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...
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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 = 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==!. 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, $++ , V. ,.
" #
_WWX`aX5). `Y {(t, θ)} Z[X R aYX5Zbbc$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), & ;: 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, +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 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
.
˜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 §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 ? * 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 $4 '
4 &k& (
A g , 2 i 1
2 % I9 &+ 7 .@
e+!\ 2 + fF ^FG, + ) e! \©, &! e !\ &! 02= & GFG 5 &+
+ 4 e I9
f ^ , GFG *&+ % F FG qc(
: 3 ! α
i
˜ n. Min(Γ1 ) = Min(Zk ) = M
∈ Γ1 , ϕ ∈ C(Γ1 ),
δαi (ϕ(x)) = d(ϕx, αi ϕ(x)) = d(ϕx, ϕαi (x)) = δαi (x),
Min(αi )
C(Γ1 )
,
˜ n, M ˜n Min(αi ) = M ˜ 2 , g˜2 ). (Rk , g0 ) × (M
αi ∈ Γ1
Γ1
⊂ C(Γ1 )Min(αi ) ⊂ Min(αi ), ˜n ˜ n , g˜) = Min(Γ1 ) = Min(Zk ) = M (M
,
, C(Γ1 )/Γ1
C(Γ1 )
).
˜ 1 = Rk , ρ i M ˜ →M ˜i M ϕ ∈ Γ(
,
ρ2 (Γ1 )
Γ
,
ϕ = (ρ1 (ϕ), ρ2 (ϕ)). ˜2 M ρ2 (C(Γ1 )))
, ρ2 (Γ )( ˜ 2 /C(Γ1 ) M ,
˜ n /C(Γ1 ) = M ˆn M
Γ = π1 (M n )
Γ1
˜ n = Rk × M ˜2 M
ϕ ∈ C(Γ1 )
˜ M
˜2 M
.
ϕ ∈ C(Γ1 ))
ρi Γ → Iso(Mi ).
.
˜2 Rk × M
Γ1
6.17
.
C(Γ1 ){x0 }
.
,
(
Γ = π1 (M )
C(Γ1 )
˜ n /C(Γ1 ) M Mn ˜ n = C(Γ1 ){x0 } M
˜ n , g˜) (M
(3)
x0 ∈ Min(αi ).
.
ˆ 2. Tk × M
ˆ2 = M ˜ 2 /C(Γ1 ), M
.
.
˜ , g˜) = (Rk , g˜0 ) × (M ˜ 2 , g˜2 ) (M
,
6.17
π1 (M n )
Zk .
6.19(Eberlein ˜ , Mn Mn ˜ 2 , g˜2 ) (Rk , g0 ) × (M
(M n , g)
)
, g˜
g
.
Riemann n ˜ (M , g˜) deRham
k
R (k 1)
π1 (M n )
Zk .
Γ1
π1 (M n )
˜ n , g˜) (M
ˆ M
,
Rk (k 1), ˜2 → Mn Rk × M Mn
Γ1
˜ n , g˜0 ) = (Rk , g0 ) × (M ˜ 2 , g˜0 ), (M ˆ 2. Tk × M
˜ n , g˜) (M
deRham
Mn
.
.
Zk
,
6.18
M
˜ 2 , g˜2 ) (Rk , g0 ) × (M ˆ ˆ 2. M T k ×M
˜ = π M
π(Rk × {y})
QnKQÆSK@PK % o:& 7 c , op .,
#
! 7 m . ##:&
3& & *
# ,- f ^ , !8 G FG i . ]"$%" G FG !8 ,
! .@ ^, -f^,-f^ , ! f ^,f #
. @ eq.@ -r*]"$%" E n( \ / r r * ] "$%" , ? * ]"$%" ! 8=
3 )% $` ! .@ ! % ! 8
&" 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 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 $, R 7.22 c (R) ⊂ ∂Ω . *[ChG] 7.10 Yim J, )N (Ω , Ω ) c >cd[. R 7.22 6'?@! φ G 8^. 0 j 2 7?@5, *R A.j j (., -* Cheeger-Gromoll =Æ . -. 1
1
Tˆ
+
u1
uj−1
uj−1 +ε
0
2
jiks jg)
(
u,Z
1. k
tM
PQ
3
= S 3 (1) = {(x1 , x2 , x3 , x4 ) | x21 + x22 + x23 + x24 = 1}
= Z/{kZ}
ol k lmp. \k h g R =C j
olhvmS, ijkl M
3
4
kR
4
R
3
lfmn
→ C2 , √ √ −1 2jπ −1 2jπ k k z1 , e z2 (z1 , z2 ) → e
= S 3 (1)
2
mn. nqrwo S (1)/Zˆ 3
k
ˆ k3 =M
pop, nx Zˆ
k
=
{h1 , · · · , hk }.
s (M , g) o 2n loqu$' 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$' ~|? 4 (Hopf ][). j S × S oÆ~~qloqu$' Riemann ~|? n
n+k
2
2
[ChC] [WuC] [WSY] [YaS]
, Æ. . : , 1983 , Æ. . : , 1993 , , . !". : , 1989 #$%, &'. . : (, 1988
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