II I Riemann 2003

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Transcription:

II I Remann 2003

Dfferental Geometry II Dfferental Geometry II I Dfferental Geometry I

1 1 1.1.............................. 1 1.2................................. 10 1.3.......................... 16 1.4......................... 17 2 24 2.1............................... 24 2.2............................... 27 2.3................................. 28 2.4............................ 32 3 Remann 43 3.1............................ 43 3.2......................... 45 3.3 Remann.............................. 48 3.4......................... 53 3.5.............................. 62 3.6............................... 71 4 Remann 79 4.1......................... 79 4.2............................. 84 4.3............................... 90 4.4 Smons............................ 93

1 1 1.1 1.1.1 V V R V V V R v V v(f) = f(v) (f V ) v : V R v (V ) (V ) V δ j δ j = { 1, = j 0, j V {u 1,..., u n } f (u j ) = δ j V {f } V dm V = dm V {f } {u j } {}}{{}}{ 1.1.2 V V V V V p+q V (p, q) T (p,q) (V ) T (p,q) (V ) (p, q) T (p,q) (V ) T (p,q) (V ) A T (r,s) (V ) B (A B)(g 1,..., g p+r, v 1,..., v q+s ) = A(g 1,..., g p, v 1,..., v q )B(g p+1,..., g p+r, v q+1,..., v q+s ) (g 1,..., g p+r V, v 1,..., v q+s V ) p q p+r q+s {}}{{}}{ A B : V V V V R A B V (p + r, q + s) A B A B T (1,0) (V ) = (V ) = V T (0,1) (V ) = V

2 1 V u 1,..., u p V f 1,..., f q (u 1 u p f 1 f q )(g 1,..., g p, v 1,..., v q ) = g 1 (u 1 ) g p (u p )f 1 (v 1 ) f q (v q ) (g 1,..., g p V, v 1,..., v q V ) p q {}}{{}}{ u 1 u p f 1 f q : V V V V R u 1 u p f 1 f q V (p, q) 1.1.3 V T (p,q) (V ) T (r,s) (V ) T (p+r,q+s) (V ) (A, B) A B q p {}}{{}}{ V V V V T (p,q) (V ) (u 1,..., u p, f 1,..., f q ) u 1 u p f 1 f q 1.1.2 A B A B u 1 u p f 1 f q u f j 1.1.4 V T (V ) = T (p,q) (V ) p,q=0 T (0,0) (V ) = R 1.1.2 T (p,q) (V ) T (r,s) (V ) T (p+r,q+s) (V ) (A, B) A B T (V ) T (V ) T (V ) T (V ) V 1.1.5 V n u 1,..., u n V f 1,..., f n u 1 u p f j 1 f j q (1 1,..., p, j 1,..., j q n) T (p,q) (V ) T (p,q) (V ) n p+q

1.1. 3 u 1 u p f j 1 f jq (1 1,..., p, j 1,..., j q n) 1,...,p=1 j 1,...,jq=1 a 1 p j 1 j q u 1 u p f j 1 f j q = 0 (a 1 p j 1 j q R) 1 k 1,..., k p, l 1,..., l q n k 1,..., k p, l 1,..., l q (f k 1,..., f kp, u l1,..., u lq ) a k 1 k p l 1 l q = 0 u 1 u p f j 1 f jq u 1 u p f j 1 f jq (1 1,..., p, j 1,..., j q n) T (p,q) (V ) T (p,q) (V ) A V v v = f j (v)u j j=1 V g g = g(u )f g 1,..., g p V v 1,..., v q V 1 =1 =1 A(g 1,..., g p, v 1,..., v q ) = A g 1 (u 1 )f 1,..., g p (u p )f p, = = 1,...,p=1 j 1,...,jq=1 1,...,p=1 j 1,...,jq=1 p =1 f j 1 (v 1 )u j1,..., j 1 =1 f jq (v q )u jq j q =1 g 1 (u 1 ) g p (u p )f j 1 (v 1 ) f j q (v q )A(f 1,..., f p, u j1,..., u jq ) A(f 1,..., f p, u j1,..., u jq ) (u 1 u p f j 1 f jq )(g 1,..., g p, v 1,..., v q ). A = 1,...,p=1 j 1,...,jq=1 A(f 1,..., f p, u j1,..., u jq )u 1 u p f j 1 f j q u 1 u p f j 1 f j q T (p,q) (V )

4 1 u 1 u p f j 1 f j q (1 1,..., p, j 1,..., j q n) T (p,q) (V ) T (p,q) (V ) n p+q 1.1.6 1.1.5 T (p,q) V A A = A(f 1,..., f p, u j1,..., u jq )u 1 u p f j 1 f j q 1,...,p=1 j 1,...,jq=1 A A(f 1,..., f p, u j1,..., u jq ) A 1.1.7 A = A(f 1,..., f p, u j1,..., u jq )u 1 u p f j 1 f j q Ensten A 1 p j 1 j q = A(f 1,..., f p, u j1,..., u jq ) A A = A 1 p j 1 j q u 1 u p f j 1 f j q A = (A 1 p j 1 j q ) 1.1.8 V n u 1,..., u n V f 1,..., f n T (p,q) (V ) A A = A 1 p j 1 j q u 1 u p f j 1 f jq V ū 1,..., ū n f 1,..., f n A = Āk 1 k p l 1 l q ū k1 ū kp f l 1 f l q u 1,..., u n ū 1,..., ū n g = (g k ) ḡ = (ḡ l j) ū k = g ku, ḡ kg k j = δ j. Ā k 1 k p l 1 l q = A 1 p j 1 j q ḡ k 1 1 ḡ kp p g j 1 l 1 g jq l q

1.1. 5 f l = h l jf j δk l = f l (ū k ) = h l jf j (gku ) = h l jgkf j (u ) = h l jgkδ j = hl gk. h = (h l j) g f l = ḡjf l j. ū k = gk u ḡa k k ḡaū k k = ḡag k ku = δau = u a. f l = ḡ l jf j g b l l g b l f l = g b l ḡ l jf j = δ b jf j = f b. u = ḡ k ū k, f j = g j l f l V A A = A 1 p j 1 j q u 1 u p f j 1 f jq = A 1 p j 1 j q ḡ k 1 1 ū k1 ḡ k p p ū kp g j 1 l 1 f l 1 g j q l q = A 1 p j 1 j q ḡ k 1 1 ḡ k p p g j 1 l 1 g j q l q ū k1 ū kp f l 1 f l q. f l q Ā k 1 k p l 1 l q = A 1 p j 1 j q ḡ k 1 1 ḡ k p p g j 1 l 1 g j q l q 1.1.9 V V u 1,..., u n f 1,..., f n T (p,q) (V ) A = (A 1 p j 1 j q ) T (r,s) (V ) B = (B k 1 k r l 1 l s ) A B (A B) 1 pk 1 k r j 1 j q l 1 l s = A 1 p j 1 j q B k 1 k r l 1 l s

6 1 A = A 1 p j 1 j q u 1 u p f j 1 f jq B = B k 1 k r l 1 l s u k1 u kr f l 1 f l s 1.1.3 A B = (A 1 p j 1 j q u 1 u p f j 1 f j q ) (B k 1 k r l 1 l s u k1 u kr f l 1 f ls ) = A 1 p j 1 j q B k 1 k r l 1 l s u 1 u p u k1 u kr f j 1 f j q f l 1 f l s (A B) 1 p k 1 k r j 1 j ql 1 l s = A 1 p j 1 j q B k 1 k r l 1 l s 1.1.10 V u 1,..., u n f 1,..., f n A T (p,q) (V ) 1 r p, 1 s q r, s p 1 q 1 {}}{{}}{ C (r,s) A : V V V V R (C (r,s) A)(g 1,..., g p 1, v 1,..., v q 1 ) = A(g 1,..., g r 1, f, g r,..., g p 1, v 1,..., v s 1, u, v s,..., v q 1 ) C (r,s) A T (p 1,q 1) (V ) C (r,s) A A 1.1.11 1.1.10 V V u 1,..., u n f 1,..., f n (C (r,s) A) 1 p 1 j 1 j q 1 = A 1 r 1 r p 1 j 1 j s 1 j s j q 1 ū k = gk u 1.1.8 f l = ḡjf l j A(g 1,..., g r 1, f k, g r,..., g p 1, v 1,..., v s 1, ū k, v s,..., v q 1 ) = A(g 1,..., g r 1, ḡ k j f j, g r,..., g p 1, v 1,..., v s 1, g ku, v s,..., v q 1 ) = ḡ k j g ka(g 1,..., g r 1, f j, g r,..., g p 1, v 1,..., v s 1, u, v s,..., v q 1 ) = δ ja(g 1,..., g r 1, f j, g r,..., g p 1, v 1,..., v s 1, u, v s,..., v q 1 ) = A(g 1,..., g r 1, f, g r,..., g p 1, v 1,..., v s 1, u, v s,..., v q 1 ).

1.1. 7 C (r,s) A (C (r,s) A) 1 p 1 j 1 j q 1 = (C (r,s) A)(f 1,..., f p 1, u j1,..., u jq 1 ) = A(f 1,..., f r 1, f, f r,..., f p 1, u j1,..., u js 1, u, u js,..., u jq 1 ) = A 1 r 1 r p 1 j 1 j s 1 j s j q 1. 1.1.12 V W p {}}{{}}{ φ : V V V V W Φ(v 1 v p g 1 g q ) = φ(v 1,..., v p, g 1,..., g q ) (v V, g j V ) q Φ : T (p,q) (V ) W 1.1.5 u 1,..., u n V f 1,..., f n u 1 u p f j 1 f j q (1 1,..., p, j 1,..., j q n) T (p,q) (V ) Φ(u 1 u p f 1 f q ) = φ(u 1,..., u p, f 1,..., f q ) Φ T (p,q) (V ) v V, g j V Φ(v 1 v p g 1 g q ) = Φ ( f 1 (v 1 )u 1 f p (v p )u p g 1 (u j1 )f j 1 g q (u jq )f j q ) = Φ ( f 1 (v 1 ) f p (v p )g 1 (u j1 ) g q (u jq )u 1 u p f j 1 f j q ) = f 1 (v 1 ) f p (v p )g 1 (u j1 ) g q (u jq )Φ(u 1 u p f j 1 f j q ) = f 1 (v 1 ) f p (v p )g 1 (u j1 ) g q (u jq )φ(u 1,..., u p, f j 1,..., f jq ) = φ ( f 1 (v 1 )u 1,..., f p (v p )u p, g 1 (u j1 )f j 1,..., g q (u jq )f j q ) = φ(v 1,..., v p, g 1,..., g q ) Φ Φ T (p,q) (V ) Φ Φ V

8 1 1.1.13 V V V End(V ) φ : V V End(V ) φ(u, f)(v) = f(v)u (u, v V, f V ) φ 1.1.12 Φ(u f) = φ(u, f) (u V, f V ) Φ : T (1,1) (V ) End(V ) V u 1,..., u n f 1,..., f n Φ(u f j )(u k ) = φ(u, f j )(u k ) = f j (u k )u = δ j k u Φ(u f j ) u j u u k 0 {Φ(u f j ) 1, j n} End(V ) 1.1.5 Φ T (1,1) (V ) End(V ) T (1,1) (V ) End(V ) A T (1,1) (V ) A j A = A ju f j Φ(A) = A jφ(u f j ). Φ(A)u k = A jφ(u f j )u k = A jδ j k u = A ku (A j) Φ(A) u 1,..., u n C (1,1) A = A = tr(φ(a)) C (1,1) A = tr(φ(a)) T (1,1) (V ) End(V ) T (1,1) (V ) 1.1.14 V W F : V W F (p,0) : T (p,0) (V ) T (p,0) (W )

