II 2000
1 Euclid 1 1.1..................................... 1 1.2..................................... 8 1.3 Euclid............. 19 1.4 3 Euclid............................ 22 2 28 2.1 Lie Lie.................................. 28 2.2................................... 46 2.3................................. 52 3 58 3.1............................ 58 3.2................................ 62 3.3............................ 73 3.4........... 81 4 Riemann 89 4.1..................................... 89 4.2 Riemann.................................. 96 i
1 Euclid Euclid 1.1 1.1.1 V, W p V p V p W ω( ) i j i j i ω(..., v i,..., v j,...) + ω(..., v j,..., v i,...) = 0 (v k V ) ω W V p W V p p (V, W ) p (V, W ) W W = R p (V, R) = p V p V V p 0 (V, W ) = W j 1.1.2 p S p p ({1,..., p} ) σ S p sgn(σ) V, W ω p (V, W ), σ S p ω(v σ(1),..., v σ(p) ) = sgn(σ)ω(v 1,..., v p ) (v i V ) σ ω(v σ(1),..., v σ(p) ) = ω(v 1,..., v p ) (v i V ) 1
2 1 Euclid σ S p ω(v σ(1),..., v σ(p) ) = sgn(σ)ω(v 1,..., v p ) (v i V ) 1.1.3 V 1, V 2, W p f : V 1 V 2 f : p (V 2, W ) p (V 1, W ) ω p (V 2, W ) (f ω)(v 1,..., v p ) = ω(f(v 1 ),..., f(v p )) (v i V 1 ) f : 0 (V 2, W ) = W 0 (V 1, W ) = W f : p (V 2, W ) p (V 1, W ) 1.1.4 V, W 1, W 2, W 3 A : W 1 W 2 W 3 ω p (V, W 1 ) η q (V, W 2 ) A(ω η) p+q (V, W 3 ) v i V A(ω η)(v 1,..., v p+q ) = 1 sgn(σ)a(ω(v σ(1),..., v σ(p) ), η(v σ(p+1),..., v σ(p+q) )) p!q! σ S p+q A(ω η) p+q (V, W 3 ) A( ) : p (V, W 1 ) q (V, W 2 ) p+q (V, W 3 ) A A(ω η) ω η 1.1.5 1/p!q! 1.1.6 V, V, W 1, W 2, W 3 A : W 1 W 2 W 3 f : V V f (A(ω η)) = A((f ω) (f η)) (ω p (V, W 1 ), η q (V, W 2 )) v i V f (A(ω η))(v 1,..., v p+q ) = (A(ω η))(f(v 1 ),..., f(v p+q )) = 1 sgn(σ)a(ω(f(v σ(1) ),..., f(v σ(p) )), η(f(v σ(p+1) ),..., f(v σ(p+n) ))) p!q! σ S p+q = 1 sgn(σ)a((f ω)(v σ(1),..., v σ(p) ), (f η)(v σ(p+1),..., v σ(p+q) )) p!q! σ S p+q = A((f ω) (f η))(v 1,..., v p+q ).
1.1 3 1.1.7 V, W 1, W 2 A : W 1 W 1 W 2 A ( A(X, Y ) = A(Y, X)) A(ω η) = ( 1) pq A(η ω) (ω p (V, W 1 ), η q (V, W 1 )) A ( A(X, Y ) = A(Y, X)) A(ω η) = ( 1) pq+1 A(η ω) (ω p (V, W 1 ), η q (V, W 1 )) S p+q τ τ = ( 1 p p + 1 p + q q + 1 p + q 1 q ) sgn(τ) = ( 1) pq v 1,..., v p+q V A(ω η)(v 1,..., v p+q ) = 1 sgn(στ)a(ω(v στ(1),..., v στ(p) ), η(v στ(p+1),..., v στ(p+q) )) p!q! σ S p+q = 1 p!q! sgn(τ) sgn(σ)a(ω(v σ(q+1),..., v σ(p+q) ), η(v σ(1),..., v σ(q) )). σ S p+q A = 1 p!q! ( 1)pq sgn(σ)a(η(v σ(1),..., v σ(q) ), ω(v σ(q+1),..., v σ(p+q) )) σ S p+q = ( 1) pq A(η ω)(v 1,..., v p+q ) A = 1 p!q! ( 1)pq+1 σ S p+q sgn(σ)a(η(v σ(1),..., v σ(q) ), ω(v σ(q+1),..., v σ(p+q) )) = ( 1) pq+1 A(η ω)(v 1,..., v p+q ) 1.1.8 V W W W W W W V ω p (V, W ), η q (V, W ), ζ r (V, W ) (ω η) ζ = ω (η ζ)
4 1 Euclid S p+q = {τ S p+q+r τ(i) = i (p + q + 1 i p + q + r)} v 1,..., v p+q+r V = = = = ((ω η) ζ)(v 1,..., v p+q+r ) 1 sgn(σ)(ω η)(v σ(1),..., v σ(p+q) ) ζ(v σ(p+q+1),..., v 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)ω(v στ(1),..., v στ(p) ) η(v στ(p+1),..., v στ(p+q) ) p!q! τ S p+q ζ(v σ(p+q+1),..., v σ(p+q+r) ) 1 1 sgn(στ) p!q!r! (p + q)! σ S p+q+r τ S p+q (ω(v στ(1),..., v στ(p) ) η(v στ(p+1),..., v στ(p+q) )) ζ(v σ(p+q+1),..., v σ(p+q+r) ) 1 sgn(σ) p!q!r! σ S p+q+r (ω(v σ(1),..., v σ(p) ) η(v σ(p+1),..., v σ(p+q) )) ζ(v σ(p+q+1),..., v σ(p+q+r) ). = (ω (η ζ))(v 1,..., v p+q+r ) 1 sgn(σ) p!q!r! σ S p+q+r ω(v σ(1),..., v σ(p) ) (η(v σ(p+1),..., v σ(p+q) ) ζ(v σ(p+q+1),..., v σ(p+q+r) )) (ω η) ζ = ω (η ζ) 1.1.9 V W 1.1.8 ω p (V, W ), η q (V, W ), ζ r (V, W ) (ω η) ζ = ω (η ζ) ω η ζ W = R R R R 1.1.10 V ω 1,..., ω p 1 V v 1,..., v p V (ω 1 ω p )(v 1,..., v p ) = σ S p sgn(σ)ω 1 (v σ(1) ) ω p (v σ(p) ) = det(ω i (v j ))
