1 Ricci V, V i, W f : V W f f(v ) = Imf W ( ) f : V 1 V k W 1

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Transcription:

1 Ricci V, V i, W f : V W f f(v = Imf W ( f : V 1 V k W 1

{f(v 1,, v k v i V i } W < Imf > < > f W V, V i, W f : U V L(U; V f : V 1 V r W L(V 1,, V r ; W L(V 1,, V r ; W (f + g(v 1,, v r = f(v 1,, v r + g(v 1,, v r (kf(v 1,, v r = kf(v 1,, v r U, V W : U V W W =< Im >=< (U V > X f : U V X g : W X 2

f = g W U V U V W = U V U V W (= U V g f X (u, v = u v, (u U, v V (2 g g 1, g 2 : W X g 1 = g 2 u U, v V g 1 (u v = f(u, v, g 2 (u v = f(u, v W u v g 1, g 2 u, v g 1 = g 2 g 3

U V W L(U, V ; W W U, V U, V U u, V v u v L(U, V ; R α U, β V u v(α, β = u(αv(β u(α, v(β : U V L(U, V ; R (u 1 + u 2 v(α, β = (u 1 + u 2 (αv(β = u 1 (αv(β + u 2 (αv(β = u 1 v(α, β + u 2 v(α, β u v (u U, v V L(U, V ; R W W L(U, V ; R W {e i }, {f i } U, V U, V {e i }, {f i } {e i }, {f i } u v = u(e i e i v(f j f j = u(e i v(f j e i f j e i f j W g L(U, V ; R g(e i, f j = g ij e i e j g = g ij e i e j g(α, β = g(α i e i, β j e j = α i β j g ij = g ij e i (αe j (β = g ij e i e j (α, β 4

L(U, V ; R W W = L(U, V ; R W f : U V X f u U, v V g : W X g(u v = f(u, v f = g g W = U V U, V u U, v V u, v u v (u 1 + u 2 v = u 1 v + u 2 v u (v 1 + v 2 = u v 1 + u v 2 (ku v = u (kv = k(u v u v U V : U V W 5

: U V W f : W W g : W W f =, g = U V W f g W W, W, f g, g f (W,, (W, U V (= W U V L(U, V ; R u v(α, β = α(uβ(v, α U, β V u, v U V (1, (2 ( (1, (2 6

U V W (= U V g f X f L(U, V : X g L(U V ; X Φ : L(U, V ; X L(u V ; X Φ L(U, V ; X = L(U V ; X X R L(U, V ; R = L(U V ; R = (U V (U V = U V U V = L(U, V ; R U V = L(U, V ; R = (U V L(U, V ; W = L(U V ; W = (U V W = U V W V 1 V 2 V k V 1,, V k W 7

: V 1 V k W W =< Im >=< (V 1 V k > X f : V 1 V k X g : W X f = g W W = V 1 V k W = L(V1,, V ; R k V R V V, (1 x, y V V x + y (2 x V, a R V ax x, y, z V, a, b R a(x + y = ax + ay, (a + bx = ax + bx (abx = a(bx 8

V (32 V Intrinsic Intrinsic Intrinsic Intrinsic V {e 1, e 2,, e n } V x x = x 1 e 1 + x 2 e 2 + + x n e n {e 1, e 2,, e n } x = (x 1, x 2,, x n V e i V v = v 1 e 1 + v n e n = v i e i 9

i V V V e i V w = w 1 e 1 + w n e n = w i e i V {e 1,, e n } {e 1,, e n } V V V V {e 1,, e n } v V v = x i e i n {x i } v = (x 1,, x n V {f 1,, f n } {e i } {f i } f i = a j i e j 10

(f 1 f n = a (e 1 1 a 1 n 1 e n ( a 1 1 a 1 n A = a n 1 a n n a n 1 a n n v = x i e i = y i f i (f 1 f n y 1 y n ( (e 1 e n A (e 1 e n A y 1 y n = x 1 x n ( = y 1 y n ( = e 1 e n x 1 x n e 1 e n x 1 x n V {e 1,, e n }, {f 1,, f n } 11

(f 1 f n = (e 1 e n A v V v = x i e i = y i f i A y 1 y n = x 1 x n a j i yi = x j 1 2 {e i } {f i } V 3 4 {e i } {f i } (f 1 f n = (e 1 e n A, f 1 e 1 = At f n e n 12

V V V {e i } {e i } v V v = x i e i n {x i } v = (x 1,, x n V V {e i } {f i } V (f 1 f n = (e 1 e n A e 1 f (e 1 1 e n = (f 1 f n = E e n f n E e 1 e n = A (f 1 f n = f 1 f n (f 1 f n e 1 e n (e 1 e n A = A (f 1 f n 13