1.1. 9 v 1,..., v p V F (p,0) (v 1 v p ) = F (v 1 ) F (v p ) F (0,q) : T (0,q) (W ) T (0,q) (V ) f 1,..., f p W F (0,q) (f 1 f q ) = (f 1 F ) (f q F ) T (p,0) V A F (p,0) (A)(f 1,..., f p ) = A(f 1 F,..., f p F ) (f 1,..., f p W ) F (p,0) (A) T (p,0) (W ) F (p,0) f 1,..., f p W F (p,0) (v 1 v p )(f 1,..., f p ) = (v 1 v p )(f 1 F,..., f p F ) = (f 1 F )(v 1 ) (f p F )(v p ) = (F (v 1 ) F (v p ))(f 1,..., f p ) F (p,0) (v 1 v p ) = F (v 1 ) F (v p ) 1.1.5 T (p,0) (V ) F (p,0) T (0,q) W B F (0,q) (B)(v 1,..., v q ) = B(F (v 1 ),..., F (v q )) (v 1,..., v q V ) F (0,q) (B) T (0,q) (V ) F (0,q) v 1,..., v q V F (0,q) (f 1 f q )(v 1,..., v q ) = (f 1 f q )(F (v 1 ),..., F (v q )) = f 1 (F (v 1 )) f q (F (v q )) = ((f 1 F ) (f q F ))(v 1,..., v q ) F (0,q) (f 1 f q ) = (f 1 F ) (f q F ) 1.1.5 T (0,q) (W ) F (0,q)

10 1 1.2 1.2.1 V T (p,0) (V ) A 1 < j p (t,j A)(f 1,..., f p ) = A(f 1,..., f j,..., f,..., f p ) (f 1,..., f p V ) t,j : T (p,0) (V ) T (p,0) (V ) p V = {A T (p,0) (V ) t,j A = A (1 < j p)} p {1,..., p} S p p V A q V B (A B)(g 1,..., g p+q ) = 1 p!q! j σ S p+q sgn(σ)(a B)(g σ(1),..., g σ(p+q) ) (g 1,..., g p+q V ) A B T p+q,0 (V ) A B p+q V A B A B 1.2.2 V (r, 0) T T (g 1,..., g r ) = σ Sr sgn(σ)t (g σ(1),..., g σ(r) ) (g 1,..., g r V ) T T r,0 (V ) T r V 1.2.1 A B p+q V p V q V p+q V (A, B) A B C r V V u 1,..., u p (A B) C = A (B C) u 1 u p = σ S p sgn(σ)u σ(1) u σ(p)

1.2. 11 T r V 1 < j r j τ S r (t,j T )(g 1,..., g r ) = T (g 1,..., g j,..., g,..., g r ) = T (g τ(1),..., g τ(r) ) j = σ S r sgn(σ)t (g τσ(1),..., g τσ(r) ) = sgn(τ) σ S r sgn(τσ)t (g τσ(1),..., g τσ(r) ) = σ S r sgn(σ)t (g σ(1),..., g σ(r) ) = T (g 1,..., g r ). t,j T = T T r V A T (p,0) (V ) B T (q,0) (V ) A B T (p+q,0) (V ) A B p+q V p V q V p+q V (A, B) A B (A, B) A B ( 1.1.3) T T A p V, B q V, C r V (A B) C = A (B C) S p+q = {τ S p+q+r τ() = (p + q + 1 p + q + r)} = = ((A B) C)(g 1,..., g p+q+r ) 1 sgn(σ)(a B)(g σ(1),..., g σ(p+q) ) C(g σ(p+q+1),..., g σ(p+q+r) )) (p + q)!r! σ S p+q+r 1 sgn(σ) (p + q)!r! σ S p+q+r 1 sgn(τ)a(g στ(1),..., g στ(p) ) B(g στ(p+1),..., g στ(p+q) ) p!q! τ S p+q C(g σ(p+q+1),..., g σ(p+q+r) )

12 1 = = 1 1 p!q!r! (p + q)! σ S p+q+r τ S p+q sgn(στ) (A(g στ(1),..., g στ(p) ) B(g στ(p+1),..., g στ(p+q) )) C(g στ(p+q+1),..., g στ(p+q+r) ) 1 sgn(σ) p!q!r! σ S p+q+r (A(g σ(1),..., g σ(p) ) B(g σ(p+1),..., g σ(p+q) )) C(g σ(p+q+1),..., g σ(p+q+r) ) = (A (B C))(g 1,..., g p+q+r ) 1 sgn(σ) p!q!r! σ S p+q+r A(g σ(1),..., g σ(p) ) (B(g σ(p+1),..., g σ(p+q) ) C(g σ(p+q+1),..., g σ(p+q+r) )) (A B) C = A (B C) σ S p sgn(σ 1 ) = sgn(σ) V u 1,..., u p (u 1 u p )(g 1,..., g p ) = σ S p sgn(σ)u 1 (g σ(1) ) u p (g σ(p) ) = σ S p sgn(σ)u σ 1 (1)(g 1 ) u σ 1 (p)(g p ) = σ S p sgn(σ 1 )u σ 1 (1)(g 1 ) u σ 1 (p)(g p ) = σ S p sgn(σ)u σ(1) (g 1 ) u σ(p) (g p ) = σ S p sgn(σ)(u σ(1) u σ(p) )(g 1,..., g p ). u 1 u p = σ S p sgn(σ)u σ(1) u σ(p)

1.2. 13 1.2.3 V V = p V p=0 0 V = R 1.2.1 p V q V p+q V (A, B) A B V V V V V 1.2.4 V p {}}{ V V p V (u 1,..., u p ) u 1 u p u 1,..., u p V 1 < j p u 1 j u j u u p = u 1 u p p A = (A j) v j = A ju v 1 v p = (det A)u 1 u p 1.1.3 (u 1,..., u p ) u 1 u p u 1,..., u p V 1 < j p u 1 j u j u u p = u 1 u p j τ S p u 1 j u j u u p = σ S p sgn(σ)u τσ(1) u τσ(p) = sgn(τ) σ S p sgn(τσ)u τσ(1) u τσ(p) = σ S p sgn(σ)u σ(1) u σ(p) = u 1 u p.

14 1 u 1 j u j u u p = u 1 u p u 1,..., u p u 1 u p = 0 S p σ S p u σ(1) u σ(p) = sgn(σ)u 1 u p p A = (A j) v j = A ju v 1 v p = ( ) ( A 1 1 u 1 A p ) p u p = σ S p A σ(1) 1 u σ(1) A σ(p) p u σ(p) ( 0 ) = sgn(σ)a σ(1) 1 A σ(p) u 1 u p σ S p p = (det A)u 1 u p 1.2.5 u 1,..., u n V u 1 u p (1 1 < < p n) ( ) n p V dm( p V ) = p u 1 u p (1 1 < < p n) 1 < < p a 1 p u 1 u p = 0 (a 1 p R) 1 k 1 < < k p n k 1,..., k p (f k 1,..., f k p ) a k1 k p = 0 u 1 u p u 1 u p (1 1 < < p n) p V p V A V g g 1,..., g p V g = g(u j )f j A(g 1,..., g p ) = A ( g 1 (u j1 )f j 1,..., g p (u jp )f jp)

1.2. 15 = g 1 (u j1 ) g p (u jp )A(f j 1,..., f jp ) = A(f j 1,..., f j p )(u j1 u jp )(g 1,..., g p ) = A(f jσ(1),..., f jσ(p) )(u jσ(1) u jσ(p) )(g 1,..., g p ) j 1 < <j p σ S p = sgn(σ)a(f j1,..., f jp )(u jσ(1) u jσ(p) )(g 1,..., g p ) j 1 < <j p σ S p = A(f j1,..., f jp )(u j1 u jp )(g 1,..., g p ). j 1 < <j p A = j 1 < <j p A(f j1,..., f jp )u j1 u jp u 1 u p p V u 1 u p (1 1 < < p n) p V p V ( ) n p 1.2.6 V W F : V W 1.1.14 F (p,0) : T (p,0) (V ) T (p,0) (W ) F (p,0) ( p V ) p W F (p,0) : p V p W 1.2.1t V,j : T (p,0) (V ) T (p,0) (V ), t W,j : T (p,0) (W ) T (p,0) (W ) F (p,0) F (p,0) t V,j = t W,j F (p,0) F (p,0) F (p,0) ( p V ) p W F (p,0) : p V p W

16 1 1.3 1.3.1 V W A : p {}}{ V V W u 1,..., u p V 1 < j p j A(u 1,..., u j,..., u,..., u p ) = A(u 1,..., u p ) A p W = R p 1.3.2 V W φ : p {}}{ V V W Φ(v 1 v p ) = φ(v 1,..., v p ) (v V ) Φ : p V W Φ : p V W φ(v 1,..., v p ) = Φ(v 1 v p ) (v V ) φ : p {}}{ V V W φ {φ : p {}}{ V V W φ } p V W Hom( p V, W ) 1.2.5 u 1,..., u n V p V u 1 u p (1 1 < < p n) Φ(u 1 u p ) = φ(u 1,..., u p )

1.4. 17 Φ p V u 1,..., u n f 1,..., f n v V Φ(v 1 v p ) = Φ ( f 1 (v 1 )u 1 f p (v p )u p ) = Φ ( f 1 (v 1 ) f p (v p )u 1 u p ) = f 1 (v 1 ) f p (v p )Φ(u 1 u p ) = f 1 (v 1 ) f p (v p )φ(u 1,..., u p ) = φ ( f 1 (v 1 )u 1,..., f p (v p )u p ) = φ(v 1,..., v p ) Φ Φ T (p,q) (V ) Φ Φ V Φ Hom( p V, W ) φ(v 1,..., v p ) = Φ(v 1 v p ) (v V ) φ : {φ : p {}}{ V V W p {}}{ V V W φ } Hom( p V, W ) 1.3.3 V p V V p 1.3.2 ( p V ) 1.4 1.4.1 V T (0,2) (V ) A V V α A(x, y) = (α(x))(y) (x, y V ) T (0,2) (V ) V V Hom(V, V ) α Hom(V, V ) T (0,2) (V ) A 0 x V (α(x))(x) > 0 A V

18 1 α Hom(V, V ) A(x, y) = (α(x))(y) (x, y V ) A T (0,2) (V ) A T (0,2) (V ) α Hom(V, V ) T (0,2) (V ) Hom(V, V ) A 0 x V (α(x))(x) > 0 A(x, x) > 0 A V 1.4.2 V,, 1.4.1 Hom(V, V ) α p V ( p V ) 1.2.6 α : V V α (p,0) : p V p V = ( p V ) T (0,2) ( p V ) p V p V ( p V ) p V φ ( p V ) Φ φ(v 1,..., v p ) = Φ(v 1... v p ) (v 1,..., v p V ) α (p,0) T (0,2) ( p V ) A V u 1,..., u p v 1,..., v p A(u 1 u p, v 1 v p ) = (α (p,0) (u 1 u p ))(v 1 v p ) = (α(u 1 ) α(u p ))(v 1 v p ) = (α(u 1 ) α(u p ))(v 1,..., v p ) = σ S p sgn(σ)(α(u σ(1) ) α(u σ(p) ))(v 1,..., v p ) = σ S p sgn(σ)(α(u σ(1) ))(v 1 ) (α(u σ(p) ))(v p ) = sgn(σ) u σ(1), v 1 u σ(p), v p σ S p = det( u, v j ) 1,j p. u 1,..., u n V u 1 u p (1 1 < < p n)

