t)] − K(t) 0 λ(t) * 2

u

ˆ λ(t) 0. 0 λ(t)

-,, # ˆ ≡ λ(t) ≡ 0. λ(t)

(7.23)

Riccati ? (7.21),

) 9Ui

KM n (t) = K(t) ≡ 0.

(7.24)

. ( , 9UiG , 9Ui G % (7.24) _,W (t) . % X → R(σ (t), X)σ (t) aa, 6* aajY. ))$ Jacobi ! 2 % *. >c! {W (t)} % ,

(X, Y ) → R(σ1,ˆ s (t), X)σ (t), Y

sˆ

sˆ

sˆ

sˆ

X → RM n (σsˆ(t), X)σsˆ(t)

aa6*aajY. {W (t)} < $ Jacobi !? sˆ

J + R(σ , J)σ ≡ 0.

kg%

· 143 ·

(# {W (t)} ! ehPi ⎧

) ∂φ

1

sˆ

∂s

* (ˆ s, t)

t∈[0,l1 ]

) Jacobi ! 6'

⎨ J(0) = c (0) = ∂φ1 (ˆ s, 0), 0 ∂s ⎩ J (0) = 0,

-,)!.6", *"& (7.20) (., R 7.22 . =f@ ( 7.15 . %R0 (j − 1) 7 . JK A.j , c = φ (s, l ) , {V (s)} < V (0) = σ (0) T c >c!, @ I N (Ω , Ω ) 6 >cd[, * φ R × [0, l ] → M , j−1

j−1

+

j−1

uj−1

j

j

j

j−1

uj−1 +ε

j

n

j

(s, t) → Expcj−1 (s) [tVj (s)]

G8^. 2 j = 2. 7, 2# c (R) .MKo Ω $, R 7.22 c (R) ⊂ ∂Ω . *[ChG] 7.10 Yim J, )N (Ω , Ω ) c >cd[. R 7.22 6'?@! φ G 8^. 0 j 2 7?@5, *R A.j j (., -* Cheeger-Gromoll =Æ . -. 1

1

Tˆ

+

u1

uj−1

uj−1 +ε

0

2

jiks jg)

(

u,Z

1. k

tM

PQ

3

= S 3 (1) = {(x1 , x2 , x3 , x4 ) | x21 + x22 + x23 + x24 = 1}

= Z/{kZ}

ol k lmp. \k h g R =C j

olhvmS, ijkl M

3

4

kR

4

R

3

lfmn

→ C2 , √ √ −1 2jπ −1 2jπ k k z1 , e z2 (z1 , z2 ) → e

= S 3 (1)

2

mn. nqrwo S (1)/Zˆ 3

k

ˆ k3 =M

pop, nx Zˆ

k

=

{h1 , · · · , hk }.

s (M , g) o 2n l&#oqu$' Riemann ($,K 1. ps σ gS → M oqtrspqrr, t P gT (M ) → T (M ) os σ u ytus, nx σ(e v e ∈ S ktv. w 2n

2.

Mn

σ

iθ

1

σ(1)

2n

σ(1)

2n

1

2n

iθ

)

%" &#"#$' ($

· 144 · (i) (ii)

x|y.

\k P ohvotnu. wv dim(M ) = 2n kzvuws M {vx, yw v ⊥ σ (1) xx v k P hz σ

2n

2n

σ

w M {vx, {xnyzy\k π (M ) = 1, { M popky|p. (iv) (Synge vH) z\k π (M ) }ky|}k Z = Z/{2Z}. (v) z{|~|pop ][}zvl}{}~? 3. (Cheeger-Gromoll x||jg) ws (iii)

2n

1

1

2n

2n

2n

2

Rk → M n+k → S n (1)

ol}Tnu S (1) u qlx||. j M oÆ~~ql&#"#qu$' ~|? 4 (Hopf ][). j S × S oÆ~~ql&#oqu$' Riemann ~|? n

n+k

2

2

[ChC] [WuC] [WSY] [YaS]

, Æ. . : , 1983 , Æ. . : , 1993 , , . !". : , 1989 #$%, &'. . : (, 1988

[An] Anosov D V. Geodesic ﬂow on Riemannian manifolds of negative curvature. Proceedings of the Steklov Institute of Mathematics, No.90. Translated from the Russian by S. Feder. Providence: American Mathematical Society, 1969 [Av] Avez A. Vari´et´ es riemanniennes sans points focaux. C. R. Acad. Sc. Paris, 1970, 270: 188∼191 [BBB] Ballmann W, Brin M, Burns K. On surfaces with no conjugate points. J. Diﬀerential Geom., 1987, 25: 249∼273 [BCG] Besson G, Courtois G, Gallot S. Entropier et rigidits des espaces localement symmetriques de curbure strictement negative. Geom. Anal. Functional Anal. (GAFA), 1995, 5: 731∼799 [Beg] Berger M. Riemannian geometry during the second half of the twentieth century. Reprint of the 1998 original. University Lecture Series, vol 17. Providence: American Mathematical Society, 2000 [Be1] Besse A L. Manifolds all of whose geodesics are closed. New York: Springer-Verlag, 1978 [Be2] Besse A L. Einstein Manifolds. New York: Springer-Verlag, 1987 [BGM] Ballmann W, Gromov M, Schroeder V. Manifolds of non-positive curvature. Progress in Math., vol 61. Berlin: Birkh ¨ auser, 1985 [BI] Burago D, Ivanov S. Remannian tori without conjugate points are ﬂat. Geom. Anal. Functional Anal. (GAFA), 1994, 4: 259∼269 [Bu1] Buyalo S. Homotopy invariance of some geometric property of non-positively curved three manifolds. St. Petersburg Math. J., 1992, 3: 791∼808 [Bu2] Buyalo S. Three dimensional manifolds with Cr-structure. In: Some Questions of Geometry in the large. AMS Translations, 1996, 176: 1∼26 [Ca] Calabi E. Hopf’s maximum principle with an application to Riemannian geometry. Duke Math. J., 1957, 18: 45∼56 [CaS1] Cao J, Shaw M. The smoothness of Riemannian submersions with nonnegative sectional curvature. Comm. Contemporary Math., 2005, 7: 137∼144 [CaS2] Cao J, Shaw M. A new proof of the Cheeger-Gromoll soul conjecture and the Takeuchi theorem. Preprint, 2003, sfubmitted [CCR1] Cao J, Cheeger J, Rong X. Splittings and Cr-structure for manifolds with non-positive sectional curvature. Invent. Math., 2001, 144: 139∼167 [CCR2] Cao J, Cheeger J, Rong X. Local splitting structures on nonpositively curved manifolds and semirigidity in dimension 3. In: A special volume in honor of P. Li. Comm. Anal. Geom., 2004, 12: 391∼417 [Cha] Chavel I. Riemannian Geometry—A modern introduction. Cambridge: Cambridge University Press, 1993 [ChE] Cheeger J, Ebin D. Comparison theorems in Riemannian geometry. New York: American Elsevier Publishing Co., 1975

· 146 ·

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