1、 + F 0l / h1 = 1y2(dx2 +dy2);YV9 (H;h1) w q1. (H;h1)w bW .3. R2l / g = 4(1+x2 +y2)2(dx2 +dy2);YV9 (R2;g) w q+1.4. bWHn = (x1; ;xn) Rn |xn 0l / h = 1(xn)2ni=1dxi dxi;1.1+ Q99 (H;h) w q.(7) 0 !( M;g) , MM0 , gg7 .:( M;g)Levi-Civita . M M_ X, Y,:XYXYM M_g,5XYl (M;g)Levi-Civita . 7II(X;Y) = XY XY;5II1X,
2、 Y (f L,V7l VM M bWE bWL,M= T. g5MB T.= T1X, Y :II(X;Y)II(X;Y) = (XY YX)(XY YX) = X;YX;Y = 0:! ME_ (h ;Xi = 0 i M_ X ), 7II (X;Y) = h ;II(X;Y)i;5II M =xMf ,1E_ = T.l, T VII (X;Y) = h ; XYi = hX ;Yi;:A (X)X M M_g,5 T VII (X;Y) = hA X;Yi;0A 1E_ 0. TM w (dimM = dim M 1),E_ ,5hX ; i = 12Xh ; i = 0X f ,
3、cf5,5M = f1(c)M w .| = gradf,5 KM H E_ . |M M_ X, Y,5II (X;Y) = h ; XYi = hgradf; XYi= XYf = YXf XYf= 2f(X;Y);10Bc vL II = 2f. TM= T ,5MM 0 . !M 0 , MBHL,5_ = _ = 0; 9 ML.Q, T SMM MML ( cM,5M 0 ./ 0 M w q M w qW1“. !X, Y,ZM M_ .5Y XZ = Y(XZ +II(X;Z)= YXZ +II(Y;XZ)+ Y(II(X;Z); VX YZ = XYZ +II(X;YZ)+
4、X(II(Y;Z); X;YZ = X;YZ +II(X;Y;Z) VR(X;Y)Z = R(X;Y)Z +II(Y;XZ)II(X;YZ)+II(X;Y;Z)+ Y(II(X;Z) X(II(Y;Z): (1.15) I n TM M_E_s .5 I n M_. |M M_ W,5hY(II(X;Z);Wi = hII(X;Z); YWi = hII(X;Z);II(Y;W)i; hX(II(Y;Z);Wi = hII(Y;Z);II(X;W)i; (1.15)R(X;Y;Z;W) = R(X;Y;Z;W)hII(X;Z);II(Y;W)i+hII(Y;Z);II(X;W)i: (1.16
5、) TMGaussZ. I n(1.15)ME_s ,5R(X;Y)Z = II(Y;XZ)II(X;YZ)+II(X;Y;Z)+ Y(II(X;Z)X(II(Y;Z): (1.17) TMCodazziZ.M w H, CodazziZ VTBte.N H, |ME_ ,5II(X;Y) = II (X;Y) VY(II(X;Z) = Y(II (X;Z) +II (X;Z)Y ;1.1+ Q11 VX(II(Y;Z)s.N HCodazziZ VR(X;Y)Z = (YII )(X;Z)(XII )(Y;Z) :+Y,M w q H, TP , H(XII )(Y;Z)1X, Y, Z .