1.4. 19 p V 1 1 < < p n 1 j 1 < < j p n A(u 1 u p, u j1 u jp ) = δ 1 j 1 δ pj p A p V 1.4.3, V p V 1.4.2 A A, u = u, u 1.4.4 1.4.2 V u 1,..., u p v 1,..., v p u 1 u p, v 1 v p = det[ u, v j ] 1,j p V e 1,..., e n p V e 1 e p (1 1 < < p n) 1.4.5 1.4.4 V u 1, u 2 u 1 u 2 2 u 1, u 1 u 1, u 2 = u 1 u 2, u 1 u 2 = u 2, u 1 u 2, u 2 = u 1 2 u 2 2 u 1, u 2 2 u 1 u 2 θ u 1, u 2 = u 1 u 2 cos θ u 1 u 2 u 1 2 u 2 2 sn 2 θ = u 1 2 u 2 2 (1 cos 2 θ) = u 1 2 u 2 2 u 1, u 2 2 = u 1 u 2 2 u 1 u 2 u 1 u 2 2 V V 3 1.4.6 R n u u m n R n u 1,..., u m, v 1,..., v m u = [u j ], v = [v j ] u 11 u 1n.. v 11 v m1.. = u 1. [v 1 v m] = [u v j ] = [ u, v j ]. u m1 u mn v 1n v mn u m

20 1 m 1.4.4 u 11 u 1n v 11 v m1 det.... = det[ u, v j ] u m1 u mn v 1n v mn = u 1 u m, v 1 v m. R n e 1,..., e n u = [u 1... u n ] = u j e j. ( 1.2.4) ( ) ( ) u 1 u m = u 1j1 e j1 u 1jm e jm = = v 1 v m = j 1 =1 #{j 1,...,j m }=m j 1 < <j m j 1 < <j m u 1 u m, v 1 v m = j 1 < <j m u 11 u 1n det.. j=1 j m =1 u 1j1 u mjm e j1 e jm u 1j1 u 1jm.. u mj1 u mjm v 1j1 v 1jm.. v mj1 v mjm u 1j1 u 1jm.. u mj1 u mjm e j1 e jm e j1 e jm v 11 v m1.. v 1j1 v 1jm.. v mj1 v mjm = = j 1 < <j m j 1 < <j m u m1 u mn u 1j1 u 1jm.. u mj1 u mjm u 1j1 u 1jm.. u mj1 u mjm v 1n v mn v 1j1 v 1jm.. v mj1 v mjm v 1j1 v mj1.. v 1jm v mjm

1.4. 21 m = n 1.4.7 V, V u 1,..., u p p u 1 u p u u 1,..., u p u v w v 1 = 0, w 1 = u 1 v 1, w 1 v w =1 u = v + w, v span{u 1,..., u 1 }, w span{u 1,..., u 1 } v w w 2 u 2 u 1 u = w 1 w (1 p) u 1 u p = w 1 w p < j u, w j = 0 1.4.4 u 1 u p 2 = w 1 w p 2 = w 1 w p, w 1 w p = det( w, w j ) w 1, w 1 0 0. = det 0............. 0 0 0 w p, w p p p p = w, w u, u = u 2. =1 u 1 u p =1 p u w, w = u, u v = 0 u 1,..., u p =1 =1

22 1 1.4.8 V W m n (m n) F : V W JF = sup{ F (u 1 ) F (u n ) u 1,..., u n V } F JF = 0 F (kerf ) v 1,..., v n JF = F (n,0) (v 1 v n ) v 1 v n = F (v 1) F (v n ) v 1 v n F dm(mf ) < n V u 1,..., u n F (u 1 ),..., F (u n ) F (u 1 ) F (u n ) = 0 JF = 0 F (kerf ) v 1,..., v n (kerf ) u 1,..., u n (a j ) v j = a j u 1.2.4 =1 v 1 v n = det(a j )u 1 u n, F (n,0) (v 1 v n ) = det(a j )F (n,0) (u 1 u n ). F (n,0) (v 1 v n ) v 1 v n = det(a j) F (n,0) (u 1 u n ) det(a j ) u 1 u n = F (n,0) (u 1 u n ) u 1 u n = F (n,0) (u 1 u n ). 1.4.4 u 1 u n = 1 JF F (n,0) (u 1 u n ) = F (n,0) (v 1 v n ). v 1 v n V w 1..., w n w = w 1 + w 2, w 1 (kerf ), w 2 kerf

1.4. 23 w 1 w = 1 F (n,0) (w 1 w n ) = F (w 1 ) F (w n ) = F (w 1 1) F (w 1 n) = F (n,0) (w 1 1 w 1 n). w1, 1..., wn 1 1.2.4 F (n,0) (w 1 1 w 1 n) = 0 (kerf ) F (n,0) (w1 1 wn) 1 = F (n,0) (u w1 1 wn 1 1 u n ) 1.4.7 F (n,0) (w 1 w n ) = F (n,0) (w 1 1 w 1 n) = w 1 1 w 1 n F (n,0) (u 1 u n ) w 1 1 w 1 n F (n,0) (u 1 u n ) F (n,0) (u 1 u n ). JF F (n,0) (u 1 u n ) = F (n,0) (v 1 v n ). v 1 v n JF F (n,0) (v 1 v n ) v 1 v n JF = F (n,0) (v 1 v n ) v 1 v n JF.

24 2 2.1 2.1.1 π E : E M M (1) E, M π E : E M C (2) k M p p U Φ U : π 1 E (U) U Rk u π E (U) Φ U (u) U π E (u) Φ U (u) = (π E (u), φ U (u)) (u π 1 E (U)) x U π 1 E (x) φ U π 1 E (x) : π 1 E (x) Rk E M π E π 1 E (x) x k ranke 2.1.2 π : E M π : E M M φ : E E π = π φ x M φ Ex : E x E x φ E E V M V M M V M M E M V E M T M M

2.1. 25 2.1.3 M x M M T x M T M = T x M x M u T M u T x M x M π(u) = x π : T M M M p p (U; x 1,..., x n ) U x x 1,..., x x n T x M π 1 (U) u u = ξ x π(u) x Φ U (u) = (π(u), ξ 1,..., ξ n ) (u π 1 (U)) Φ U : π 1 (U) U R n π 1 (U) (x 1,..., x n, ξ 1,..., ξ n ) (V ; y 1,..., y n ) v π 1 (V ) v = η y π(v) π 1 (V ) (y 1,..., y n, η 1,..., η n ) (x 1,..., x n, ξ 1,..., ξ n ) η = ξ j y x j. ) (y 1,..., y n, ξ j y1 yn,..., ξj xj x j C T M π π(x 1,..., x n, ξ 1,..., ξ n ) = (x 1,..., x n )

26 2 π : T M M C Φ U Φ U (u) U π(u) ( ) φ U ξ = (ξ 1,..., ξ n ) x x U φ U π 1 (x) : π 1 (x) R n π : T M M M 2.1.4 π E : E M M C σ : M E π E σ = 1 M E E Γ(M, E) Γ(E) 2.1.5 M p M T p M, p M C X, Y X, Y M C, M Remann (M,, ) Remann Remann Remann Eucld 2.1.6 ι : M M M Remann ( M, g) M x ι dι x : T x M T ι(x) M M Remann g dι g = ι g M Remann (M, g) ( M, g) Remann 2.1.7 2.1.6 M Remann M Remann M Remann Remann (M, g) ( M, g) C ι M x dι x : T x M T ι(x) M ι (M, g) ( M, g) Remann 2.1.8 ι : M ( M, g) Remann ( M, g) Remann x M T x M = {u T ι(x) M u, dιx (T x M) = 0} T M = x M T x M T M u T M u T x M x M π(u) = x π : T M M π : T M M π : T M M Remann M T M M

2.2. 27 2.1.9 E M, E E s, t s, t (x) = s(x), t(x) (x M) M s, t C, E (E,, ) 2.1.10 2.1.5 Remann Remann Remann Remann 2.2 2.2.1 M x M T x M (p, q) T (p,q) (T x M) T x (p,q) M T (p,q) M = T x (p,q) M x M T (p,q) M M ( 2.2.2) T (p,q) M C (p, q) 2.2.2 T (p,q) M M u T (p,q) M u T x (p,q) M x M π(u) = x π : T (p,q) M M M x x (U; x 1,..., x n ) π 1 (U) u u = u 1 p j 1 j q x 1 π(u) x dx j 1 p π(u) dxj q π(u) π(u) Φ U (u) = (π(u), u 1 p j 1 j q ) (u π 1 (U)) Φ U : π 1 (U) U R np+q

28 2 π 1 (U) (x 1,..., x n, u 1 p j 1 j q ) (V ; y 1,..., y n ) v π 1 (V ) v = v 1 p j 1 j q y 1 π(v) y p dy j 1 π(u) dyj q π(v) π 1 (V ) (y 1,..., y n, v 1 p j 1 j q ) π(v) v k 1 k p l 1 l q = u 1 p y k 1 j 1 j q x ykp x j 1 1 x p y xjq l 1 y. l q (x 1,..., x n, u 1 p j 1 j q ) ( ) y 1,..., y n, u 1 p y k 1 j 1 j q x ykp x j1 1 x p y xjq l 1 y lq C T (p,q) M π π(x 1,..., x n, u 1 p j 1 j q ) = (x 1,..., x n ) π : T (p,q) M M C Φ U Φ U (u) U π(u) ( φ U u 1 p j 1 j q x ) 1 x dxj 1 dx j q = (u 1 p p j 1 j q ) x U φ U π 1 (x) : π 1 (x) R np+q π : T (p,q) M M 2.2.3 Remann (0, 2) 2.3 2.3.1 M n M x p Tx (M) ω x ω ω M p ( )M (U; x 1,..., x n ) ( ) x ω x x 1,, x x p x 1,..., p U C 2.3.2 M 0 M C

2.3. 29 2.3.3 f M C f df M 1 n = dm M M (U; x 1,..., x n ) f C ( ) x df x x = f x x (x) U C df M 1 2.3.4 ω V n M p M (U; x 1,..., x n ) x U x x (1 n) T x (M) (dx ) x (1 n) 1.2.5 x U ω x p Tx (M) ω x = ( ) x j 1,, x x j p (dx j 1 ) x (dx j p ) x x j 1 < <j p ω x ω x (dx j 1 ) x (dx jp ) x C 2.3.1 () ω (dx j 1 ) x (dx j p ) x 2.3.5 M p Ω p (M) f, g C (M), ω, η Ω p (M) x M (fω + gη) x = f(x)ω x + g(x)η x ( p (T x (M)) ) Ω p (M) C (M) C (M) φ Ω p (M), ψ Ω q (M) x M φ ψ x = φ x ψ x ( p (Tx (M)) q (Tx (M)) ) φ ψ Ω p+q (M) : Ω p (M) Ω q (M) Ω p+q (M) C (M) Ω p (M) C (M) f, g C (M), ω, η Ω p (M) fω + gη Ω p (M) p (Tx (M))

