1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1

1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1

f, g C P, λ R (1) v(f + g) = v(f) + v(g) (2) v(λf) = λv(f) (3) v(fg) = v(f)g(p ) + f(p )v(f) {v} { x 1,..., x m } m (1),(2) (3) f x 0 R m x 0 = (x 1 0,..., x m 0 ) g i ( x 0 ) = f x i ( x 0 ) m {g i } x 0 x = (x 1,, x m ) f( x ) = f( x 0 ) + (x i x i 0)g i ( x ) ϕ ϕ(t) = f ( x 0 + t( x x 0 )) f( x ) f( x 0 ) = ϕ(1) ϕ(0) 2

1 = = = 0 1 ϕ (t)dt d 0 dt f ( x 0 + t( x x 0 ) 1 (x i x i 0) 1 0 x i f ( x 0 + t( x x 0 )) g i ( 1 x ) = 0 x i f ( x 0 + t( x x 0 )) {g i } g i ( x 0 ) = x i f( x 0 ) P M P (U, ϕ; x i ) P ϕ(u) R m ϕ(p ) P M v 3 v : C P R f, g C P, λ R (1) v(f + g) = v(f) + v(g) (2) v(λf) = λv(f) (3) v(fg) = v(f)g(p ) + f(p )v(f) f(x) 1 v(f(x)) = v(1) = v(1 1) = 2v(1) v(1) = 0 f(x) k 3

v(f(x)) = v(k) = kv(1) = 0 ϕ(p ) = x 0 f C x0 g i ( x 0 ) = f x i ( x 0 ) g i C x0 f( x ) = f( x 0 ) + (x i x i 0)g i ( x ) v(f) = v(f( x 0 ) + (x i x i 0)g i ( x )) = v(x i ) f x i ( x 0 ) v ( v(x 1 ),..., v(x m ) ) v(x i ) = a i v(f) = v = a i a i v(x i ) = a i x i x = x0 f x i x = x0 R m R m f x 0 a = (a 1,..., a m ) D a f( x 0 ) D a f( x 0 ) = d dt t=0f( x 0 + t a ) 4

= a i f x i ( x 0 ) x 0 v x 0 (v(x 1 ),..., v(x m )) M P T P M T P M v, v 1, v 2 T P (M), λ R (v 1 + v 2 )(f) = v 1 (f) + v 2 (f) v(λf) = λv(f) f C P T P M M P M m T P M m (U, ϕ; x i ) v { x 1 P,..., x m P } v = a i x i P v = a i x i P = 0 a i = v(x i ) = 0 { x 1 P,..., x m P } 5

2. M m M P T P M M X M T P M X : P X(P ) T P M M X M M P P (U; x i ) X X = X i x i X i C X M (V ; y α ) U V φ X = Y α y α α=1 Y α C M X (M) X (M) 6

X M f C (M) fx fx(p ) = f(p )X(P ), P M fx X (M) X (M) C (M) X (M) X M f C (M) X(f)(P ) = X(P )(f) X : C (M) C (M), X : C (M) C (M) f, g C (M) λ R 3 (1) X(f + g) = X(f) + X(g) (2) X(λf) = λx(f) (3) X(fg) = X(f)g + fx(g) 3 X : C (M) C (M) 1 ( ) X X (M) M P (U; x i ) X = X i x i X i C (U) 7

X(f) U = X i f x i P X(f) C (M). X : C (M) C (M) ( ) C (M) P v C p X : C (M) C (M) X P : C P R f C P g U 1, g M\V = 0 g f = fg P U V U V f f C (M) X P (f) = (X(f))(P ) (U; x i ) X = X i xi 8

X i = X(x i ) C (U) X M M X X : C (M) C (M) f, g C (M) λ R 3 (1) X(f + g) = X(f) + X(g) (2) X(λf) = λx(f) (3) X(fg) = X(f)g + fx(g) Intrinsic X, Y X (M) X, Y : C (M) C (M) X Y : C (M) C (M) X Y X Y (fg) = X (Y (f)g + fy (g)) = X Y (f)g + Y (f)x(g) + X(f)Y (g) + fx Y (g) (1) 9