1.1 5 p p = 1 p = q p = q + 1 (ω 1 ω q+1 )(v 1,..., v q+1 ) = 1 sgn(σ)(ω 1 ω q )(v σ(1),..., v σ(q) ) ω q+1 (v σ(q+1) ) q! σ S q+1 = 1 sgn(σ) sgn(τ)ω 1 (v στ(1) ) ω q (v στ(q) ) ω q+1 (v σ(q+1) ) q! σ S q+1 τ S q = 1 sgn(στ)ω 1 (v στ(1) ) ω q (v στ(q) ) ω q+1 (v σ(q+1) ) q! σ S q+1 τ S q = sgn(σ)ω 1 (v σ(1) ) ω q (v σ(q) ) ω q+1 (v σ(q+1) ) σ S q+1 = det(ω i (v j )). 1.1.11 V n e 1,..., e n V ω 1,..., ω n 0 V = R n < p p V = {0} 1 p n ( ) ω j 1 ω j p (1 j 1 < < j p n) p V dim p V = ( ) n ω p V 1 j p 1 < < j p n a j1...j p = ω(e j1,..., e jp ) ω = a j1...j p ω j 1 ω jp j 1 < <j p 1.1.1 0 V = R p > 0 ω p V v 1,..., v p V v i = n j=1 b j i e j ω(v 1,..., v p ) = j 1,...,j p =1 b j 1 1 b j p p ω(e j1,..., e jp ) j k = j l ω(e j1,..., e jp ) = 0 n < p k, l ω = 0 p V = {0} 1 p n j 1,..., j p ω(v 1,..., v p ) = = j 1 < <j p j 1 < <j p b jσ(1) 1 b j σ(p) σ S p sgn(σ)b jσ(1) 1 b j σ(p) σ S p p ω(e jσ(1),..., e jσ(p) ) p ω(e j1,..., e jp ).
6 1 Euclid b j σ(i) i = ω j σ(i) (vi ) 1.1.10 sgn(σ)b jσ(1) 1 b j σ(p) p = σ S p σ S p sgn(σ)ω j σ(1) (v 1 ) ω j σ(p) (v p ) = (ω j 1 ω j p )(v 1,..., v p ). ω(v 1,..., v p ) = a j1...j p (ω j 1 ω jp )(v 1,..., v p ) j 1 < <j p ω = a j1...j p ω j 1 ω j p. j 1 < <j p ( ) p V ( ) c j1...j p R j 1 < <j p c j1...j p ω j 1 ω j p = 0 (e k1,..., e kp ) (k 1 < < k p ) 1.1.10 0 = c j1...j p ω j 1 ω j p (e k1,..., e kp ) j 1 < <j p = c j1...j p sgn(σ)ω j σ(1) (e k1 ) ω j σ(p) (e kp ) j 1 < <j p σ S p = c k1...k p. ( ) p V ( ) n p dim p V = ( ) n p 1.1.12 V n W m e 1,..., e n V ω 1,..., ω n f 1,..., f m W 0 (V, W ) = W n < p p (V, W ) = {0} 1 p n v i V ( ) ω j 1 ω jp f k (1 j 1 < < j p n, 1 k m) (ω j 1 ω j p f k )(v 1,..., v p ) = (ω j 1 ω j p )(v 1,..., v p )f k ( ) p (V, W ) dim p (V, W ) = ( ) n p m ω p (V, W ) 1 j 1 < < j p n m a k j 1...j p f k = ω(e j1,..., e jp ) W k=1
1.1 7 ω = m a k j 1...j p ω j 1 ω j p f k j 1 < <j p k=1 p V m m p V p (V, W ) S m S(φ 1,..., φ m ) = φ k f k ((φ 1,..., φ m ) m p V ) k=1 S S f 1,..., f m W Sω p (V, W ) m ω(v 1,..., v p ) = ξ k (v 1,..., v p )f k (v 1,..., v p V ) k=1 ω p (V, W ) kξ k p V S(ξ 1,..., ξ m ) = ω S 1.1.11 m p V ( ) p (V, W ) dim p (V, W ) = ( ) n p m j 1 < < j p m ω(e j1,..., e jp ) = ξ k (e j1,..., e jp )f k k=1 a k j 1...j p = ξ k (e j1,..., e jp ) 1.1.11 ω = = = m ξ k f k k=1 m k=1 ξ k (e j1,..., e jp )ω j 1 ω jp f k j 1 < <j p m a k j 1...j p ω j 1 ω jp f k. j 1 < <j p k=1 1.1.13 V, W 1, W 2, W 3 A : W 1 W 2 W 3 ω 1 (V, W 1 ) η 1 (V, W 2 ) A(ω, η) v 1, v 2 V A(ω, η)(v 1, v 2 ) = A(ω(v 1 ), η(v 2 )) A(ω, η) V V W 3 (ω, η) A(ω, η) A A(ω, η) ωη
8 1 Euclid 1.2 1.2.1 V M n M x p (T x (M), V ) ω x ωω V M p R ()M (U; x 1,..., x n ) ( ) x ω x x i,, 1 x x i p x i 1,..., i p U V C 1.2.2 V M 0 M V C 1.2.3 f M V C f df V M 1 n = dim M M (U; x 1,..., x n ) f C ( ) x df x x i x = f x i (x) U V C df V M 1 1.2.4 ω V n M p M (U; x 1,..., x n ) x U x i x (1 i n) T x (M) (dx i ) x (1 i n) 1.1.12 x Uω x p (T x (M), V ) ω x = j 1 < <j p ω x ( x j,, 1 x x jp x ) (dx j 1 ) x (dx j p ) x ω x (dx j 1 ) x (dx j p ) x V C 1.2.1 () ω (dx j 1 ) x (dx j p ) x 1.2.5 V M p Ω p (M; V ) f, g C (M), ω, η Ω p (M; V ) x M (fω + gη) x = f(x)ω x + g(x)η x ( p (T x (M), V ) )