30 2 (U; x 1,..., x n ) M x U ω x = a 1 p (x)(dx 1 ) x (dx p ) x, 1 < < p η x = b 1 p (x)(dx 1 ) x (dx p ) x 1 < < p 2.3.4 a 1 p, b 1 p U C (fω + gη) x = (f(x)a 1 p (x) + g(x)b 1 p (x))(dx 1 ) x (dx p ) x. 1 < < p fa 1 p + gb 1 p U C fω + gη φ Ω p (M), ψ Ω q (M) φ ψ Ω p+q (M) C (M) x U φ x = c 1 p (x)(dx 1 ) x (dx p ) x, 1 < < p ψ x = d j1 j q (x)(dx j1 ) x (dx jq ) x j 1 < <j q 2.3.4 c 1 p d j1 j q U C x U = = (φ ψ) x 1 < < p 1 < < p j 1 < <j q (c 1 p (x)(dx 1 ) x (dx p ) x ) (d j1 j q (x)(dx j 1 ) x (dx j q ) x ) j 1 < <j q c 1 p (x)d j1 j q (x) (dx 1 ) x (dx p ) x (dx j 1 ) x (dx jq ) x (φ ψ) x (dx k 1 ) x (dx k p+q ) x c 1 p (x)d j1 j q (x) U C φ ψ M p + q 2.3.6 F M N C x M F df x : T x (M) T F (x) (N) 1.1.14 df x (0,p) ω Ω p (N) (F ω) x = (df x ) (0,p) ω F (x) (F ω) x p (T x (M)) F ω Ω p (M)

2.3. 31 m = dm M, n = dm N F ω Ω p (M) F (U) U M N (U; x 1,..., x m ) (U ; y 1,..., y n ) ( ) x (F ω) x x 1,, x x p x 1,..., p U C x U ( ) df x (y j F ) x = (x) x x y j j=1 F (x) ( ) (F ω) x x 1,, x x ( ( ) p x ( )) = ω F (x) df x x 1,, df x x x p x ( (y j 1 F ) = (x) (yj p F ) (x) ω x 1 x p F (x) y j 1,, F (x) y j p j 1,,j p =1 F (x) U C F ω Ω p (M) 2.3.7 2.3.6 F ω ω F 2.3.8 V F : M M, G : M M C ω Ω p (M ) (G F ) ω = F (G ω) ) d(g F ) x = dg F (x) df x v T x (M) ((G F ) ω) x (v) = ω G F (x) (d(g F ) x (v)) = ω G(F (x)) (dg F (x) df x (v)) = (G ω) F (x) (df x (v)) = (F (G ω)) x (v) (G F ) ω = F (G ω) 2.3.9 F M N C φ Ω p (N), ψ Ω q (N) F (φ ψ) = (F φ) (F ψ)

32 2 x M F (φ ψ) x = (df x ) ((φ ψ) F (x) ) = (df x ) (φ F (x) ψ F (x) ) = (df x ) φ F (x) (df x ) ψ F (x) = (F φ) x (F ψ) x = ((F φ) (F ψ)) x F (φ ψ) = (F φ) (F ψ) 2.4 2.4.1 M n ω Ω p (M) dω Ω p+1 (M) () M (U; x 1,..., x n ) ω ω x = a 1 p (x)(dx 1 ) x (dx p ) x 1 < < p (dω) x = 1 < < p =1 a 1 p (x)(dx ) x x (dx 1 ) x (dx p ) x 1 < < p =1 a 1 p (x)(dx ) x x (dx 1 ) x (dx p ) x p+1 (Tx (M)) a 1 p (x) ω x ( ) a 1 p (x) = ω x x 1,, x x p (1 1 < < p n) x 1,..., p a 1 p (x) (V ; y 1,..., y n ) M 1 ( ) b 1 p (x) = ω x y 1,, x y p x

2.4. 33 ω x = 1 < < p b 1 p (x)(dy 1 ) x (dy p ) x x U V T x (M) 2 x x y x x j y = x y (x) x j j=1 (dy y ) x = x j (x)(dxj ) x j=1 x b 1 p (x) = j 1,,j p=1 x j 1 xjp (x) y 1 y (x)a p j 1 j p (x) ( ) = 1 p! = 1 p! = 1 p! 1 < < p =1, 1,..., p=1, 1,..., p =1 1,..., p =1 =1 x j y (x) y x (x) = δ k jk b 1 p (x)(dy ) y x (dy 1 ) x (dy p ) x b 1 p (x)(dy ) y x (dy 1 ) x (dy p ) x j,j 1,...,jp=1 k,k 1,...,kp=1 x j y (x) { x j 1 } xjp (x) x j y 1 y (x)a j p 1 j p (x) y x k (x)(dxk ) x y1 x k 1 (x)(dxk 1 ) x yp x kp (x)(dxk p ) x j 1,...,jp=1 k,k 1,...,kp=1 { x j 1 xjp (x) x k y 1 y (x)a j p 1 j p (x) } (dx k ) x y 1 x k 1 (x)(dxk 1 ) x yp x kp (x)(dxkp ) x. r =1 x jr y r (x) yr x k r (x) = δ j rk r

34 2 r =1 ( x j r ) y x k y (x) r r x (x) = x j r k r y (x) r x k r =1 x j r = (x) r y r=1 2 y r ( y r ) x (x) k r (x) x k kr x k k r (dx k ) x (dx k 1 ) x (dx kp ) x k k r ( ) = 1 p! = 1 p! = 1,..., p =1 k,k 1,...,k p =1 k 1 < <k p k=1 j 1,...,jp=1 k,k 1,...,kp=1 x j 1 xjp (x) y 1 y (x) a j 1 j p (x) p x k (dx k ) x y 1 x k 1 (x)(dxk 1 ) x yp x k p (x)(dxkp ) x a k1 k p (x)(dx k ) x k x (dx k 1 ) x (dx k p ) x a k1 k p x k (x)(dx k ) x (dx k 1 ) x (dx kp ) x. dω dω dω p + 1 dω 2.4.2 M 2.4.1 d : Ω p (M) Ω p+1 (M) 2.4.3 F M N C F F : Ω p (N) Ω p (M) m = dm M, n = dm N ω Ω p (N) F (U) U M N (U; x 1,..., x m ) (U ; y 1,..., y n ) ( ) a j1 j p (y) = ω y y j 1,, y y jp ω U ω y = a j1 j p (y)(dy j1 ) y (dy jp ) y j 1 < <j p y

2.4. 35 2.3.6 x U ( ) (F ω) x x 1,, x x p x (y j 1 F ) = (x) (yj p F ) (x) a x j1 j 1 x p p (F (x)). j 1,,j p=1 = (d(f ω)) x m j 1,,j p =1 1 < < p =1 { (y j 1 F ) x x 1 (dx ) x (dx 1 ) x (dx p ) x. (x) (yj p } F ) (x) a x p j1 j p (F (x)) 2 (y jr F ) (x) x x r r (dx ) x (dx 1 ) x (dx p ) x r = (d(f ω)) x j 1,,j p =1 1 < < p m =1 (y j 1 F ) x 1 (x) (yj p F ) (x) a j 1 j p (F (x)) x p x = 1 p! (dx ) x (dx 1 ) x (dx p ) x m m (y j 1 F ) x 1 j 1,,j p =1 1,, p =1 =1 (dx ) x (dx 1 ) x (dx p ) x. (x) (yjp F ) (x) a j 1 j p (F (x)) x p x (dω) y = = 1 p! j 1 < <j p j=1 j 1,,j p =1 j=1 a j1 j p (y)(dy j ) y j y (dy j 1 ) y (dy j p ) y a j1 j p (y)(dy j ) y j y (dy j 1 ) y (dy jp ) y (F (dω)) x = (df x ) (dω) F (x) = 1 p! j 1,,j p =1 j=1 a j1 j p (F (x))(df y j x ) (dy j ) F (x) (df x ) (dy j 1 ) F (x) (df x ) (dy jp ) F (x).

36 2 (df x ) (dy j ) F (x) = (dy j ) F (x) df x = d(y j F ) x = m =1 (y j F ) (x)(dx ) x x = = j=1 j=1 m =1 a j1 j p (F (x))(df y j x ) (dy j ) F (x) m =1 a j1 j p (F (x)) (yj F ) y j x a j1 j p (F (x)) (x)(dx ) x x (x)(dx ) x = 1 p! (F (dω)) x m m j 1,,j p =1 1,, p =1 =1 (y j 1 F ) x 1 (dx ) x (dx 1 ) x (dx p ) x = (d(f ω)) x. F (dω) = d(f ω) (x) (yj p F ) (x) a j 1 j p (F (x)) (x) x p x 2.4.4 2.4.3 M N F : M N N M M M M N 2.4.5 M d : Ω p (M) Ω p+1 (M) 2.4.1 d 2.4.6 M φ Ω p (M), ψ Ω q (M) d(φ ψ) = dφ ψ + ( 1) p φ dψ

2.4. 37 n = dm M (U; x 1,..., x n ) M U φ ψ φ x = a 1 p (x)(dx 1 ) x (dx p ) x 1 < < p ψ x = b j1 j q (x)(dx j1 ) x (dx jq ) x j 1 < <j q (φ ψ) x = 1 < < p j 1 < <j q a 1 p (x)b j1 j q (x) (dx 1 ) x (dx p ) x (dx j 1 ) x (dx j q ) x k 1 < < k p+q { 1,..., p, j 1,..., j q } = {k 1,..., k p+q } ( ) 1 p j 1 j q sgn = 0 k 1 k p k p+1 k p+q = (φ ψ) x sgn k 1 < <k p+q 1 < < p j 1 < <j q ( 1 p j 1 j q k 1 k p k p+1 k p+q ) a 1 p (x)b j1 j q (x)(dx k 1 ) x (dx k p+q ) x = = = d(φ ψ) x sgn k 1 < <k p+q 1 < < p j 1 < <j q =1 ( 1 p j 1 j q k 1 k p k p+1 k p+q (a 1 p (x), b j1 j q (x)) (dx ) x x (dx k 1 ) x (dx k p+q ) x ( 1 p j 1 j q sgn k 1 < < p j 1 < <j 1 k p k p+1 k p+q q { a1 p (x)b x j1 j q (x) + a 1 p (x) b } j 1 j q (x) x k 1 < <k p+q =1 (dx ) x (dx k 1 ) x (dx k p+q ) x { a1 p (x)b x j1 j q (x) + a 1 p (x) b } j 1 j q (x) x 1 < < p j 1 < <j q =1 ) ) (dx ) x (dx 1 ) x (dx p ) x (dx j 1 ) x (dx j q ) x = (dφ ψ) x + ( 1) p (φ dψ) x.