[X, Y ] = X Y Y X [X, Y ] (1) X, Y [X, Y ](fg) = [X, Y ](f)g + f[x, Y ](g) [X, Y ] X, Y [, ] : X (M) X (M) X (M) (X, Y ) [X, Y ] X, Y, Z X (M), λ R, f, g C (M) (1) [X + Y, Z] = [X, Z] + [Y, Z] (2) [λx, Y ] = λ[x, Y ] (3) [X, Y ] = Y, X (4) [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0 (5) [fx, gy ] = fx(g)y gy (f)x + fg[x, Y ] M P T P M T P M P ω TP M ω ω : T P M R v T P M, ω T P M 10

ω(v) =< ω, v >=< v, ω > <, >: T P M TP M R f CP df : T P M R df(v) = v(f) P df P df(p ) (U; x i ) P { } x 1 P,..., x m P {dx 1 P,..., dx m P } df(p ) : T P M R {dx 1 P,..., dx m P } df(p ) = f i dx i P ( ) f i = df x i P df(p ) = = f x i (P ) f x i (P )dxi P df 11

f 3. X Y Y X 3 X, Y, Z Z Y X Y Z X 0 n 2 Z Y X Y Z X n 12

M P T P M TP M P r s r s {}}{{}}{ Ts r (P ) = T P M T P M TP M TP M P (r, s) M r s M T r s (P ) T : P X(P ) T r s (P ) M r s (r, s) T M M P P (U; x i ) X T = T i 1...i r j 1...j s x i... 1 x i dxj 1 dx j s r X i 1...i r j 1...j s C X M M r s T r s (M) T0 1 (M) = X (M) T1 0 (M) 1 (M) 13

T r s (M) C (M) (1, 2) (1, 3) (1, 3) M T M (1, 3) 3 T : X (M) X (M) X (M) X (M) P M, X, Y, Z (M) T (X, Y, Z)(P ) = T (P ) (X(P ), Y (P ), Z(P )) T X, Y, Z C (M) f, g C (M), X, Y X (M) T (, fx + gy, ) = fx(, X, ) + gx(, Y, ) 3 T : X (M) X (M) X (M) X (M) C (1) T M (1, 3) M P (1, 3) T (P ) 3 T (P ) : T P (M) T P (M) T P (M) T P (M) T (P ) (X(P ), Y (P ), Z(P )) X(P ), Y (P ), Z(P ) f (P ) f P f(p ) 14

P (1) T (P ) P X(P ), Y (P ), Z(P ) M (U; x i ) X = X i x i, Y = Y i x i, Z = Zi x i T (X, Y, Z) = X i Y j Z k T ( x j, x k ) x i, T ( x i, x j, x k ) T (X, Y, Z) X i (P ), Y j (P ), Z k (P ) T (1,3) [, ] [, ] : X (M) X (M) X (M) [fx, gy ] = fx(g)y gy (f)x + fg[x, Y ] X = X i x i, Y = Y j x i [X, Y ] = (X i Y j ) x i Y i Xj x i x j ( ) [X, Y ] P X p, Y p ( ) Y j x i, Xj x i 15

X P, Y P 4. M R 3 f 2 f x i = f u k u k x i 2 f x i x j = 2 f u k u l u k u l x i x j + f 2 u k u k x i x j ( ) ( ) R 3 ( ) 16

M f Hess(f) D D(df) ddf = 0 R 3 0 d M r (0, r) T : X (M) X (M) C }{{} r X, Y X (M) T (, X,, Y, ) = T (, Y,, X, ) 17

T r r ( ) 1 Pfaff M r r (M) 0 (M) = C (M) 1 (M) = T 0 1 (M) M m m + 1 M m i (M) = i=0 (M), (U; x 1,..., x m ) T P (M) {dx1,..., dx m } U p (M) {dx i 1 dx i p i 1 < < i p } p (M) T U i 1 < <i p T i1...i p dx i1 dx ip = 1 p! T i 1...i p dx i 1 dx i p i 1,..., i p 1 m i 1,..., i p T i1...i p = 0 X i = X j i x j T = T i1...i p dx i1 dx ip i 1 < <i p 18