A f 1 f n = v V e 1 e n v = x i e i = y i f i (x 1 x n e 1 e n (x 1 x n A ( = f 1 f n (x 1 x n A = x i a i j = y j ( = y 1 y n f 1 f n y 1 y n (y 1 y n f 1 f n V {e 1,, e n }, {f 1,, f n } (f 1 f n = (e 1 e n A {e 1,, e n }, {f 1,, f n } {e 1,, e n }, {f 1,, f n } f 1 A f n = e 1 e n v V 14

v = x i e i = y i f i (y 1 y n = (x 1 x n A y j = a i j x i V V V {e 1,, e n }, {f 1,, f n } (f 1 f n = (e 1 e n A A B = (b i j v V v = x i e i = y i f i y 1 A y n = y 1 y n = B x 1 x n x 1 x n v V v = x i e i = y i f i (1 (y 1 y n = (x 1 x n A (2 y i = x j a j i V 15

V V V f : V V V {e 1,, e n } f(e i = f j i e j n 2 {f j i } {d 1,, d n } (d 1 d n = (e 1 e n A d i = a j i e j f(d i = f j i d j { f j i } {f j i } f(d i = f(a j i e j = a j i f k j e k f(d i = f j i d j = f j i ak j e k f j i ak j = aj i f k j (a j i (bj i bl k k f j i ak j bl k = aj i f k j bl k f l i = f k j aj i bl k (3 {f j i } 16

V V n 2 {f j i } (3 f : V V F : V V R α V, v V F (α, v = α(f(v F V {e i } {e i } f(v = f(e i, ve i well defined 30 f j i f(e i = f j i e j f(e i = F (e j, e i e j f j i = F (ej, e i f j i F (α, v = α(f(v F : V V R 17

f j i = F (ej, e i f j i {e i }, {d i } d i = a j i e j d i = b i j ej f j i = F (dj, d i = F (b j k ek, a l i e l = b j k al i f k l 1 V n V r + s r s {}}{{}}{ f : V V V V R r,s (r,s V {e i } {e i } n r+s f i 1i r y1 j s = f(e i 1,, e i r, e j1,, e jr f {e i } f i 1i r y1 j s 18

( 2 V n V n r+s n r+s r,s {e i } n r+s {f i 1i r y1 j s } {d i } n r+s f i 1i r j1 j s d i = a j i e j f k 1k rl1 l s = b k 1 i 1 b k r i r a j 1 l1 a j s ls f i 1i r j1 j s (b j i (aj i 1 2 d i = a j i e j d i = b i j ej f k 1k rl1 l s = f(d k i,, d k r, d l1,, d ls = b k 1 i 1 b k r i r a j 1 l1 a j s ls f i 1i r j1 j s 2 1 (2, 2 V {e i } n 2+2 f i 1i 2j1 j 2 19

f k 1k 2l1 l 2 = b k 1 i 1 b k 2 i 2 a j 1 l1 a j 2 l2 f i 1i 2 j1 j 2 u, v V, α, β V u = u i e i, v = v i e i, α = α i e i, β = β i e i, f(α, β, u, v f(α, β, u, v = α i1 β i2 u j 1 v j 2 f i 1i 2 j1 j 2 V d i = a j i e i v i e i = v i d i v i = b i j vj α i e i = ᾱ i d i ᾱ i = a j i α j ᾱ i1 βi2 ū j1 v j 2 i f 1 i 2 j1 j 2 = (a k 1 i 1 α k1 (a k 2 i 2 β k2 (b j 1 l1 u l 1 (b j2 l 2 v l 2 f i 1i 2 j1 j 2 = α k1 β k2 u l 1 v l 2 f k 1k 2l1 l 2 20

r s {}}{{}}{ f : V V V V R r s {}}{{}}{ L( V,, V, V,, V ; R Vs r r s f, g V r s, k R (f + g(v = f(v + g(v (kf(v = kf(v f + g, kf V r s r s {}}{{}}{ Vs r = L( V,, V, V,, V ; R r = 2, s = 2 V 2 2 = L(V, V, V, V ; R V {e i } V {e i } e i e j e k e l V 2 2 α = α i e i, β = β i e i, u = u i e i, v = v i e i, e i e j e k e l (α, β, u, v = α(e i β(e j e k (ue l (v = α i β j u k v l 21