38 2 d(φ ψ) = dφ ψ + ( 1) p φ dψ 2.4.7 M d : Ω p (M) Ω p+1 (M) d d = 0 n = dm M ω Ω 0 (M) d 2 ω = 0 (U; x 1,..., x n ) M x U (dω) x = j=1 ω x j (x)(dxj ) x (d 2 ω) x = =1 j=1 2 ω x x j (x)(dx ) x (dx j ) x. 2 ω (x) j (dx ) x x j x (dx j ) x j (d 2 ω) x = 0 d 2 ω = 0 p > 0 ω Ω p (M; V ) U ω x = a 1 p (x)(dx 1 ) x (dx p ) x 1 < < p (dω) x = = 1 < < p =1 a 1 p (x)(dx ) x x (dx 1 ) x (dx p ) x 1 < < p (da 1 p ) x (dx 1 ) x (dx p ) x 2.4.1 (d 2 ω) x = (d 2 a 1 p ) x (dx 1 ) x (dx p ) x 1 < < p + (da 1 p ) x 1 < < p = 0. p ( 1) j (dx 1 ) x (d 2 x j ) x (dx p ) x j=1 (d 2 ω) x = 0 d 2 ω = 0

2.4. 39 2.4.8 n M Z p (M) = {ω Ω p (M) dω = 0} B p (M) = {dη η Ω p 1 (M)} d d 2 = 0 B p (M) Z p (M) H p (M) = Z p (M)/B p (M) H p (M) M p de Rham χ(m) = ( 1) p dm H p (M) p=0 χ(m) M Euler M C p C p (M) Z p (M) = {c C p (M) c = 0} B p (M) = { c c C p+1 (M)} 2 = 0 B p (M) Z p (M) H p (M) = Z p (M)/B p (M) H p (M) M C p 2.4.9 (de Rham) ω Z p (M) ω H p (M) [ω] Z p (M) R ; c ω H p (M) H p (M) H p (M) 2.4.10 M ω Z p (M) ω H p (M) [ω] [ω] H p (M) [η] H q (M) [ω η] ω, η [ω] [η] c ω Ω p 1 (M) η Ω q 1 (M) (ω + dω ) (η + dη ) = ω η + ω dη + dω η + dω dη d(ω η ) = dω η + ( 1) p ω dη = ( 1) p ω dη, d(ω η) = dω η + ( 1) p 1 ω dη = dω η, d(ω dη ) = dω dη + ( 1) p 1 ω d 2 η = dω dη B p+q (M) [(ω + dω ) (η + dη )] = [ω η] [ω η] ω, η [ω] [η]

40 2 2.4.11 n M H (M) = H p (M) 2.4.10 H (M) M de Rham p=0 2.4.12 M Z 0 (M) 0 M R H 0 (M) = Z 0 (M) M R R 1 de Rham Z 1 (R) = Ω 1 (R) C f(x) f(x)dx f(t)dt df = f(x)dx Z 1 (R) = B 1 (R) H 1 (R) = {0} H (R) = R R R 1 S 1 de Rham F (x) = x 0 S 1 = {(x, y) R 2 x 2 + y 2 = 1} (cos θ, sn θ) θ Z 1 (S 1 ) = Ω 1 (S 1 ) 2π C f(θ) f(θ)dθ I : Z 1 (S 1 ) R ; f(θ)dθ 2π 0 f(θ)dθ I I Ω 0 (S 1 ) 2π C g(θ) dg = g (θ)dθ I(dg) = 2π 0 g (θ)dθ = g(2π) g(0) = 0. B 1 (S 1 ) ker I f(θ)dθ ker I F (θ) = θ f(t)dt 0 F (θ) θ C 2π f(t)dt = 0 0 F (θ) 2π C B 1 (S 1 ) df = F (θ)dθ = f(θ)dθ ker I B 1 (S 1 ) B 1 (S 1 ) = ker I H 1 (S 1 ) = Z 1 (S 1 )/B 1 (S 1 ) = Z 1 (S 1 )/ ker I = mi = R. 2.4.13 M ω Ω p (M) p = 0 X X(M) dω(x) = Xω. p = 1 X, Y X(M) dω(x, Y ) = X(ω(Y )) Y (ω(x)) ω([x, Y ]).

2.4. 41 p = 0 X X(M) dω(x) = Xω C dω p = 1 ω ω = a dx ω dω dω =,j a x j dxj dx X, Y X = ξ x, Y = j η j x j dω(x, Y ) =,j a x j (ξj η η j ξ ) ω(y ) = a η X(ω(Y )) =,j ξ j x (a η ) = j,j ( ) ξ j a η x j η + a. x j X Y Y (ω(x)) =,j ( ) η j a ξ x j ξ + a x j [X, Y ] =,j =,j ξ ηj x x j,j (ξ j η ξ ηj xj x j η j ξ x j ) x x

42 2 ω([x, Y ]) =,j ( a ξ j η ξ ηj xj x j ). dω(x, Y ) = X(ω(Y )) Y (ω(x)) ω([x, Y ]) 2.4.14 2.4.13 p > 0 X 1,... X p+1 X(M) = dω(x 1,..., X p+1 ) p+1 ( 1) 1 X (ω(x 1,..., ˆX,..., X p+1 )) =1 + ( 1) +j ω([x, X j ], X 1,..., ˆX,..., ˆX j,..., X p+1 ). <j

43 3 Remann 3.1 2 R 2 D R 3 dp : R 2 R 3 D 2 p R 2 D u, v p u = p u, p v = p v p u, p v dp p p u, p v 2 u p u v p v R 3 p u, p v 2 dp I = dp, dp I (0, 2) X, Y I(X, Y ) = dp(x), dp(y ) dp I Remann dp = p u du + p v dv I(X, Y ) = p u du(x) + p v dv(x), p u du(y ) + p v dv(y ) = p u, p u du(x)du(y ) + p u, p v du(x)dv(y ) + p v, p u dv(x)du(y ) + p v, p v dv(x)dv(y ) = p u, p u du(x)du(y ) + p u, p v (du(x)dv(y ) + dv(x)du(y )) + p v, p v dv(x)dv(y ) = p u, p u du du(x, Y ) + p u, p v (du dv + dv du)(x, Y ) + p v, p v dv dv(x, Y ).

44 3 Remann (0, 1) φ, ψ X, Y (φ ψ)(x, Y ) = 1 (φ(x)ψ(y ) + ψ(x)φ(y )) 2 (0, 2) φ ψ I = p u, p u du du + 2 p u, p v du dv + p v, p v dv dv I E = p u, p u, F = p u, p v, G = p v, p v Remann I = Edu du + 2F du dv + Gdv dv. p u, p v p u p v e = p u p v p u p v e II = dp, de II (0, 2) X, Y II(X, Y ) = dp(x), de(y ) de = e u du + e v dv II(X, Y ) = p u du(x) + p v dv(x), e u du(y ) + e v dv(y ) = p u, e u du(x)du(y ) p u, e v du(x)dv(y ) p v, e u dv(x)du(y ) p v, e v dv(x)dv(y ). e p u, p v p u, e = 0, p v, e = 0 u v p uu, e + p u, e u = 0, p uv, e + p u, e v = 0, p vu, e + p v, e u = 0, p vv, e + p v, e v = 0.

3.2. 45 p uv = p vu p u, e v = p v, e u II = p u, e u du du 2 p u, e v du dv p v, e v dv dv = p uu, e du du + 2 p uv, e du dv + p vv, e dv dv II L = p uu, e, M = p uv, e, N = p vv, e II = Ldu du + 2Mdu dv + Ndv dv. p 0 = p(u 0, v 0 ) e 0 e 0 f f(u, v) = p(u, v), e 0. p 0 e 0 df = p u du + p v dv, e 0 = p u, e 0 du + p v, e 0 dv = 0 f p 0 f 2 f u u (p 0) = p uu, e 0 = L, 2 f v v (p 0) = p vv, e 0 = N 2 f u v (p 0) = 2 f v u (p 0) = p uv, e 0 = M, f p 0 Hesse II II LN M 2 > 0 II LN M 2 < 0 II 3.2 3.2.1 M E M : C (T M) C (E) C (E); (X, φ) X φ (1) (4) E

46 3 Remann (1) X+Y φ = X φ + Y φ, (X, Y C (T M), φ C (E)) (2) X (φ + ψ) = X φ + X ψ, (X C (T M), φ, ψ C (E)) (3) fx φ = f X φ, (X C (T M), φ C (E), f C (M)) (4) X (fφ) = f X φ + (Xf)φ. (X C (T M), φ C (E), f C (M)) X C (T M) X φ = 0 φ C (E) 3.2.2, M E, E E X φ, ψ = X φ, ψ + φ, X ψ (X C (T M), φ, ψ C (E)), 3.2.3 M E X, Y C (T M) φ C (E) R (X, Y )φ = X Y φ Y X φ [X,Y ] φ E R (X, Y )φ R A 2 (T M, End(E)) C A 2 (U, V ) U U V X, Y C (T M) φ C (E) M C f R (fx, Y )φ R (X, Y )φ = R (Y, X)φ = fx Y φ Y fx φ [fx,y ] φ = f X Y φ Y (f X φ) f[x,y ] φ + (Y f)x φ = f X Y φ f Y X φ (Y f) X φ f [X,Y ] φ + (Y f) X φ = fr (X, Y )φ.

3.2. 47 R (fx, Y )φ = fr (X, Y )φx, Y R (X, fy )φ = fr (X, Y )φ R (X, Y )fφ = X Y (fφ) Y X (fφ) [X,Y ] fφ = X (f Y φ + (Y f)φ) Y (f X φ + (Xf)φ) f [X,Y ] φ ([X, Y ]f)φ = f X Y φ + (Xf) Y φ + (Y f) X φ + (XY f)φ f Y X φ (Y f) X φ (Xf) Y φ (Y Xf)φ f [X,Y ] φ ([X, Y ]f)φ = fr (X, Y )φ. R (X, Y )fφ = fr (X, Y )φ R (X, Y )φ X, Y, φ R A 2 (T M, End(E)) C 3.2.4 3.2.3 R R 3.2.5 M E R R (X, Y ) + R (Y, X) = 0 (X, Y C (T M)) E,, R (X, Y )φ, ψ + φ, R (X, Y )ψ = 0 (X, Y C (T M), φ, ψ C (E)) R A 2 (T M, End(E)) [X, Y ] = XY Y X 3.2.2 0 = XY φ, ψ Y X φ, ψ [X, Y ] φ, ψ = X Y φ, ψ + X φ, Y ψ Y X φ, ψ Y φ, X ψ [X,Y ] φ, ψ φ, [X,Y ] ψ = X Y φ, ψ + Y φ, X ψ + X φ, Y ψ + φ, X Y ψ Y X φ, ψ X φ, Y ψ Y φ, X ψ φ, Y X ψ [X,Y ] φ, ψ φ, [X,Y ] ψ = R (X, Y )φ, ψ + φ, R (X, Y )ψ.

48 3 Remann 3.3 Remann 3.3.1 M p M T p M, p M C X, Y X, Y M C, M Remann (M,, ) Remann Remann Remann Eucld 1.4 3.3.2 Remann (M,, ) X Y Y X = [X, Y ] (X, Y C (T M)) Remann, Remann M X, Y, Z X Y, Z + Y, X Z = X Y, Z Y Z, X + Z, Y X = Y Z, X Z X, Y X, Z Y = Z X, Y. X Y Y X = [X, Y ] X Y, Z + [X, Z], Y + [Y, Z], X + Z, [Y, X] + Z, X Y = X Y, Z + Y Z, X Z X, Y. X Y, Z = 1 (X Y, Z + Y Z, X Z X, Y 2 + [X, Y ], Z [Y, Z], X + [Z, X], Y ) X+Y Z = X Z + Y Z (X, Y, Z C (T M)) X (Y + Z) = X Y + X Z (X, Y, Z C (T M)) M C f fx Y, Z = 1 (fx Y, Z + Y Z, fx Z fx, Y 2 + [fx, Y ], Z [Y, Z], fx + [Z, fx], Y )

3.3. Remann 49 = 1 (fx Y, Z + (Y f) Z, X + fy Z, X 2 (Zf) X, Y fz X, Y +f [X, Y ], Z (Y f) X, Z f [Y, Z], X +f [Z, X], Y + (Zf) X, Y ) = 1 f(x Y, Z + Y Z, X Z X, Y 2 + [X, Y ], Z [Y, Z], X + [Z, X], Y ) = f X Y, Z. fx Y, Z = f X Y, Z fx Y = f X Y X (fy ), Z = 1 (X fy, Z + fy Z, X Z X, fy 2 + [X, fy ], Z [fy, Z], X + [Z, X], fy ) = 1 ((Xf) Y, Z + fx Y, Z + fy Z, X 2 (Zf) X, Y fz X, Y +f [X, Y ], Z + (Xf) Y, Z f [Y, Z], X + (Zf) Y, X + [Z, X], fy ) = f X Y, Z + (Xf) Y, Z = f X Y + (Xf)Y, Z. X (fy ), Z = f X Y + (Xf)Y, Z X (fy ) = f X Y + (Xf)Y T M X Y, Z + Y, X Z = 1 (X Y, Z + Y Z, X Z X, Y 2 + [X, Y ], Z [Y, Z], X + [Z, X], Y ) + 1 (X Z, Y + Z Y, X Y X, Z 2 + [X, Z], Y [Z, Y ], X + [Y, X], Z ) = X Y, Z.