T (X 1,..., X p ) = T i1...i p dx i1 dx ip (X 1,..., X p ) i 1 < <i p = X i 1 1 X i 1 p T i1...i p ( ) i 1 < <i p X i p 1 X i p p dx i 1 dx i p = δ j 1...j p i 1...i p dx j 1 dx j p ( ) ( ) ( ) ( ) T T (..., X i + Y i,... ) = T (..., X i,... ) + f(..., Y i,... ) f C (M) T (..., fx,... ) = ft (..., X,... ) T (, X,, Y, ) = T (, Y,, X, ) (M) M d : (M) (M) 19

(1) d R T, S (M), λ R d(t + S) = dt + ds, d(λt ) = λdt (2) f 0 (M) = C (M) df f (3) T r (M), T (M) d(t T ) = dt T + ( 1) r T dt (4) d 2 = d d = 0 (U; x i ) d () d T = fdx i 1 (f C (U)) (4) d(dx i ) = 0 (3) dt = df dx i 1 T = f dt = df. T = fdx i 1 dt = df dx i 1 d (1),(2),(3),(4). (1),(2) 20

(3) T = fdx i 1, T = gdx j 1 dx j s d(t T ) = d(fg))dx i 1 dx j 1 dx j s = (df dx i 1 ) (gdx j 1 dx j s ) +( 1) r fdx i 1 ) (dg dx j 1 dx j s = dt T + ( 1) r T dt (3) (4) dt = df dx i 1 f x j dxj dx i 1 d(dt ) = = 1 2 = 0 2 f x j x k dxk dx j dx i 1 2 f x k x j ( 2 f x j x k ) dx k dx j dx i 1 d T = fdx i 1 (f C (U)) dt = df dx i 1 Intrinsic. T p (M), X 1,..., X p+1 X (M) 21

dt (X 1,..., X p+1 ) p+1 = ( 1) i+1 X i (T (X 1,..., ˆX i,..., X p+1 )) + i<j( 1) i+j T ([X i, X j ], X 1,..., ˆX i,..., ˆX j,..., X p+1 ) ˆX i X i d p + 1 (p + 1 ) dx j i dx j p+1 S(X 1,..., X p+1 ) S(..., X i + Y i,... ) = S(..., X i,... ) + S(..., Y i,... ) S(..., X i,..., X j,... ) = S(..., X j,..., X i,... ) f C (M) S(X 1,..., fx i,..., X p+1 ) = fs(x 1,..., X i,..., X p+1 ) i i = 1 S(fX 1, X 2,..., X p+1 ) p+1 = fx 1 (T (X 2,..., X p+1 ))+ ( 1) i+1 X i (T (fx 1, X 2,..., ˆX i,..., X p+1 )) i=2 22

( 1) i+1 T ([fx 1, X i ],..., ˆX i,..., X p+1 ) p+1 + i=2 + 2 i<j ( 1) i+j T ([X i, X j ], fx 1, X 2,..., ˆX i,..., ˆX j,..., X p+1 ) X i (T (fx 1, X 2,..., ˆX i,..., X p+1 )) = (X i f)t (..., ˆX i,... ) + fx i (T (..., ˆX i,... )) T ([fx 1, X i ],..., ˆX i,..., X p+1 ) = T (f[x 1, X i ] (X i f)x 1, X 2,..., ˆX i,... ) (X i f)t (..., ˆX i,... ) = ft (X 1,..., X p1 ) p + 1 T = 1 p! T i 1...i p dx i 1... dx i p dt ( x j,..., 1 x j ) p+1 S( x j,..., 1 x j ) p+1 dt ( x j,..., 1 x j p+1 ) T = 1 p! T i 1...i p dx i 1... dx i p 23

dt = 1 T i1...i p p! x i dx i dx i 1... dx i p dt ( x j,..., 1 x j ) p+1 T i1...i p = 1 p! x i δ ii 1...i p j 1...j p+1 i j 1 j p+1 i 1,..., i p p! p+1 = ( 1) T k+1 j 1... jˆ k...j p+1 k=1 x j k S( x j,..., 1 x j ) p+1 [ ] x i, x j = 0 p+1 = ( 1) k+1 x j (T ( k x j,... ˆ 1 x j..., k x j )) p+1 k=1 p+1 = ( 1) T k+1 j 1... ˆ x j k k=1 j k...j p+1 dt ( x j,..., 1 x j ) p+1 = S( x j,..., 1 x j ) p+1 ( ) 24