f V 2 2 e i e j e k e l f(α, β, u, v = f(α i e i, β j e j, u k e k, v l e l = f ij kl α iβ j u k v l f(e i, e j e k, e l = f ij kl = f ij kl e i e j e k e l (α, β, u, v α, β, u, v f = f ij kl e i e j e k e l f V 2 2 n4 {e i e j e k e l } V 2 2 {e i e j e k e l } f ij kl e i e j e k e l = 0 f ij kl = f(ei, e j, e k, e l = 0 {e i e j e k e l } V 2 2 dim V 2 2 = n 4 L(V, V, V, V = V V V V ( u v α β(u, v V, α, β V V V V V ( Vs r e i1 e ir e j 1 e j s f Vs r 22

f i 1i r j 1 j s e i1 e ir e j 1 e j s dim V r s = n r+s V {e i }, {f i } {e i }, {f i } f j = a i j e i A A = (a i j A B B = (b i j A i j a i j b i j ai j V ( x i e i = y i f j y i = b i j xj V ( x i e i = y i f i 23

y i = a j i x j (2,2 T V {e i }, {f i } T ij pq kl, T rs T pq rs = b p i bq j ak ra l st ij kl (b i j y i = b i j xj a j i yi = x j T rs pq = b p i bq j ak ra l st ij kl a i pa j pq q T rs = a k ra l st ij kl 24

(u, v = g(u, v (u, v V g : V V R, (u, v g(u, v (1 g g (2 g g(u, v = g(v, u, u, v V (3 g g(u, u 0, u V u = 0 g g g V g V {e 1, e n } {ω 1,, ω n } g g = g ij ω i ω j g ij = g(e i e j u = u i e i, v = v i e i 25

g(u, v = g(u i e i, v j e j = u i v j g(e i, e j = g ij u i v j = g ij ω i (uω j (v = g ij ω i ω j (u, v g {e i } n 2 {g ij } g ē i ω i ē i = a j i e j (A g = ḡ ij ω i ω j ḡ ij = g(ē i, ē j = g(a k i e k, a l j e l = a k i al j g kl (B g {e i } n 2 {g ij }, {ē i } n 2 {ḡ ij } (A g ij ḡ ij (B {e i } n 2 {g ij } {ē i } n 2 {ḡ ij } (B V g : V V R V {e 1,, e n }, {ē 1,, ē n } ē i = a j i e j u, v V u = u i e i = ū i ē i, v = v i e i = v i ē i u i = a i jūj, v i = a i j vj 26

g(u, v = g ij u i v j = g ij a i kūk a j l ūl = ḡ kl ū k v l g(u, v g {e i } {ω i } g = g ij ω i ω j g ij ω i ω j (u k e k, v l e l = g ij ω i (u k e k ω j (v l e l = g ij u i u j {ē i } ω i g = g ij ω i ω j = ḡ kl ω k ω l g ij ω i ω j = g ij a i k aj l ωk ω l ḡ kl = g ij a i k aj l g ij, ḡ ij {g ij } {g ij } {g ij } V {e i } n 2 {ē i } ē i = e j a j i g ij = g(e i, e j, ḡ ij = g(ē i, ē j {g ij }, {ḡ ij } {g ij }, {ḡ ij } {a i j } {bi j } a i b i i a i b i = a j b j = a p b p 27

ḡ kl = g ij a i k aj l b k i b k i ḡkl = b k i g sta s k at l = δs i at l g st = a j l g ij ḡ lr g si b k i ḡklḡ lr g si = b k i δr k gsi = b r i gsi a j l g ijḡ lr g si = a j l δs j ḡlr = a s l ḡlr b r i gsi = a s l ḡlr a k r g sk = a s l ak rḡ lr {g ij }, {g ij } g ij = a i k aj l ḡkl (1 ḡ ij = a k i aj l g kl (2 {g ij } (1 V G : V V R {e i } {ω i },{ē i } { ω i } ē i = a j i e j ω i = a i j ωj α, β V α = α i ω i = ᾱ i ω i, β = β i ω i = β i ω i ᾱ i = α j a j i, β i = β j a j i G(α, β = g ij α i β j = a i k aj l ḡkl α i β j = ḡ kl ᾱ k βl G 28

G = g ij e i e j {g ij } {g ij } {g ij } {g ij } V f : V V V {e i } f(e i = f j i e j n 2 {f j i } V {ē i } ē i = e j a j i f(ē i = f j i ēj f(ē i = f(a k i e k = a k i f j k e j f k i ēk = f k i aj k e j a k i f j k = aj k f k i (1 {f j i }, { f j i } (1 V v = v i e i = v i ē i f(v = v i f j i e j = a i k vk f j i e j = f l k aj l vk e j = f l k vk ē l (1 29