50 3 Remann Remann, X Y, Z Y X, Z = 1 (X Y, Z + Y Z, X Z X, Y 2 + [X, Y ], Z [Y, Z], X + [Z, X], Y ) 1 (Y X, Z + X Z, Y Z Y, X 2 + [Y, X], Z [X, Z], Y + [Z, Y ], X ) = [X, Y ], Z X Y Y X = [X, Y ] 3.3.3 3.3.2 Remann Lev-Cvta X Y Y X 3.3.4 Remann (M,, ) Lev-Cvta X, Y, Z X Y, Z = 1 (X Y, Z + Y Z, X Z X, Y 2 + [X, Y ], Z [Y, Z], X + [Z, X], Y ) (U; x 1,..., x n ) x 3.3.5 Remann (M, g) (U; x 1,..., x n ) j = Γ k j k U C Γ k j X = X, Y = Y j j X Y X Y = (XY k + Γ k jx Y j ) k Remann g = g j dx dx j (g j ) g j Γ k j = 1 2 gkl ( g jl + j g l l g j ) (V ; y 1,..., y n ) y p p p q = Γ r pq r U V Γ r pq = yr x x j x k y p y q Γk j + 2 x k y r y p y q x k

3.3. Remann 51 X Y = X (Y j j ) = (XY j ) j + Y j X j = (XY j ) j + Y j X j = (XY j ) j + X Y j j = (XY k ) k + X Y j Γ k j k = (XY k + Γ k jx Y j ) k. 3.3.4 g( j, l ) = 1 2 ( g( j, l ) + j g( l, ) l g(, j )) = 1 2 ( g jl + j g l l g j ). g( j, l ) = g(γ k j k, l ) = Γ k jg( k, l ) = Γ k jg kl. Γ k jg kl = 1 2 ( g jl + j g l l g j ). Γ m j = Γ k jg kl g lm = 1 2 glm ( g jl + j g l l g j ) Γ k j = 1 2 gkl ( g jl + j g l l g j ) p p = y = x p y p x = x y p Γ r pq r = p q ) x = ( k p y + x x j q Γk j y p y q k ( ) 2 x k = y p y + x x j y r q Γk j y p y q x k r Γ r pq = yr x x j x k y p y q Γk j + 2 x k y r y p y q x k 3.3.6 3.3.5 Γ k j Chrstoffel

52 3 Remann 3.3.7 Remann Chrstoffel 3.3.8 Remann (M, ) c X ( ) c (t)x = c (t)x k + Γ k dx (c(t)) j X j k dt c c (t)x (U; x 1,..., x n ) (V ; y 1,..., y n ) 3.3.5 X = X = X p p c (t)x k = d dt Xk (c(t)) = Xk dx (c(t)) x dt 3.3.5 Chrstoffel ( ) c (t)x k + Γ k dx (c(t)) j X j k dt ( ) X k dx (c(t)) = + Γ k dx (c(t)) x j X j k dt dt ( ( ) ) y s x k = x y s y X x q dy p (c(t)) + Γ k x dy p (c(t)) x j q y p j dt y p dt y X y q r q x k r ( y s 2 x k q x dy = X p (c(t)) + ys x k X q x dy p (c(t)) x y s yq y p dt x y q y s y p dt ) +Γ k x dy p (c(t)) x j j y p dt y X y q r q x k r ( 2 x k = y p y X q dyp (c(t)) + xk X ) q dy p (c(t)) + Γ k x dy p (c(t)) x j q dt y q y p j dt y p dt y X y q r q x k r ( ( ) ) Xr dy p (c(t)) 2 x k = + y p dt y p y + x x j y r dy p (c(t)) q Γk j X q y p y q x k r dt ( ) = c (t) X r + Γ r dy p (c(t)) pq X q r dt c (t)x 3.3.9 3.3.8 c (t)x X c (t)x = 0 X X dx k (c(t)) dt + Γ k j dx (c(t)) X j = 0 (1 k n) dt

3.4. 53 X k c c [a, b] u T c(a) M X(a) = u X u X(b) T c(b) M T c(a) M T c(b) M τ c 3.3.10 Remann c X, Y d dt X, Y = c (t) X, Y = c (t)x, Y + X, c (t)y = 0. 3.3.11 Remann X, Y x c(0) = x c (0) = X x c c c(t) c(0) τ t 0 1 ( X Y )(x) = lm t 0 t (τ 0Y t c(t) Y x ) T x M e 1,..., e n c e (t) Y (c(t)) = Y (t)e (t) ( X Y )(x) = c (0)(Y (t)e (t)) = dy dt (0)e. 1 lm t 0 t (τ 0Y t 1 c(t) Y x ) = lm t 0 t (Y (t) Y (0))e = dy dt (0)e. 1 ( X Y )(x) = lm t 0 t (τ 0Y t c(t) Y x ) 3.4 3.4.1 Remann X T X T

54 3 Remann (1) X S, T X (S T ) = X S T + S X T. (2) C f X f = Xf Y X Y Lev-Cvta X (0, 1) 1 ω (1, 0) Y ω Y ( C (1,1) (ω Y ) = C (1,1) ω Y j dx ) = ω x j Y = ω(y ) ( X ω)(y ) = C (1,1) ( X ω Y ) = C (1,1) ( X (ω Y ) ω X Y ) = X C (1,1) (ω Y ) ω( X Y ) = X (ω(y )) ω( X Y ) = X(ω(Y )) ω( X Y ). ( X ω)(y ) = X(ω(Y )) ω( X Y ) X ω (0, 0) (1, 0) (0, 1) X (p, q) T 1 ω 1,..., ω p X 1,..., X q C C(T ω 1 ω p X 1 X q ) = T (ω 1,..., ω p, X 1,..., X q ) ( X T )(ω 1,..., ω p, X 1,..., X q ) = C( X T ω 1 ω p X 1 X q ) = C ( X (T ω 1 ω p X 1 X q ) p T ω 1 X ω ω p X 1 X q =1 ) q T ω 1 ω p X 1 X X j X q j=1

3.4. 55 = X(T (ω 1,..., ω p, X 1,..., X q )) p T (ω 1,..., X ω,..., ω p, X 1,..., X q ) =1 q T (ω 1,..., ω p, X 1,..., X X j,..., X q ). j=1 X T X T X T C f ( X ω)(fy ) = X(ω(fY )) ω( X (fy )) = X(fω(Y )) ω((xf)y + f X Y ) = (Xf)ω(Y ) + fx(ω(y )) (Xf)ω(Y ) fω( X Y ) = f( X ω)(y ) X ω (0, 1) : C (T M) C (T (0,1) M) C (T (0,1) M); (X, ω) X ω T (0,1) M ( fx ω)(y ) = fx(ω(y )) ω( fx Y ) = fx(ω(y )) fω( X Y ) = f( X ω)(y ) fx ω = f X ω. ( X (fω))(y ) = X(fω(Y )) fω( X Y ) = (Xf)ω(Y ) + fx(ω(y )) fω( X Y ) = (Xf)ω(Y ) + f( X ω)(y ) = ((Xf)ω + f( X ω))(y ) X (fω) = (Xf)ω + f( X ω).

56 3 Remann T (0,1) M (p, q) T X T (p, q) X (fω k ) = (Xf)ω k + f X ω k ( X T )(ω 1,..., fω k,..., ω p, X 1,..., X q ) = X(fT (ω 1,..., ω p, X 1,..., X q )) p f T (ω 1,..., X ω,..., ω p, X 1,..., X q ) =1 (Xf)T (ω 1,..., ω p, X 1,..., X q ) q f T (ω 1,..., ω p, X 1,..., X X j,..., X q ) j=1 = f( X T )(ω 1,..., ω p, X 1,..., X q ). ( X T )(ω 1,..., ω p, X 1,..., fx l,..., X q ) = X(fT (ω 1,..., ω p, X 1,..., X q )) p f T (ω 1,..., X ω,..., ω p, X 1,..., X q ) f =1 q T (ω 1,..., ω p, X 1,..., X X j,..., X q ) j=1 (Xf)T (ω 1,..., ω p, X 1,..., X q ) = f( X T )(ω 1,..., ω p, X 1,..., X q ). X T (p, q) (p, q) S (r, s) T X (S T )(ω 1,..., ω p+r, X 1,..., X q+s ) = X((S T )(ω 1,..., ω p+r, X 1,..., X q+s )) p+r (S T )(ω 1,..., X ω,..., ω p+r, X 1,..., X q+s ) =1 q+s (S T )(ω 1,..., ω p+r, X 1,..., X X j,..., X q+s ) j=1 = X(S(ω 1,..., ω p, X 1,..., X q )T (ω p+1,..., ω p+r, X q+1,..., X q+s ))

3.4. 57 p S(ω 1,..., X ω,..., ω p, X 1,..., X q )T (ω p+1,..., ω p+r, X q+1,..., X q+s ) =1 p+r =p+1 S(ω 1,..., ω p, X 1,..., X q )T (ω p+1,..., X ω,..., ω p+r, X q+1,..., X q+s ) q S(ω 1,..., ω p, X 1,..., X X j,..., X q )T (ω p+1,..., ω p+r, X q+1,..., X q+s ) j=1 q+s j=q+1 S(ω 1,..., ω p, X 1,..., X q )T (ω p+1,..., ω p+r, X q+1,..., X X j,..., X q+s ) = ( X S)(ω 1,..., ω p, X 1,..., X q )T (ω p+1,..., ω p+r, X q+1,..., X q+s ) +S(ω 1,..., ω p, X 1,..., X q )( X T )(ω p+1,..., ω p+r, X q+1,..., X q+s ) = ( X S T + S X T )(ω 1,..., ω p+r, X 1,..., X q+s ). X (S T ) = X S T + S X T X T (p, q) (C (r,s) T )(ω 1,..., ω p 1, X 1,..., X q 1 ) = T (ω 1,..., dx a..., ω p 1, X 1,..., a..., X q 1 ) r (C (r,s) X T )(ω 1,..., ω p 1, X 1,..., X q 1 ) r = ( X T )(ω 1,..., dx a..., ω p 1, X 1,..., a..., X q 1 ) r = X(T (ω 1,..., dx a..., ω p 1, X 1,..., a..., X q 1 )) p 1 r s T (ω 1,..., X ω,..., dx a..., ω p 1, X 1,..., a..., X q 1 ) =1 r T (ω 1,..., X (dx a )..., ω p 1, X 1,..., a..., X q 1 ) q 1 r s T (ω 1,..., dx a..., ω p 1, X 1,..., X X j,..., a..., X q 1 ) j=1 r T (ω 1,..., dx a..., ω p 1, X 1,..., X a..., X q 1 ). s s X a = X b Γ c ba c s s s