1.2 9 Ω p (M; V ) C (M) C (M) V 1, V 2, V 3 A : V 1 V 2 V 3 φ Ω p (M; V 1 ), ψ Ω q (M; V 2 ) x M A(φ ψ) x = A(φ x ψ x ) ( A( ) p (T x (M), V 1 ) q (T x (M), V 2 ) ) A(φ ψ) Ω p+q (M; V 3 ) C (M) A( ) : Ω p (M; V 1 ) Ω q (M; V 2 ) Ω p+q (M; V 3 ) Ω p (M; V ) C (M) f, g C (M), ω, η Ω p (M; V ) fω + gη Ω p (M; V ) p (T x (M); V ) (U; x 1,..., x n ) M x U ω x = η x = a i1 i p (x)(dx i 1 ) x (dx ip ) x, i 1 < <i p b i1 i p (x)(dx i 1 ) x (dx ip ) x i 1 < <i p 1.2.4 a i1 i p, b i1 i p U V C (fω + gη) x = (f(x)a i1 i p (x) + g(x)b i1 i p (x))(dx i 1 ) x (dx i p ) x. i 1 < <i p fa i1 i p + gb i1 i p U V C fω + gη φ Ω p (M; V 1 ), ψ Ω q (M; V 2 ) A(φ ψ) Ω p+q (M; V 3 ) A( ) C (M) x U φ x = ψ x = c i1 i p (x)(dx i 1 ) x (dx i p ) x, i 1 < <i p d j1 j q (x)(dx j 1 ) x (dx j q ) x j 1 < <j q 1.2.4 c i1 i p d j1 j q U V 1 V 2 C x U = A(φ ψ) x i 1 < <i p j 1 < <j q A((c i1 i p (x)(dx i 1 ) x (dx ip ) x )
10 1 Euclid (d j1 j q (x)(dx j 1 ) x (dx j q ) x )) = i 1 < <i p j 1 < <j q A(c i1 i p (x), d j1 j q (x)) (dx i 1 ) x (dx i p ) x (dx j 1 ) x (dx j q ) x A(φ ψ) x (dx k 1 ) x (dx k p+q ) x A(c i1 i p (x), d j1 j q (x)) U V 3 C A(φ ψ) V 3 M p + q 1.2.6 V F M N C x M F df x : T x (M) T F (x) (N) 1.1.6 ω Ω p (N; V ) (F ω) x = (df x ) ω F (x) (F ω) x p (T x (M); V ) F ω Ω p (M; V ) m = dim M, n = dim NF ω Ω p (M; V ) F (U) U M N (U; x 1,..., x m ) (U ; y 1,..., y n ) ( ) x (F ω) x x i,, 1 x x i p x i 1,..., i p U V C x U ( ) (y j F ) df x x i = (x) x i y j x j=1 F (x) ( ) (F ω) x x i,, 1 x x ip ( ( ) x ( )) = ω F (x) df x x i,, df 1 x x x ip x (y j 1 F ) = (x) (yj p F ) (x) ω x i F (x) 1 x ip y j,, 1 F (x) j 1,,j p =1 y jp F (x) U V C F ω Ω p (M; V ) 1.2.7 1.2.6 F ωω F
1.2 11 1.2.8 V F : M M, G : M M C ω Ω p (M, V ) (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) 1.2.9 V 1, V 2, V 3 A : V 1 V 2 V 3 F M N C φ Ω p (N; V 1 ), ψ Ω q (N; V 2 ) F (A(φ ψ)) = A((F φ) (F ψ)) 1.1.6 x M F (A(φ ψ)) x = (df x ) (A(φ ψ) F (x) ) = (df x ) (A(φ F (x) ψ F (x) )) = A(((dF x ) φ F (x) ) ((df x ) ψ F (x) )) = A((F φ) x (F ψ) x ) = A((F φ) (F ψ)) x F (A(φ ψ)) = A((F φ) (F ψ)) 1.2.10 V M n ω Ω p (M; V ) dω Ω p+1 (M; V ) () M (U; x 1,..., x n ) ω ω x = a i1 i p (x)(dx i 1 ) x (dx ip ) x i 1 < <i p (dω) x = i 1 < <i p i=1 a i1 i p (x)(dx i ) x i x (dx i 1 ) x (dx i p ) x
12 1 Euclid i 1 < <i p i=1 a i1 i p (x)(dx i ) x i x (dx i 1 ) x (dx ip ) x p+1 (T x (M), V ) 1.1.12 a i1 i p (x) ω x ( ) a i1 i p (x) = ω x x i,, 1 x x i p x (1 i 1 < < i p n) i 1,..., i p a i1 i p (x) (V ; y 1,..., y n ) M 1 ( ) b i1 i p (x) = ω x y i,, 1 x y i p x ω x = b i1 i p (x)(dy i 1 ) x (dy ip ) x i 1 < <i p x U V T x (M) 2 x i x y i x y i = x j=1 (dy i ) x = j=1 x j y i (x) x j x y i x j (x)(dxj ) x b i1 i p (x) = j 1,,j p =1 i=1 x j 1 xjp (x) y i 1 y (x)a ip j 1 j p (x) x j yi (x) yi x (x) = δ k jk ( ) = 1 p! i 1 < <i p i=1 i,i 1,...,i p =1 b i1 i p (x)(dy i ) y i x (dy i 1 ) x (dy ip ) x b i1 i p (x)(dy i ) y i x (dy i 1 ) x (dy ip ) x
= 1 p! = 1 p! i,i 1,...,i p=1 i 1,...,i p =1 j,j 1,...,jp=1 k,k 1,...,kp=1 1.2 13 x j y (x) { } x j 1 xjp (x) i x j y i 1 y (x)a i j p 1 j p (x) y i x k (x)(dxk ) x yi1 x k 1 (x)(dxk 1 ) x yip j 1,...,jp=1 k,k 1,...,kp=1 { x j 1 xjp (x) x k y i 1 y (x)a ip j 1 j p (x) x k (x)(dxk p ) p x } (dx k ) x yi 1 x k 1 (x)(dxk 1 ) x yip x k p (x)(dxkp ) x. i r =1 i r =1 x j r yir (x) ir y x (x) = δ kr j r k r ( ) x j r y i r (x) x k ir y x (x) = n x j ( ) r y i r (x) (x) kr ir i r =1 y x k kr x x jr = y (x) 2 y ir i r x k x (x) k r i r =1 k k r (dx k ) x (dx k 1 ) x (dx k p ) x k k r ( ) = 1 p! = 1 p! = i 1,...,i 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 i 1 y (x)a j 1 j p (x) i p x k (dx k ) x yi 1 x k 1 (x)(dxk 1 ) x yip x k p (x)(dxk p ) 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 k p ) x. dω dω dω p + 1 dω 1.2.11 1.2.10 p = 0 ω Ω 0 (M; V ) M V C dω (dω) x = i=1 ω x i (x)(dxi ) x
