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?@'+7 08 $ 5 9: ; % E0? @' # AB 00? CDEFGFG )H KL&M ( 'N *HC ' I 1 JI J ) 67 OKC LM%N *+,

6; P-; Q O0PQRSR?. 8 /TUS$5 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 )$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

) ) 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 > ) 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 #* 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
%

= (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 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 = gîj . ∇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_bêf 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]ô \^ 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==!. 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 .

'
?. 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. ,.

" #

_WWXàX5). `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 Poincaré 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 Poincaré 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), & ;: 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. ;>??+&@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, +QXàQ?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 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

/ $ 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 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
.. % 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+ % %> 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 ( 5 >@G 7 HHJ A, *

83 & . !8IC .> C.B 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

.

ñ ϕ Y1 × [0, a] → M

.

: . & - > < ! .i5c 22

,

.

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

- $ 4 6 & !

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

=* ' .;I* .;* 4 & % 8 7#6 97 *% ac 1 §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)}.

.

ñ 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 )

ñ 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 ,

ñ → 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 mìj & ! - 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 ? * 91 5 54 . @

# 5 $ , 4 2 c -$4 ? "& > < mh` (% !-$4 &k 4 &k

-$4 &k ll ti+ & f ^ , ^ f^, d hn n ) -.@\ # ? < f^1$& " -/ e!\ 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 ñ Min(αi ) = M ˜ 2 , g˜2 ). (Rk , g0 ) × (M

αi ∈ Γ1

Γ1

⊂ C(Γ1 )Min(αi ) ⊂ Min(αi ), ñ ˜ 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

&" 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

Poincaré

3

3

M

Poincaré

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 #ùzS ]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 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 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 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 ,( ( . #] 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 *+, -, 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]). 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:; 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. Qc!. "& (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, 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. Variét´ 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

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