58 3 Remann X (dx a )( c ) = X(dx a ( c )) dx a ( X c ) = dx a (X b Γ d bc d ) = X b Γ a bc X (dx a ) = X b Γ a bcdx c. r T (ω 1,..., X (dx a )..., ω p 1, X 1,..., a..., X q 1 ) r T (ω 1,..., dx a..., ω p 1, X 1,..., X a..., X q 1 ) r = X b Γ a bct (ω 1,..., dx c,..., ω p 1, X 1,..., a..., X q 1 ) = 0. r X b Γ c bat (ω 1,..., dx a..., ω p 1, X 1,..., c..., X q 1 ) s s s s (C (r,s) X T )(ω 1,..., ω p 1, X 1,..., X q 1 ) r = X(T (ω 1,..., dx a..., ω p 1, X 1,..., a..., X q 1 )) p 1 r s T (ω 1,..., X ω,..., dx a..., ω p 1, X 1,..., a..., X q 1 ) =1 q 1 r s T (ω 1,..., dx a..., ω p 1, X 1,..., X X j,..., a..., X q 1 ) j=1 s = X((C (r,s) T )(ω 1,..., ω p 1, X 1,..., X q 1 )) p 1 s (C (r,s) T )(ω 1,..., X ω,..., ω p 1, X 1,..., a..., X q 1 ) =1 q 1 (C (r,s) T )(ω 1,..., ω p 1, X 1,..., X X j,..., X q 1 ) j=1 = ( X C (r,s) T )(ω 1,..., ω p 1, X 1,..., X X j,..., X q 1 ) C (r,s) X T = X C (r,s) T X

3.4. 59 3.4.2 (p, q) T ( X T )(ω 1,..., ω p, X 1,..., X q ) = X(T (ω 1,..., ω p, X 1,..., X q )) p T (ω 1,..., X ω,..., ω p, X 1,..., X q ) =1 q T (ω 1,..., ω p, X 1,..., X X j,..., X q ) j=1 3.4.3 3.4.1 : C (T M) C (T (p,q) M) C (T (p,q) M); (X, T ) X T T (p,q) M (0, 0) (1, 0) (0, 1) 3.4.1 (p, q) 3.4.2 ( fx T )(ω 1,..., ω p, X 1,..., X q ) = fx(t (ω 1,..., ω p, X 1,..., X q )) p T (ω 1,..., fx ω,..., ω p, X 1,..., X q ) =1 q T (ω 1,..., ω p, X 1,..., fx X j,..., X q ) j=1 = fx(t (ω 1,..., ω p, X 1,..., X q )) p f T (ω 1,..., X ω,..., ω p, X 1,..., X q ) f =1 q T (ω 1,..., ω p, X 1,..., X X j,..., X q ) j=1 = f( X T )(ω 1,..., ω p, X 1,..., X q ). fx T = f X T.

60 3 Remann ( X (ft ))(ω 1,..., ω p, X 1,..., X q ) = X(fT (ω 1,..., ω p, X 1,..., X q )) p ft (ω 1,..., X ω,..., ω p, X 1,..., X q ) =1 q ft (ω 1,..., ω p, X 1,..., X X j,..., X q ) j=1 = (Xf)T (ω 1,..., ω p, X 1,..., X q ) +fx(t (ω 1,..., ω p, X 1,..., X q )) p f T (ω 1,..., X ω,..., ω p, X 1,..., X q ) f =1 q T (ω 1,..., ω p, X 1,..., X X j,..., X q ) j=1 = (Xf)T (ω 1,..., ω p, X 1,..., X q ) f( X T )(ω 1,..., ω p, X 1,..., X q ) = ((Xf)T + f( X T ))(ω 1,..., ω p, X 1,..., X q ). X (ft ) = (Xf)T + f( X T ) T (p,q) M 3.4.4 (p, q) T T ( T )(ω 1,..., ω p, X 1,..., X q ; X) = ( X T )(ω 1,..., ω p, X 1,..., X q ) T (p, q + 1) (ω C (T (0,1) M), X j, X C (T (1,0) M)) 3.4.5 C f f (0, 1) f X gradf, X = df(x) gradf gradf f gradf df f 1 f Remann gradf Remann

3.4. 61 1 f gradf f f gradf gradf = (gradf) (gradf) k g kj = gradf, j = j f (gradf) = (gradf) k g kj g j = g j j f. 1 ω g j ω j X 1 g j X j 3.4.6 T = 0 3.4.7 Remann (M, g) Lev-Cvta ( X g)(y, Z) = X(g(Y, Z)) g( X Y, Z) g(y, X Z) = 0 g = 0 Remann 3.4.8 (p, q) T T = T 1 p j 1 j q 1 p dx j 1 dx j q T = T 1 p j 1 j q;k 1 p dx j 1 dx j q dx k k T 1 p j 1 j q = k T 1 p j 1 j q 1 p dx j 1 dx j q dx k = k T 1 p j 1 j q + p a=1 Γ a kl T 1 l p j 1 j q q b=1 Γ m kj b T 1 p j 1 m j q l a m b 3.4.2 k dx = Γ kadx a

62 3 Remann k T 1 p j 1 j q = ( k T )(dx 1,..., dx p, j1,..., jq ) = k (T (dx 1,..., dx p, j1,..., jq )) p T (dx 1,..., k dx a,..., dx p, j1,..., jq ) a=1 q T (dx 1,..., dx p, j1,..., k jb,..., jq1) b=1 = k T 1 p j 1 j q p T (dx 1,..., Γ a kl dx l,..., dx p, j1,..., jq ) a=1 q T (dx 1,..., dx p, j1,..., Γ m kj b m,..., jq1) b=1 = k T 1 p j 1 j q + p a=1 Γ a kl T 1 l p j 1 j q q b=1 Γ m kj b T 1 p j 1 m j q 3.5 3.5.1 V, W V q V V W L q (V, W ) M L q (T M, T M) = L q (T x M, T x M) x M 2.2.2 M L q (T M, T M) C T T (ω, X 1,..., X q ) = ω(t (X 1,..., X q )) (ω C (T M), X C (T M)) T T M (1, q) M (1, q) L q (T M, T M) C L q (T M, T M) C M (1, q) T T ( j1,..., jq ) = T j 1 j q (1, q) T T j 1 j q = T (dx, j1,..., jq ) = dx (T ( j1,..., jq ))

3.5. 63 = dx (T k j 1 j q k ) = T j 1 j q T T T 3.5.2 Remann M Lev-Cvta R Remann L 3 (T M, T M) C 3.5.1 M (1, 3) 3.5.3 Remann R X, Y, Z, W (1) (5) (1) R(X, Y )Z + R(Y, X)Z = 0, (2) R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0, (3) R(X, Y )Z, W + Z, R(X, Y )W = 0, (4) R(X, Y )Z, W = R(Z, W )X, Y, (5) ( X R)(Y, Z)W + ( Y R)(Z, X)W + ( Z R)(X, Y )W = 0. (2) 1Banch (5) 2Banch (1) (3) 3.2.5 (2) Lev-Cvta R(X, Y )Z = X Y Z Y X Z [X,Y ] Z X Y Y X = [X, Y ] = X ( Z Y + [Y, Z]) Y ( Z X + [X, Z]) [X,Y ] Z = X Z Y + [Y,Z] X + [X, [Y, Z]] Y Z X [X,Z] Y [Y, [X, Z]] [X,Y ] Z. R(Y, Z)X R(Z, X)Y R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = R(X, Y )Z + Y Z X Z Y X [Y,Z] X + Z X Y X Z Y [Z,X] Y = [X, [Y, Z]] + [Y, [Z, X]] + Z ( X Y Y X) [X,Y ] Z = [X, [Y, Z]] + [Y, [Z, X]] + Z [X, Y ] [X,Y ] Z = [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0.

64 3 Remann Jacob (4) 3.5.4 V V (1, 3) R (1) R(X, Y )Z + R(Y, X)Z = 0, (2) R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0, (3) R(X, Y )Z, W + Z, R(X, Y )W = 0 R R(X, Y )Z, W R(X, Y )Z, W = R(Z, W )X, Y = R(Y, Z)X, W R(Z, X)Y, W ((2) ) = R(Y, Z)W, X + R(Z, X)W, Y ((3) ) = R(Z, W )Y, X R(W, Y )Z, X R(X, W )Z, Y R(W, Z)X, Y ((2) ) = 2 R(Z, W )X, Y + R(W, Y )X + R(X, W )Y, Z ((1) (3) ) = 2 R(Z, W )X, Y R(Y, X)W, Z ((2) ) = 2 R(Z, W )X, Y R(X, Y )Z, W. ((1) (3) ) R(X, Y )Z, W = R(Z, W )X, Y 3.5.3(5) 3.5.5 Remann X, Y T R(X, Y )T = ( X Y Y X [X,Y ] )T (1) R(X, Y ) S, T R(X, Y )(S T ) = R(X, Y )S T + S R(X, Y )T.

3.5. 65 (2) C f R(X, Y )f = 0 (p, q) T (R(X, Y )T )(ω 1,..., ω p, X 1,..., X q ) p = T (ω 1,..., R(X, Y )ω,..., ω p, X 1,..., X q ) =1 q T (ω 1,..., ω p, X 1,..., R(X, Y )X j,..., X q ) j=1 R(X, Y ) 3.4.1 R(X, Y ) S, T X Y (S T ) = X ( Y S T + S Y T ) = X Y S T + Y S X T + X S Y T + S X Y T R(X, Y )(S T ) = R(X, Y )S T + S R(X, Y )T. C f R(X, Y )f = XY f Y Xf [X, Y ]f = 0. R(X, Y ) (p, q) T 0 = R(X, Y )(T (ω 1,..., ω p, X 1,..., X q )) = (R(X, Y )T )(ω 1,..., ω p, X 1,..., X q ) p + T (ω 1,..., R(X, Y )ω,..., ω p, X 1,..., X q ) + =1 q T (ω 1,..., ω p, X 1,..., R(X, Y )X j,..., X q ). j=1 (R(X, Y )T )(ω 1,..., ω p, X 1,..., X q ) p = T (ω 1,..., R(X, Y )ω,..., ω p, X 1,..., X q ) =1 q T (ω 1,..., ω p, X 1,..., R(X, Y )X j,..., X q ) j=1

66 3 Remann 3.5.6 (Rcc ) Remann (p, q) T X, Y ( 2 T )(ω 1,..., ω p, X 1,..., X q ; X; Y ) ( 2 T )(ω 1,..., ω p, X 1,..., X q ; Y ; X) = (R(X, Y )T )(ω 1,..., ω p, X 1,..., X q ) 3.4.2 ( T )(ω 1,..., ω p, X 1,..., X q ; X) = X(T (ω 1,..., ω p, X 1,..., X q )) p T (ω 1,..., X ω,..., ω p, X 1,..., X q ) =1 q T (ω 1,..., ω p, X 1,..., X X j,..., X q ) j=1 ( 2 T )(ω 1,..., ω p, X 1,..., X q ; X; Y ) = Y (( T )(ω 1,..., ω p, X 1,..., X q ; X)) p ( T )(ω 1,..., Y ω k,..., ω p, X 1,..., X q ; X) =k q ( T )(ω 1,..., ω p, X 1,..., Y X l,..., X q ; X) l=1 ( T )(ω 1,..., ω p, X 1,..., X q ; Y X) = Y X(T (ω 1,..., ω p, X 1,..., X q )) (5.1) p Y (T (ω 1,..., X ω,..., ω p, X 1,..., X q )) (5.2) + + =1 q Y (T (ω 1,..., ω p, X 1,..., X X j..., X q )) (5.3) j=1 p X(T (ω 1,..., Y ω k,..., ω p, X 1,..., X q )) (5.4) k=1 p T (ω 1,..., X ω,..., Y ω k,..., ω p, X 1,..., X q ) (5.5) k=1 k p T (ω 1,..., X Y ω k,..., ω p, X 1,..., X q ) (5.6) k=1