6、(8) wL_ ! : a;b M; wL, W TM _ . 8 , _ V t7V(t) T (t)M. _ , US“xini=1, !(t) = (x1(t);x2(t); ;xn(t);5 V (t) =ni=1xi(t) xi(t):B, _ V VV(t) =ni=1Vi(t) xi(t); (1.18)TVi(t) (t;f ,5V ;_ .!TM ,5 W TM . 8 , !V _ , V U (1.18),5l_ V / _ ( Einstein p)_ V(t) = Vi(t) xi(t)+ xi(t)Vj(t)kij (t) xk(t); (1.19)_ VV xM
7、, l US |1.Y L , !yjnj=1 6B US“,: (t) = xj(t) yj; V(t) = Vj(t) yj;5xj(t) = xi(t)yjxi (t);Vj(t) = Vi(t)yjxi (t):12Bc vL Vj(t) yj +xi(t)Vj(t)kij yk= Vi(t)yjxiyj +Vi(t) xl(yjxi)xl(t) yj + xl(t)yixlVm(t) yjxm yiyj= Vi(t) xi +Vi(t)xl(t) xl(yjxi) yj + xl(t)Vm(t) yjxm xlyj= Vi(t) xi + xl(t)Vm(t) xlxm= Vi(t)
8、 xi + xl(t)Vm(t)klm xk;yN_ V US |1. , (t) = 0 H,xM _ V(t)B ,. , (t) pH, V(t)TpMBH wL, _ V(t) H wL M_ .TLevi-Civita ,5 M : !V, W ( _ ,5ddthV;Wi = h_ V; Wi+hV; _ Wi: (1.20)Y L ,:V(t) = Vi(t) xi; W(t) = Wj(t) xj; g = gijdxi dxj;5ddthV;Wi =ddtVi(t)Wj(t)gij (t)= Vi(t)Wj(t)gij (t)+Vi(t) Wj(t)gij (t)+Vi(t)
9、Wj(t)( xkgij)xk(t)= Vi(t)Wj(t)gij (t)+Vi(t) Wj(t)gij (t)+Vi(t)Wj(t)xk(t)(h xkxi;xji+hxi; xkxji)= h Vi(t) xi +Vi(t)xk(t) xkxi; Wj(t) xji+hVi(t) xi; Wj(t) xj +Wj(t)xk(t) xkxji= h_ V; Wi+hV; _ Wi:! : a;bc;d M; , US“xini=1V U(t;s) = (x1(t;s);x2(t;s); ;xn(t;s):s% H, (t;s) 1t wL, M_ : t.t% , (t;s) 1s wL,
10、M_ : s. US“, t =xitxi(t;s); s = xisxi(t;s):1.2L 13 t s =2xitsxi(t;s)+ xitxis kij xk(t;s); (1.21) t s = s t; T t; s = 0.x1.2L / L !(M;g) . V wL 7 S) .! : a;b Ms ; wL, L( )lL( ) = ba| (t)|dt;| (t)| = g( (t); (t) M_ (t)./ 5, x f bW, K wLL .51.2.1. ! : a;b Rns ; wL,5L( ) | (b) (a)|;O| H (a), (b)L. . T,
11、! ; wL.ZE1.: (t) = ( 1(t); ; n(t),5 (t) 6= 0 H,| |0 = (h (t); (t)i)0 = 1| (t)|ni=1i(t) i(t); (t) = 0 H, | |0(t) = | (t)|.9, Cauchy-Schwarz T V| |0 | (t)|; t a;b: L( ) = ba| (t)|dt ba| |0 dt= ba| |0dt=| (b)| (a)| ;T (a) = 05 5;5 VYV M| (a)M V( M ).14Bc vLZE2. s M X ?, i /9 M,yNL ! (a) = 0, (b) = (l;0
12、; ;0).N HL( ) = ba| (t)|dt ba| 1(t)|dt ba 1(t)dt= | 1(b) 1(a)|= |l| = | (b) (a)|;| H 2(t) = = n(t) = 0. = , s x f bW.l1.2.1 ( ). ! : (M;g) (N;h) WCk (k 1). Th = g, ip M, p : TpM T(p)N ( =,5 . TN H 3 ,5 3 .B, T Bs ,5 .1954 M, John Nash Ck (k 3) ( V 3 x f bWRN, N = 12n(3n+11). N, Gromov k 3 H V |N = n
13、2 + 10n+ 3,k 4 H V |N = 12(n+ 2)(n+ 3).1987 M, Gunther eZE N V |max12n(n+3)+5; 12n(n+5).1.2.1.w bWK wL.!p = (0;y0), q = (0;y1) H ,H | h1 =y2(dx2 +dy2). ! : a;b H p, qs ; wL,5L( ) ln y1y0:Y L ,: (t) = (x(t);y(t),5L( ) = ba| (t)|dt = ba1y(t)|x(t)|2 +|y(t)|2dt ba1y(t)|y(t)|dt bay(t)y(t)dt= ln y1y0:i O|