14 1 Euclid 1.2.12 V M 1.2.10 d : Ω p (M; V ) Ω p+1 (M; V ) 1.2.13 V F M NΓ F F : Ω p (N; V ) Ω p (M; V ) m = dim M, n = dim Nω Ω p (N; V ) 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 j p ω U ω y = a j1 j p (y)(dy j 1 ) y (dy j p ) y j 1 < <j p 1.2.6 x U ( ) (F ω) x x i,, 1 x x i p x (y j 1 F ) = (x) (yj p F ) (x) a x i 1 x i p j1 j p (F (x)). = (d(f ω)) x j 1,,j p=1 m j 1,,j p =1 i 1 < <i p i=1 { (y j 1 F ) x i x i 1 (dx i ) x (dx i 1 ) x (dx i p ) x. y } (x) (yjp F ) (x) a x i p j1 j p (F (x)) 2 (y j r F ) (x) i i x i r (dx i ) x (dx i 1 ) x (dx i p ) x i i r x ir = (d(f ω)) x m j 1,,j p =1 i 1 < <i p i=1 (y j 1 F ) x i 1 (x) (yjp F ) (x) a j 1 j p (F (x)) x i p x i = 1 p! (dx i ) x (dx i 1 ) x (dx i p ) x m m (y j 1 F ) x i 1 j 1,,j p =1 i 1,,i p =1 i=1 (dx i ) x (dx i 1 ) x (dx ip ) x. (x) (yj p F ) (x) a j 1 j p (F (x)) x ip x i
1.2 15 (dω) y = 1.1.6 = 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 j p ) y (F (dω)) x = (df x ) (dω) F (x) = 1 a j1 j p (F (x))(df p! y j x ) (dy j ) F (x) j 1,,j p =1 j=1 (df x ) (dy j 1 ) F (x) (df x ) (dy j p ) F (x). (df x ) (dy j ) F (x) = (dy j ) F (x) df x = d(y j F ) x = m i=1 (y j F ) (x)(dx i ) x i x = = a j1 j p (F (x))(df j=1 y j x ) (dy j ) F (x) m a j1 j p (F (x)) (yj F ) (x)(dx i ) j=1 i=1 y j x i x m a j1 j p (F (x)) (x)(dx i ) i=1 x i x = 1 p! (F (dω)) x m m j 1,,j p =1 i 1,,i p =1 i=1 (y j 1 F ) x i 1 (dx i ) x (dx i 1 ) x (dx i p ) x = (d(f ω)) x. F (dω) = d(f ω) (x) (yjp F ) (x) a j 1 j p (F (x)) (x) x ip x i 1.2.14 1.2.13 M N F : M N N M M M M N
16 1 Euclid 1.2.15 V M d : Ω p (M; V ) Ω p+1 (M; V ) 1.2.10 d 1.2.16 V 1, V 2, V 3 A : V 1 V 2 V 3 Mφ Ω p (M; V 1 ), ψ Ω q (M; V 2 ) da(φ ψ) = A(dφ ψ) + ( 1) p A(φ dψ) n = dim M (U; x 1,..., x n ) M Uφ ψ φ x = ψ x = a i1 i p (x)(dx i 1 ) x (dx i p ) x i 1 < <i p b j1 j q (x)(dx j 1 ) x (dx j q ) x j 1 < <j q A(φ ψ) x = i 1 < <i p j 1 < <j q A(a i1 i p (x), b j1 j q (x)) (dx i 1 ) x (dx i p ) x (dx j 1 ) x (dx j q ) x k 1 < < k p+q {i 1,..., i p, j 1,..., j q } {k 1,..., k p+q } ( ) i1 i p j 1 j q sgn = 0 k 1 k p k p+1 k p+q = A(φ ψ) x sgn k 1 < <k p+q i 1 < <i p j 1 < <j q ( i1 i p j 1 j q k 1 k p k p+1 k p+q A(a i1 i p (x), b j1 j q (x))(dx k 1 ) x (dx k p+q ) x ) da(φ ψ) x
1.2 17 = = = sgn k 1 < <k p+q i 1 < <i p j 1 < <j q i=1 ( i1 i p j 1 j q k 1 k p k p+1 k p+q A(a i1 i p (x), b j1 j q (x)) (dx i ) x i x (dx k 1 ) x (dx k p+q ) x ( i1 i p j 1 j q sgn k 1 < <k p+q i 1 < <i p j 1 < <j q { ( ai1 i p i=1 A x i (x), b j1 j q (x) k 1 k p k p+1 k p+q ) ( )} + A a i1 i p (x), b j 1 j q (x) x i (dx i ) x (dx k 1 ) x (dx k p+q ) x { ( ) ( ai1 i A p (x), b x i j1 j q (x) + A a i1 i p (x), b )} j 1 j q (x) x i i 1 < <i p j 1 < <j q i=1 ) ) (dx i ) x (dx i 1 ) x (dx i p ) x (dx j 1 ) x (dx j q ) x = A(dφ ψ) x + ( 1) p A(φ dψ) x. da(φ ψ) = A(dφ ψ) + ( 1) p A(φ dψ) 1.2.17 V M d d = 0 d : Ω p (M; V ) Ω p+1 (M; V ) n = dim Mω Ω 0 (M; V ) d 2 ω = 0 (U; x 1,..., x n ) M x U (d 2 ω) x = (dω) x = i=1 j=1 j=1 ω x j (x)(dxj ) x 2 ω x i x j (x)(dxi ) x (dx j ) x. 2 ω (x) i j (dx i ) x i x j x (dx j ) x i j (d 2 ω) x = 0 d 2 ω = 0 p > 0 ω Ω p (M; V ) U ω x = a i1 i p (x)(dx i 1 ) x (dx i p ) x i 1 < <i p (dω) x = = i 1 < <i p i=1 a i1 i p (x)(dx i ) x i x (dx i 1 ) x (dx ip ) x i 1 < <i p (da i1 i p ) x (dx i 1 ) x (dx ip ) x