3.5. 67 + + + + p k=1 j=1 q T (ω 1,..., Y ω k,..., ω p, X 1,..., X X j,..., X q ) (5.7) q X(T (ω 1,..., ω p, X 1,..., Y X l,..., X q )) (5.8) l=1 q l=1 p T (ω 1,..., X ω,..., ω p, X 1,..., Y X l,..., X q ) (5.9) =1 q T (ω 1,..., ω p, X 1,..., Y X j,..., X X l,..., X q ) (5.10) l=1 j l q T (ω 1,..., ω p, X 1,..., X Y X l,..., X q ) (5.11) l=1 ( X Y )(T (ω 1,..., ω p, X 1,..., X q )) (5.12) p + T (ω 1,..., Y Xω k,..., ω p, X 1,..., X q ) (5.13) + k=1 q T (ω 1,..., ω p, X 1,..., Y XX l,..., X q ). (5.14) l=1 X Y (1) [X, Y ](T (ω 1,..., ω p, X 1,..., X q )) (2) (4) (3) (8) (5) X Y (6) p T (ω 1,..., ( X Y Y X )ω k,..., ω p, X 1,..., X q ) k=1 (7) (9) (10) X Y (11) q T (ω 1,..., ω p, X 1,..., ( X Y Y X )X l,..., X q ) l=1 (12) (1) (13) ( X Y Y X)(T (ω 1,..., ω p, X 1,..., X q )) = [X, Y ](T (ω 1,..., ω p, X 1,..., X q )) p T (ω 1,..., X Y Y Xω k,..., ω p, X 1,..., X q ) k=1

68 3 Remann = p T (ω 1,..., [X,Y ] ω k,..., ω p, X 1,..., X q ) k=1 (14) = q T (ω 1,..., ω p, X 1,..., X Y Y XX l,..., X q ) l=1 q T (ω 1,..., ω p, X 1,..., [X,Y ] X l,..., X q ) l=1 3.5.5 = ( 2 T )(ω 1,..., ω p, X 1,..., X q ; X; Y ) ( 2 T )(ω 1,..., ω p, X 1,..., X q ; Y ; X) p T (ω 1,..., R(X, Y )ω,..., ω p, X 1,..., X q ) =1 + q T (ω 1,..., ω p, X 1,..., R(X, Y )X j,..., X q ) j=1 = (R(X, Y )T )(ω 1,..., ω p, X 1,..., X q ) 3.5.7 C f (0, 2) 2 f ( 2 f)(x; Y ) ( 2 f)(y ; X) = (R(X, Y )f) = 0 2 f 2 f C f Hessan ( 2 f)(x; Y ) = ( Y f)(x) = Y (( f)(x)) f( Y X) = Y (df(x)) df( Y X) = Y Xf ( Y X)f 3.5.3(5) 1 ω 3.5.6 3.5.5 ( 2 ω)(w ; X; Y ) ( 2 ω)(w ; Y ; X) = (R(X, Y )ω)(w ) = ω(r(x, Y )W ) = (C (1,1) (ω R))(X, Y, W ).

3.5. 69 Z ( 3 ω)(w ; X; Y ; Z) ( 3 ω)(w ; Y ; X; Z) = ( Z C (1,1) (ω R))(X, Y, W ) = (C (1,1) ( Z ω R + ω Z R))(X, Y, W ) = ( Z ω)(r(x, Y )W ) + ω(( Z R)(X, Y )W ). ω 3.5.6 3.5.5 ( 3 ω)(w ; X; Y ; Z) ( 3 ω)(w ; X; Z; Y ) = (R(Y, Z) ω)(w ; X) = ( ω)(r(y, Z)W ; X) + ( ω)(w ; R(Y, Z)X). X, Y, Z ( 3 ω)(w ; Y ; Z; X) ( 3 ω)(w ; Y ; X; Z) = ( ω)(r(z, X)W ; Y ) + ( ω)(w ; R(Z, X)Y ) ( 3 ω)(w ; Z; X; Y ) ( 3 ω)(w ; Z; Y ; X) = ( ω)(r(x, Y )W ; Z) + ( ω)(w ; R(X, Y )Z). ( ω)(r(x, Y )W ; Z) + ( ω)(r(y, Z)W ; X) + ( ω)(r(z, X)W ; Y ) +( ω)(w ; R(X, Y )Z) + ( ω)(w ; R(Y, Z)X) + ( ω)(w ; R(Z, X)Y ) = ( 3 ω)(w ; X; Y ; Z) ( 3 ω)(w ; Y ; X; Z) +( 3 ω)(w ; Y ; Z; X) ( 3 ω)(w ; Z; Y ; X) +( 3 ω)(w ; Z; X; Y ) ( 3 ω)(w ; X; Z; Y ) = ( X ω)(r(y, Z)W ) + ( Y ω)(r(z, X)W ) + ( Z ω)(r(x, Y )W ) +ω(( X R)(Y, Z)W ) + ω(( Y R)(Z, X)W ) + ω(( Z R)(X, Y )W ) 3.5.3(2) ω(( X R)(Y, Z)W + ( Y R)(Z, X)W + ( X R)(Y, Z)W ) = ( ω)(w ; R(X, Y )Z + R(Y, Z)X + R(Z, X)Y ) = 0. 1 ω ( X R)(Y, Z)W + ( Y R)(Z, X)W + ( X R)(Y, Z)W = 0

70 3 Remann 3.5.8 Remann R(, j ) k = R l jk l R(, j ) k = j k j k [, j ] k = (Γ m jk m ) j (Γ m k m ) = ( Γ m jk) m + Γ m jk m ( j Γ m k) m Γ m k j m = ( Γ l jk) l + Γ m jkγ l m l ( j Γ l k) l Γ m kγ l jm l Rjk l = Γ l jk j Γ l k + Γ l mγ m jk Γ l jmγ m k (0, 4) R(X, Y )Z, W R(, j ) k, l = R jkl R jkl = R(, j ) k, l = Rjk m m, l = g lm Rjk. m 3.5.3 Rjk l + Rjk l = 0, Rjk l + Rjk l + Rkj l = 0, R jkl + R jlk = 0 R jkl = R klj Rjkl m + j Rkl m + k Rjl m = 0. 3.5.5 (R(, j )dx k )( l ) dx k (R(, j ) l ) = dx k (R m jl m ) = R k jl R(, j )dx k = R k jldx l. (p, q)t (R(, j )T ) 1 p j 1 j q (R(, j )T ) 1 p j 1 j q = (R(, j )T )(dx 1,..., dx p, j1,..., jq )

3.6. 71 = = p a=1 p T (dx 1,..., R(, j )dx a,..., dx p, j1,..., jq ) a=1 q T (dx 1,..., dx p, j1,..., R(, j ) b,..., jq ) b=1 R a jk T 1 k p j 1 j q q b=1 R l jj a T 1 p j 1 l j q. k a l b Rcc ( 3.5.6) T 1 p j 1 j q ;;j T 1 p j 1 j q ;j; = j T 1 p j 1 j q p = a=1 j T 1 p j 1 j q q R a jk T 1 k p j 1 j q + b=1 R l jj b T 1 p j 1 l j q. C f f = f ; = f f Hessan 2 f f ;;j = j f = j f Γ k j k f. df x = 0 x f ;;j Hessan = j f 2 f 3.6 3.6.1 2 Remann M p T p M 2 σ R(X, Y )Y, X X Y 2 (X, Y σ ) Z, W σ Z = ax + by, W = cx + dy (ad bc 0) 3.5.3 (1) (3) R(Z, W )W, Z = R(aX + by, cx + dy )(cx + dy ), ax + by

72 3 Remann = R(aX, dy )(cx + dy ), ax + by + R(bY, cx)(cx + dy ), ax + by = ad R(X, Y )cx, by + ad R(X, Y )dy, ax +bc R(Y, X)cX, by + bc R(Y, X)dY, ax = (ad bc) 2 R(X, Y )Y, X. 1.2.4 Z W = (ad bc)x Y Z W 2 = (ad bc) 2 X Y 2. R(Z, W )W, Z Z W 2 = R(X, Y )Y, X X Y 2. 3.6.2 2 Remann M p T p M 2 σ K σ = R(X, Y )Y, X X Y 2 (X, Y σ ) K σ σ M p T p M 2 σ K σ M 3.6.3 1 Remann 3.5.3 (1) 0 2 2 σ σ e 1, e 2 e 1 e 2 = 1 K σ = R(e 1, e 2 )e 2, e 1 3.6.4 2 Remann M K p M X, Y, Z T p M R(X, Y )Z = K( Y, Z X X, Z Y ) K R R(X, Y )Z = K( Y, Z X X, Z Y ) R(X, Y )Y, X = K Y, Y X X, Y Y, X = K( X, X Y, Y X, Y 2 ).

3.6. 73 1.4.4 X Y 2 = X, X Y, Y X, Y 2 K M M M K R K (X, Y )Z = K( Y, Z X X, Z Y ) M (1, 3) R K R K R K 3.5.4 (1) R K (X, Y )Z + R K (Y, X)Z = 0 R K (X, Y )Z + R K (Y, Z)X + R K (Z, X)Y = K( X, Z Y Y, Z X + Y, X Z Z, X Y + Z, Y X X, Y Z) = 0 R K 3.5.4 (2) R K (X, Y )Z, W + Z, R K (X, Y )W = K( Y, Z X, W X, Z Y, W ) + K( Z, X Y, W Z, Y X, W ) = 0 R K 3.5.4 (3) 3.5.4 R K R K (X, Y )Z, W = R K (Z, W )X, Y R K (X, Y )Z, W = K( Y, Z X, W X, Z Y, W ) = K W, X Z Z, X W, Y = R K (Z, W )X, Y. R K 3.5.3 (1) (4) 3.5.3 R (1) (4) S = R R K S (1) (4) (1, 3) S 0 X, Y S(X, Y )Y, X = 0

74 3 Remann X, Y X, Y S(X, Y )Y, X = 0 X, Y S(X, Y )Y, X = R(X, Y )Y, X R K (X, Y )Y, X = K X Y 2 K( X, X Y, Y X, Y 2 ) = 0. X, Y S(X, Y )Y, X = 0 Y Y + Z 0 = S(X, Y + Z)(Y + Z), X = S(X, Y )Y, X + S(X, Y )Z, X + S(X, Z)Y, X + S(X, Z)Z, X = S(X, Y )Z, X + S(X, Z)Y, X. S(X, Y )Z, X = S(X, Y )X, Z S(X, Z)Y, X = S(Y, X)X, Z = S(X, Y )X, Z 2 S(X, Y )X, Z = 0 S(X, Y )X = 0 X X + Z 0 = S(X + Z, Y )(X + Z) = S(X, Y )X + S(X, Y )Z + S(Z, Y )X + S(Z, Y )Z = S(X, Y )Z + S(Z, Y )X = S(X, Y )Z S(Y, X)Z S(X, Z)Y = 2S(X, Y )Z S(X, Z)Y. 2S(X, Y )Z = S(X, Z)Y.

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