14、 H, p, qY . s TLM9 h1M, s TLM V, (H;h1) i K wL1 0,yN(M;d) bW. .p | USxini=1, Pxi(p) = 0, 1 in, Oq =(x1)2 +(xn)2 = U; sl .U, g Vg =ni;j=1gij(x)dxi dxj:g0 = ni=1dxi dxi. g; V,i , 0, P2g0 g 2g0:16Bc vL ,U wL , Lg0( ) L( ) Lg0( ): (1.22) 51.2.1 VL( ) | (b) (a)|:+Y, T H p, q wL,5iKlt0, P (t0) U.N HL( ) L
15、( |a;t0) :dld(p;q) 0. . !x, y U, | (t) = x+t(yx),5(1.22)Vd(x;y) L( ) Lg0( ) = |yx|:N VB , bW(M;d) M 5 B.! : a;b Ms ; wL,:s(t) = ta| (u)|du; t a;b: uW H, s 1t9f , Qf Vt = t(s).N H, (t) = (t(s) V AsM wL. s . wL , M_ 1./ I n“5: M | p, q, i p, q wL , PL( ) = d(p;q) ? Ti, wLK wL.B, TFBL !, 5s . R2 (0;0)e
16、 0,R2 (0;0)i (1;0)(1;0)K wL. K wL A1Hq, ! : a;b M; wL, OL( ) =d( (a); (b). e n,YV , !| (t)| l 0.y LC (a) (b)WK , T VL( |t1;t2) = d( (t1); (t2); t1;t2 a;b:yNYVI n,BL ! cUS #U,US /, U Vj x f bW 7“.U, |;_ V(t), PV(a) = 0, V(b) = 0. I n; : a;b(“;“) U;(t;s) 7 (t)+sV(t);1.2L 17“ sl . B%s, s(t) = (t;s) (a)
17、 (b); wL, :L(s).L(s) = L( s) d( (a); (b) = L( ) = L(0)V, s = 0 L(s)Kl, L0(0) = 0. 9 /:L0(s) = dds ba| s(t)|dt = baddsh s(t); s(t)idt= 12 ba1| s(t)|ddsh s(t); s(t)idt= ba1| s(t)|h s t; tidt= ba1| s(t)|h t s; tidt:s = 0 H, | 0(t)| = | (t)| l, T VL0(0) = 1l bah t s; tidt= 1l baddth s; tih s; t tidt= 1l
18、 baddthV(t); (t)ihV(t);_ idt= 1lhV(t); (t)iba 1l bahV(t);_ idt; TBMs T.N HV(a) = V(b) = 0, TH ,.0 = L0(0) = 1l bahV(t);_ idt:T |V(t) = (ta)(bt)_ T,5 V A /Z_ = 0: (1.23)ZLZ, L. US“xini=1,: (t) = (x1(t); ;xn(t),5 (t) =ni=1xi(t) xi; LZxk(t)+kij (t) xi(t) xj(t) = 0; k = 1;2; ;n: (1.24)18Bc vL1.2.2. x f
19、bWL.Rn,LZ = 0, (t) = (0)+ (0)t; L.1.2.3. o L.!Sn Rn+1, ,5LZ0 = _ = (_ ) = ( ): SnE_ .yN,if (t), P = (t) (t):L= ,(t) = h ; i = h ; i0 h ; i = h ; i:ih ; i0 = 2h_ ; i = 0;yN| | ,:c.5 Z = c2 (t); (t) = (0)cos(ct)+ 1c (0)sin(ct); Sn v.VN V A,L M_ . L?L.B ,LZBzA T p.i(1.24)=sZF, sZF , p M#v TpM,i“ 0#L :
20、0;“) M, P (0) = p, (0) = v. sZF1 SHq;G , V / :51.2.3. pM,ip 7 #U# , P iq U,w TqM,|w| H,iBL w : 0;1 M, P w(0) = q, w(0) = w.1.2L 19:Bp( ) = v TpM | |v| 0, Pexpp : B(“) M BpK ( 3 . . (i)vl, expp| ,_ p.yTpM (0) = t v(0) = tvBL,yNexpp(tv) = (1) = v(t).(1.27)(expp)0(v) = ddtt=0v(t) = v;(expp)0 .(ii)yK“,
21、1 / V: p M,i“ 0#p 7 #W, Pexpq : B(“) M Bq W ( 3 . Qf B.N, I n; E : U M M; (q;w) 7 (q;expqw):20Bc vL E(p;0) U) M . dE(p;0) : T(p;0)U T(p;p)(M M) = TpM TpM.yTMM M M, N ,1 dE(p;0) V,7El#(i) . EQf V,i(p;0)TM 7 #W, PE|Ws (p;p)M M 7 #.+Y,ipM7 #W, PE(W) W W. | sl “, P(q;w) |q W; |w| “ W;5 iq W, expq : B(“)