18 1 Euclid 1.2.10 (d 2 ω) x = i 1 < <i p (d 2 a i1 i p ) x (dx i 1 ) x (dx i p ) x + (da i1 i p ) x i 1 < <i p = 0. (d 2 ω) x = 0 d 2 ω = 0 p ( 1) j (dx i 1 ) x (d 2 x i j ) x (dx i p ) x j=1 1.2.18 V Mω Ω p (M; V ) p = 0 X X(M) p = 1 X, Y X(M) dω(x) = Xω. dω(x, Y ) = X(ω(Y )) Y (ω(x)) ω([x, Y ]). p = 0 l = dim V V v 1,..., v l ω l ω = ω i v i i=1 (ω i C (M)) X X(M) l l l dω(x) = dω i v i (X) = dω i (X)v i = (Xω i )v i = Xω. i=1 i=1 i=1 p = 1 ω ω = i a i dx i ω dω dω = i,j a i x j dxj dx i X, Y X = i ξ i x, i Y = j η j x j dω(x, Y ) = i,j a i x j (ξj η i η j ξ i )
1.3 Euclid 19 ω(y ) = i a i η i X(ω(Y )) = i,j ξ j x (a iη i ) = j i,j ( ξ j ai η i ) x j ηi + a i. x j X Y Y (ω(x)) = i,j ( η j ai ξ i ) x j ξi + a i x j [X, Y ] = i,j = i,j ξ i ηj x i x η j ξi j i,j x j x i ( ) ξ j ηi ξi ηj xj x j x i ω([x, Y ]) = i,j ( ) a i ξ j ηi ξi ηj. xj x j dω(x, Y ) = X(ω(Y )) Y (ω(x)) ω([x, Y ]) 1.2.19 1.2.18 p > 0 X 1,... X p+1 X(M) = dω(x 1,..., X p+1 ) p+1 i=1 ( 1) i 1 X i (ω(x 1,..., ˆX i,..., X p+1 )) + i<j( 1) i+j ω([x i, X j ], X 1,..., ˆX i,..., ˆX j,..., X p+1 ). 1.3 Euclid N Euclid R N 1.2
20 1 Euclid M N Euclid R N M x T x M R N M R N M 0 M X X Ω 0 (M; R N ) dx Ω 1 (M; R N ) M x R 3 T x M T x M T x M M x dx dx = X + A X. X T x MA X T x M M Y ( X)(Y ) = Y X M fm Y d(fx)(y ) = Y (fx) = (Y f)x + fy X = df(y )X + fdx(y ) = (dfx + fdx)(y ) d(fx) = dfx + fdx (fx) = dfx + f X A fx = fa X M A M A A M x A : T x M T x M T x M ; (X, Y ) A X (Y ) Mξ R N M 0 dξ Ω 1 (M; R N ) Mdξ dξ = B ξ + ξ. B ξ T x M ξ T x M M X ( ξ)(x) = Xξ M fm d(fξ) = dfξ + fdξ B fξ = fb ξ (fξ) = dfξ + f ξ
1.3 Euclid 21 B B M x B : T x M T x M T x M ; (X, ξ) B ξ (X) 1.3.1 R N R N M X, Y ξ, η d(x Y ) = X Y + X Y, d(ξ η) = ξ η + ξ η. d(x Y ) = dx Y + X dy = ( X + A X ) Y + X ( Y + A Y ) = X Y + X Y. d(ξ η) = dξ η + ξ dη = (B ξ + ξ) η + ξ (B η + η) = ξ η + ξ η. 1.3.2 R N M X ξ A X ξ + X B ξ = 0. X ξ X ξ = 0 0 = d(x ξ) = dx ξ + X dξ = ( X + A X ) ξ + X (B ξ + ξ) = A X ξ + X B ξ. 1.3.3 R N M X, Y [X, Y ] = X Y Y X, A(X, Y ) = A(Y, X). A dx(y ) = X(Y ) + A X (Y ) = Y X + A X (Y )
22 1 Euclid [X, Y ] = XY Y X = dy (X) dx(y ) = X Y + A Y (X) Y X A X (Y ) = ( X Y Y X) + (A Y (X) A X (Y )). [X, Y ] = X Y Y X, A(X, Y ) = A(Y, X) 1.4 3 Euclid 3 Euclid 1.2 M U e 1, e 2 Ω 0 (U; R 3 ) θ 1, θ 2 Ω 1 (U; R) R 3 e 3 = e 1 e 2 e 3 Ω 0 (U; R 3 ) e 1, e 2, e 3 R 3 M R 3 P P M R 3 P Ω 0 (M; R 3 ) P dp Ω 1 (M; R 3 ) U dp = θ 1 e 1 + θ 2 e 2 I = dp dp M I I R 3 U I = dp dp = θ 1 θ 1 + θ 2 θ 2 U e 1, e 2, e 3 R 3 M X [de 1 (X) de 2 (X) de 3 (X)] = [e 1 e 2 e 3 ]ω(x) ω Ω 1 (U; M 3 (R)) M 3 (R) 3 d[e 1 e 2 e 3 ] = [de 1 de 2 de 3 ] = [e 1 e 2 e 3 ]ω
1.4 3 Euclid 23 σ = [e 1 e 2 e 3 ] Ω 0 (U; M 3 (R)) σ dσ = σω S S n 1 n σ σ = 1 3 σ Ω 0 (U; M 3 (R)) 1.2.16 0 = d(σ σ) = (dσ )σ + σ dσ = (σω) σ + σ σω = ω σ σ + σ σω = ω + ω. ω U 1 3 o(3) ω Ω 1 (U; o(3)) ω = (ω j i ) de 1 = ω 2 1e 2 + ω 3 1e 3, de 2 = ω 1 2e 1 + ω 3 2e 3, de 3 = ω 1 3e 1 + ω 2 3e 2 e 1 = ω 2 1e 2, e 2 = ω 1 2e 1, A e1 = ω 3 1e 3, A e2 = ω 3 2e 3 e 3 = 0, B e3 = ω 1 3e 1 + ω 2 3e 2 M X U X = X 1 e 1 + X 2 e 2 X = (X 1 e 1 ) + (X 2 e 2 ) = (dx 1 )e 1 + X 1 e 1 + (dx 2 )e 2 + X 2 e 2 = (dx 1 )e 1 + X 1 ω 2 1e 2 + (dx 2 )e 2 + X 2 ω 1 2e 1 = (dx 1 + ω 1 2X 2 )e 1 + (dx 2 + ω 2 1X 1 )e 2. M ωω 1.4.1 σ = [e 1 e 2 e 3 ] σ = [ē 1 ē 2 ē 3 ] d σ = σ ω ω Ω 1 (U; o(3)) σ = σf f Ω 0 (U; M 3 (R)) f 3 O(3) ω = f 1 df + f 1 ωf. σ = σf d σ = dσf + σdf = σωf + σdf = σ(ωf + df).