V (1,1 F : V V R {f j i }, { f j i } a k i f j k = aj f k i k (1 f : V V F : V V R, L(V ; V L(V, V ; R v V, α V,V {e i } {ω i } Φ : L(V ; V L(V, V ; R, f F F (α, v = α(f(v Ψ : L(V, V ; R L(V ; V, F f f(v = F (ω i, ve i Φ Ψ F (ω i, ve i = F (a i j ωj, ve i = F ( ω j, va j i e i = F ( ω j, vē j F Ψ Φ (Ψ Φ(f(v = Φ(f(ω i, ve i = ω i (f(ve i = f(v Ψ Φ Φ U, V 30

f : U V, u v u v Φ : L(V ; V L(V, V ; R, f F f F V V a V V f f(a R V R( a V V a V V V V V V V {e 1,, e n } {ω 1,, ω n } e i ω i f : V V, a i e i a i ω i {ē i } { ω i }, ē i = a j i e j f : ē i = a j i e j a j i ωj ω i = b i j ωj ē i ω i v V v f : V W w W f(v = w v v 1 v 2 f(v 1 f(v 2 31

V, W V, {e i } V v = v i e i v = (v 1,, v n V n {e i } n {v i } {ē i } n v i ē i = a j i e j v i = a i j vj V v i e i = a i j vj e i = v j ē j (, V V, V (Intrinsic f : V V 32

x V f(x(y = (x, y y V f(x V Intrinsic f(x = 0 x = 0 V, V f : V V V {e 1,, e n }, {ω 1,, ω n } (e i, e j = g ij f(e i = a ij ω j a ij = (a ik ω k (e j = f(e i (e j = (e i, e j = g ij f(e i = g ij ω j V e i V g ij ω j e i = g ij ω j e i, ω j e 1 g 11 g 1n ω 1 = e n g n1 g nn (g ij (g ij ω 1 g 11 g 1n e 1 = ω n g n1 g nn e i, ω i ω n e n 33

e i = g ij ω j, ω i = g ij e j V x i e i V x i g ij ω j V y i ω i V y i g ij e j V x i e i x i ω i V x i = g ij x j, x i = g ij x j f = f ijk ω i ω j ω k = f ijk (g ih e h ω j ω k = f hjk (g hi e i ω j ω k = f i jk e i ω j ω k f i jk = f hjkg hi V (, f = f i jk ei ω j k, h = h i jk ei ω j k (f, g (f, h = f i 1 j 1 k 1 h i 2 j2 k 2 (e i1, e i2 (ω j 1, ω j 2 (ω k 1, ω k 2 = f i 1 j 1 k 1 h i 2 j2 k 2 g i1 i 2 g j 1j 2 g k 1k 2 f 2 = (f, f = f i 1 j 1 k 1 f i 2 j 2 k 2 g i1 i 2 g j 1j 2 g k 1k 2 34 = f ijk f ijk

(V, g V V V {e 1,, e n } V R n, x i e i (x 1,, x n (V, g V v V α α(v V {e i } {e i } v V, α V α(v v = v i e i, α = α i e i α(v = α i e i (v j e j = α i v i n α i v i i=1 v V, α V v α v α = v i α j e i e j fj i = vi α j 35

v α = f i j e i e j α(v n α(v = v i α i = v α(e i, e i i=1 v α(e i, e i u, v V, α, β V u v α β V 2 2 u α α(uv β V 1 1 u = u i e i, v = v i e i, α = α i e i, β = β i e i u v α β = u i v j α k β l e i e j e k e l = f ij kl e i e j e k e l α(uv β = u i α i v j β l e j e l = f ij il e j e l (2,2 f ij kl T 2 2 (1,1 f ij il e j e l f V 2 2 g V 1 1 g(α, v = f(e i, α, e i, v 36

V {e i } {e i } p, q 1 p r, 1 q s (r,s f (r-1,s-1 g g(α 1, α r 1, v 1, v s 1 = f(α 1,, α p 1, e i α p,, α r 1, v 1,, v q 1, e i, e q,, v s 1 g f p q g g = contr (p,q (f f i p j q contr (ij,pq (f (i, p g = contr(f V {e i } f i 1i r j 1 j s (r,s f p q (r-1,s-1 g g i 1i p 1 i p+1 i r j 1 j q 1 i q+1 j r = f i 1i p 1 ki p+1 i r j 1 j q 1 ki q+1 j r g f p q g g = contr (p,q (f 37

f (2,2 u = u i e i, v = v i e i, α = α i e i, β = β i e i f(α, β, u, v = f ij kl α iβ j u k v l, (4, 4 f α β u v contr (1234,3412 f α β u v V f f : V V (1,1 F : V V R F (α, v = α(f(v V {e i } F contr(f = F (e i, e i = e i f(e i = fi i f f 38

( 39

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