22、 M ( 3 . .V V V A,ip 7 #, P E(q;w) |q W; |w| 0, Pexpp : B( ) M 3 .:B (p) = expp(B( ),5expp : B( ) B (p)s .!v B( ), (t) = tvB( )_L , (t) = expp(tv)Vp?_L ./ TV ,_L AK wL. ZE V 1 51.2.1.51.2.5. !(M;g) .(i) !expp : B( ) B (p)s , P, QB (p). T : a;b B (p) P, Qs ; wL,5L( ) |exp1p P|exp1p Q| : (1.28)(ii) q
23、B (p), |exp1p q| = d(p;q), p, q_L K wL. . (i): (s) = exp1p (s),5 (s) B( ) TpM exp1p Pexp1p Q wL. 7: 0;1a;b M(t;s) 7 exppt (s):1.2L 21 B%s a;b, s(t) = (t;s) (MVp?_L.:L(s) = L( s). BMs Tw,L0(s) = 101| s(t)|h t s; tidt= 1| (s)| 10h t s; tidt= 1| (s)|h s; ti10 1| (s)| 10h s; t tidt= 1| (s)|h (s); s(1)i:
24、i| s(1)| = | s(0)| = | (s)|, Cauchy-Schwarz T V|L0(s)| | (s)|; s a;b: L( ) = ba| (s)|ds ba|L0(s)|ds baL0(s)ds= |L(b)L(a)|= |exp1p P|exp1p Q| :(ii)(i) V, T B (p) p, qs ; wL,5L( ) |exp1p q|. T p, q, cB (p),5 L( ) |exp1p q|:y p, q_L |exp1p q|, l d(p;q) = |exp1p q|;_L K wL. . exp;#TpMS |TpM 0 ;, f dp :
25、B (p) R; q 7d(p;q) = |exp1p q|B (p)p ;f . !q B (p)p, X TqM. |Vq wL(s), P (0) = X. 5 9 V(dp )0(0) = L0(0) = 1|exp1p q|h (0); (1)i= hX; (1)|exp1p q|i;22Bc vL (t) = expp(texp1p q) p, q_L. dp(q) = (1)|exp1p q|;+Y, |dp(q)| = 1, O_L o Sr(p) = q M | d(p;q) = r(0 0, P t0 ; t0+ K wL.2. ! p, qs ; wL. T ,5 A;L
26、.3. , o Sn K wLAv(y7 ).4. !expp : B( ) M 3 . 51.2.5 q M |d(p;q) 0#q 7 #U, Pp U H, expp : B(“) M 3 . |t0 0. 1 R = . (QE) !R 0, Px SR(p), Od(y;x) “ H,iK L x, y.!q M, d(p;q) R+“. iK L p, q,“ . !d(p;q) R.ySR(p)“,#iq0 SR(p), Pd(q;q0) = d(q;SR(p):. d(p;q) = d(p;q0)+d(q0;q).Y L , | p, q wL (t). ! (t0) SR(p
27、),5L( ) d(p; (t0)+d( (t0);q)R+d( (t0);q) R+d(q0;q); d(p;q) R+d(q0;q) = d(p;q0)+d(q0;q) d(p;q);24Bc vL T| .+Y, d(q0;q) = d(p;q)R “.V7i q0, qK L . | p, q0K L ,5 p, qK wL, AK L. 1.2.1 (Hopf-Rinow). !(M;g) ,5/+HN:(1) (M;g)!; (2) pM, expplTpM ;(3)ipM, P expplTpM ;(4) M“ . . (1) = (2) s(i). (2) = (3) A .(
28、3) = (4): s(ii) , iq M, (iK L p, q.yN, TA“,5iK 0, PA q M |d(p;q) K expp(w | |w| K); A “0“,V79 .(4) = (1): =T . / w A .w 1.2.7. (i) !; (ii)(M;g)! H, expp : TpM M ( .1.2.4.! 0. D = (x;y) R2 | x2 +y2 0; h1 = 1(xn)2ni=1dxi dxi:4. ! : (M;g) (N;h) ,59 M.5. ! : (M;g) (M;g) 1,5 “ B Ys ( 0 .x1.3 Jacobi ?sZE
29、. -B, expp ,_ ) M ./ I nexpp iv TpM) M ,| Jacobi.N, !w Tv(TpM) = TpM,: (s) = v+sw. TpMVv? wL, S M_ w. M l,(dexpp)v(w) = ddss=0expp (s) = ddss=0exp(v+sw): (1.29): (t) = expp(tv). Tw = v ( R),5 TV (expp)v(w) = ddss=0expp(v+ sv) = ddss=0(1+ s) = (1): (1.30)Vn,“_, dexpp . I nB f, MsE4) 5.XE |M wLexpp (s) ABBL , IBBLMs.N, 7: 0;10;“ M;(t;s) 7 exppt (s):T(t;s) = t, U(t;s) = s,5T;U = TU UT = 0:U(t) = U(t;0),5U(0) = 0, O(expp)v(w) = U(1): (1.31)