24 1 Euclid d σ = σ ω = σf ω. σf ω = σ(df + ωf) f ω = df + ωf ω = f 1 df + f 1 ωf 1.4.2 M M e 1, e 2, e 3 ω e 1, e 2, e 3 1.2.17 d d = 0 1.4.3 dθ 1 = θ 2 ω 1 2, dθ 2 = θ 1 ω 2 1, θ 1 ω 3 1 + θ 2 ω 3 2 = 0. 1.2.17 1.2.16 0 = d(dp ) = d(θ 1 e 1 + θ 2 e 2 ) = dθ 1 e 1 θ 1 de 1 + dθ 2 e 2 θ 2 de 2 = dθ 1 e 1 θ 1 (ω 2 1e 2 + ω 3 1e 3 ) + dθ 2 e 2 θ 2 (ω 1 2e 1 + ω 3 2e 3 ) = (dθ 1 θ 2 ω 1 2)e 1 + (dθ 2 θ 1 ω 2 1)e 2 (θ 1 ω 3 1 + θ 2 ω 3 2)e 3. e 1, e 2, e 3 0 1.4.4 ω1, 3 ω2 3 U 1 A = ω 3 1 = b 3 11θ 1 + b 3 12θ 2, ω 3 2 = b 3 21θ 1 + b 3 22θ 2 2 i,j=1 b 3 ijθ i θ j e 3 A A = θ 1 A e1 + θ 2 A e2 = θ 1 ω1e 3 3 + θ 2 ω2e 3 3 = θ 1 (b 3 11θ 1 + b 3 12θ 2 )e 3 + θ 2 (b 3 21θ 1 + b 3 22θ 2 )e 3 = 2 b 3 ijθ i θ j e 3. i,j=1
1.4 3 Euclid 25 1.4.5 1.4.3 1.4.4 ω 3 1, ω 3 2 0 = θ 1 ω1 3 + θ 2 ω2 3 = θ 1 (b 3 11θ 1 + b 3 12θ 2 ) + θ 2 (b 3 21θ 1 + b 3 22θ 2 ) = b 3 12θ 1 θ 2 + b 3 21θ 2 θ 1 = (b 3 12 b 3 21)θ 1 θ 2. b 3 12 = b 3 21 1.3.3 1.4.6 dω 1 2 = (b 3 11b 3 22 b 3 12b 3 21)θ 1 θ 2. 1.2.17 0 = d(de 1 ) = d(ω1e 2 2 + ω1e 3 3 ) = dω1e 2 2 ω1 2 de 2 + dω1e 3 3 ω1 3 de 3 = dω1e 2 2 ω1 2 (ω2e 1 1 + ω2e 3 3 ) + dω1e 3 3 ω1 3 (ω3e 1 1 + ω3e 2 2 ) (ωj i = ω j i ωj i ω j i = 0) = (dω1 2 ω1 3 ω3)e 2 2 + (dω1 3 ω1 2 ω2)e 3 3. dω 2 1 ω 3 1 ω 2 3 = 0, dω 3 1 ω 2 1 ω 3 2 = 0 dω2 1 = dω1 2 = ω1 3 ω3 2 = ω1 3 ω2 3 = (b 3 11θ 1 + b 3 12θ 2 ) (b 21 θ 1 + b 3 22θ 2 ) = (b 3 11b 3 22 b 3 12b 3 21)θ 1 θ 2. 1.4.7 (b 3 ij) M Gauss (b 3 ij) 1/2 Gauss M Gauss K 1.4.6 dω 1 2 = Kθ 1 θ 2 M e 3 ±1 Gauss e 1, e 2, e 3 1.4.8 R 3 M x O x (M) = {(x; ẽ 1, ẽ 2 ) ẽ 1, ẽ 2 T x M } O(M) = O x (M) x M
26 1 Euclid O(M) M R 3 R 3 O(M) R 2 e 1, e 2 R 2 T x M u (u(e 1 ), u(e 2 )) O x (M) = {u u R 2 T x M } R 2 a O(2) R 2 T x M u ua O x (M) O(2) O x (M) O(2) O(M) u O x (M) x π : O(M) M π C x Mπ 1 (x) O(2) O(M) M O(M) ẽ 1, ẽ 2 ẽ 1, ẽ 2 Ω 0 (O(M); R 3 ) M U O(M) ẽ 3 = ẽ 1 ẽ 2 ẽ 3 ẽ 3 Ω 0 (O(M); R 3 ) σ = [ẽ 1 ẽ 2 ẽ 3 ] Ω 0 (O(M); M 3 (R)) d σ = σ ω ω Ω 1 (O(M); M 3 (R)) ω O(3) ω o(3) ω Ω 1 (O(M); o(3)) Mω dẽ 1 = ω 2 1ẽ 2 + ω 3 1ẽ 3, dẽ 2 = ω 1 2ẽ 1 + ω 3 2ẽ 3, dẽ 3 = ω 1 3ẽ 1 + ω 2 3ẽ 2 M U e 1, e 2, e 3 ω O(M) ẽ 1, ẽ 2, ẽ 3 ω s = (e 1, e 2 ) U O(M) C π s = 1 U s ẽ i = ẽ i s = e i s σ = σ σω = dσ = ds σ = s d σ = s ( σ ω) = σs ω. ω = s ω M U e 1, e 2 ω s = (e 1, e 2 ) U O(M) O(M) ω s O(M) ωm e 1, e 2 ωω ω ω ω
1.4 3 Euclid 27 M 1 θ 1, θ 2 O(M) R 2 1 θ θ u (X) = u 1 (dπ u (X)) (u O(M), X T u (O(M))) O(M) R 2 1 θ u O(M) R 2 T π(u) M θ = θ 1 e 1 + θ 2 e 2 M s = (e 1, e 2 ) (s θ)(ei ) = s 1 (dπ s ds(e i )) = s 1 d(π s) s (e i ) = s 1 (e i ) = e i. (s θ)(ei ) = (s θ1 )(e i )e 1 + (s θ2 )(e i )e 2 (s θj )(e i ) = δ j i s θ1, s θ2 e 1, e 2 s θi = θ i 1.4.9 d θ 1 = θ 2 ω 1 2, d θ 2 = θ 1 ω 2 1, θ1 ω 3 1 + θ 2 ω 3 2 = 0. θu O(M) dπ u = u θ u = u( θ 1 e 1 + θ 2 e 2 ) = θ 1 ue 1 + θ 2 ue 2 = θ 1 ẽ 1 + θ 2 ẽ 2 π : O(M) M R 3 π Ω 0 (O(M); R 3 ) 1.2.17 1.2.16 0 = d(dπ) = d( θ 1 ẽ 1 + θ 2 ẽ 2 ) = d θ 1 ẽ 1 θ 1 dẽ 1 + d θ 2 ẽ 2 θ 2 dẽ 2 = d θ 1 ẽ 1 θ 1 ( ω 2 1ẽ 2 + ω 3 1ẽ 3 ) + d θ 2 ẽ 2 θ 2 ( ω 1 2ẽ 1 + ω 3 2ẽ 3 ) = (d θ 1 θ 2 ω 1 2)ẽ 1 + (d θ 2 θ 1 ω 2 1)ẽ 2 ( θ 1 ω 3 1 + θ 2 ω 3 2)ẽ 3. ẽ 1, ẽ 2, ẽ 3 0 1.4.10 M s = (e 1, e 2 ) 1.4.9 s M 1.4.3
2 3 Euclid 1 2 2.1 Lie Lie Lie Lie Lie 2.1.1 G G G G; (x, y) xy, G G; x x 1 C G Lie ( e ) 2.1.2 2.1.1 Lie Lie 2.1.3 V V GL(V ) Lie GL(R n ) GL(n, R) GL(V ) dim V = n End(V ) = {f : V V f } V End(V ) End(V ) n 2 Euclid End(V ) det : End(V ) R End(V ) GL(V ) = {f End(V ) detf 0} 28
2.1 Lie Lie 29 GL(V ) End(V ) GL(V ) n 2 GL(V ) GL(V ) GL(V ); (x, y) xy End(V ) C GL(V ) GL(V ); x x 1 End(V ) C (Cramer ) 2.1.4 Lie G g L g, R g L g : G G; x gx, R g : G G; x xg gg XG g (dl g ) x (X x ) = X gx (x G) (dr g ) x (X x ) = X xg (x G) 2.1.5 Lie G e (U; x 1,..., x n ) g G (L g (U); x 1 L 1 g,..., x n L 1 g ) g 2.1.6 g [, ] : g g g X, Y, Z g [X, Y ] = [Y, X], [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0 g Lie Lie g h [, ] h g Lie 2.1.7 M X(M) Lie [, ] Lie 2.1.8 V End(V ) X, Y [X, Y ] = XY Y X End(V ) Lie Lie gl(v ) gl(r n ) gl(n, R)
30 2 [, ] : End(V ) End(V ) End(V ) End(V ) X, Y, Z [X, Y ] = XY Y X = [Y, X], [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = [XY Y X, Z] + [Y Z ZY, X] + [ZX XZ, Y ] = XY Z Y XZ ZXY + ZY X + Y ZX ZY X XY Z + XZY +ZXY XZY Y ZX + Y XZ = 0. End(V ) [, ] Lie 2.1.9 G Lie G g g Lie X(G) Lie α : g T e (G); X X e dim g = dim T e (G) = dim G g X(G) X, Y g g G (dl g ) x (X x ) = X gx (dl g ) x (Y x ) = Y gx (x G) Lie (dl g ) x ([X, Y ] x ) = [X, Y ] gx (x G) [X, Y ] g g X(G) Lie α X g, α(x) = 0 g G X g = (dl g ) e (X e ) = 0 X = 0 Kerα = 0 α X T e (G) X g = (dl g ) e (X) X α (U; x 1,..., x n ) G e G G G; (x, y) xy e V V V = {xy x, y V } U X = ξ i x i i=1 e
2.1 Lie Lie 31 2.1.5 g G (L g (V ); x 1 L 1 g,..., x n L 1 g ) g y i = x i L 1 g gx L g(v ) (x V ) X gx = (dl gx ) e (X) = d(l g L x ) e (X) = (dl g ) x (dl x ) e (X) = (dl g ) x ξ i (xj L x ) (e) i,j=1 x i x j x = ξ i (xj L x ) (e)(dl x i g ) x x j. x i,j=1 ( (dl g ) x ) ( ) x j y k = x x j (y k L g ) = δ jk x (dl g ) x x j = x y j gx X gx = i,j=1 ξ i (xj L x ) (e) x i y j. gx G G G; (x, y) xy = L x (y) C i, j C j V R; x (xj L x ) (e) x i L g (V ) R; gx i=1 ξ i (xj L x ) (e) x i C X G X α 2.1.10 Lie G Lie g Lie G Lie 2.1.11 Lie C f : G H f Lie f f 1 f 1 Lie f Lie Lie G HLie f : g h [f(x), f(y )] = f([x, Y ]) (X, Y g) f Lie f f 1 f Lie Lie g h
32 2 2.1.12 M X M IM c : I M dc dt (t) = X c(t) (t I) c X 2.1.13 M X M t 0 M x M X c : I M t 0 I, c(t 0 ) = x c 1, c 2 : I M c 1 (t 0 ) = c 2 (t 0 ) = x X c 1 = c 2 x M (U; x 1,..., x n ) X U X x = a i (x) i=1 x i (x U) x U dc dt (t) = X c(t) (t I) c i=1 d(x i c(t)) dt x i = c(t) a i (c(t)) i=1 x i c(t) d(x i c(t)) dt = a i (c(t)), x i c(t 0 ) = x i (x) (1 i n) Euclid a i C t 0 I c : I U c M (U; x 1,..., x n ) c 1 (I), c 2 (I) U x 1,..., x n dc 1 dt (t) = X c 1 (t), dc 2 dt (t) = X c 2 (t) (t I) Euclid c 1 = c 2 c 1 (I), c 2 (I) M 1 t 0 < s, s I 0 < ε t 0 < t 1 <... < t k = s I i = (t i 1 ε, t i + ε) (1 i k) i c 1 (I i ), c 2 (I i ) M 1 c 1 (t 1 ) = c 2 (t 1 ),..., c 1 (t k ) = c 2 (t k ) c 1 (s) = c 2 (s) t 0 > s, s I c 1 (s) = c 2 (s) c 1 = c 2
2.1 Lie Lie 33 2.1.14 R Lie R Lie G Lie G 2.1.15 G Lie Lie g Lie g G 1 1 X g X c : R G c(0) = e 1 c G X g c G c 2.1.9 dc (0) g dt X c (1),(2) (1) X g X c : R G c(0) = e c G (2) 2 (1) 2.1.13 δ > 0 X a : ( δ, δ) G a(0) = e s < δ/2 s 1 b 1 (t) = a(s + t), b 2 (t) = a(s)a(t) ( s < δ/2) t b 1 (t) X ( ) d d dt b 2(t) = (dl a(s) ) a(t) dt a(t) = (dl a(s) ) a(t) (X a(t) ) = X a(s)a(t) t b 2 (t) X 2.1.13 b 1 (0) = a(s) = a(s)e = a(s)a(0) = b 2 (0) b 1 (t) = b 2 (t) ( t < δ/2). a(s + t) = a(s)a(t) = a(t)a(s) ( s, t < δ/2) t R t/k < δ/2 k ( ) t k c(t) = a k c : R G t R t/k < δ/2, t/l < δ/2 k, l ( ) t k ( ) t kl ( ) t l a = a = a k kl l
34 2 c kc(0) = a(0) = e c G t R t/k < δ/2 kt I s I s/k < δ/2 s I c(s) = a(s/k) k c I C c : R G C s, t R s/k, t/k < δ/4 k s/k, t/k, (s + t)/k < δ/2 c(s)c(t) = ( ) s k ( ) t k ( a a = a k k = ( ) s + t k a = c(s + t). k ( s k ) ( )) t k a k c : R G G c X s R t (s δ, s + δ) c(t) = c(s)c(t s) = L c(s) (c(t s)) ( d dt c(t) d = (dl c(s) ) e t=s dt c(t) t=0 ) = (dl c(s) ) e (X e ) = X c(s). c X (2) X g G c dc(0) = X dt e c g X G c g X X e = dc (0) dt (1) g X G c dc(0) = X dt e c X X G c 2.1.16 GL(n, R) GL(n, R) gl(n, R) X gl(n, R) = T e (GL(n, R)) GL(n, R) X 2.1.9 X g = gx (g GL(n, R)) X GL(n, R) c dc(t) dt : e A = = c(t)x (t R), c(0) = e n=0 1 n! An c(t) = e tx 2.1.17 G Lie Lie g X g 2.1.15 X c : R G c(0) = e exp X = c(1) exp : g G exp Lie G 2.1.18 2.1.16 GL(n, R) Lie gl(n, R) X e tx GL(n, R) 2.1.19 G Lie Lie g X g 2.1.15 G t exp tx
2.1 Lie Lie 35 X g 2.1.15 G t c(t, X) s R d dt c(st, X) = sx c(st,x) (t R) t c(st, X) sx g c(st, X) = c(t, sx) t = 1 c(s, X) = c(1, sx) = exp sx. X g G t exp tx 2.1.20 G Lie Lie g G exp : g G C X 1,..., X n g u 1,..., u n u 1,..., u n g G (U; x 1,..., x n ) e U, x 1 (e) =... = x n (e) = 0 n u i X i g c(t; u 1,..., u n ) i=1 d dt c(t; u 1,..., u n ) = u i (X i ) c(t;u1,...,u n ) i=1 X i U (X i ) x = a ji (x) j=1 x j x c(t; u 1,..., u n ) d dt x j(c(t; u 1,..., u n )) = u i a ji (c(t; u 1,..., u n )) i=1 : (t, u 1,..., u n ) (x 1 (c(t; u 1,..., u n )),..., x n (c(t; u 1,..., u n ))) 0 C exp( u i X i ) = c(1; u 1,..., u n ) i=1 exp : g G 0 N C X g p 1 p X N X O 1 p O N exp(z) = exp ( ) p 1 p Z (Z O) exp O C exp : g G C
36 2 2.1.21 Lie G Lie g G exp g 0 G e X g = T 0 (g) d exp 0 (X) = d dt exp(tx) t=0 = X e = α(x) d exp e = α : g T e (G) 2.1.9 d exp e exp g 0 G e 2.1.22 G Lie Lie g X, Y g, f C (G), g G (Xf)(g) = d dt t=0 f(g exp tx) ([X, Y ]f)(g) = s t f(g exp sx exp ty (exp sx) 1 ) s=t=0 X g t g exp tx t = 0 (Xf)(g) = X g (f) = d dt f(g exp tx) t=0 (XY f)(g) = f(g exp sx exp ty ) ( ) s t s=t=0 = ( ) t s f(g exp sx exp ty (exp sx) 1 exp sx) s=0 t=0 = ( t s f(g exp sx exp ty (exp sx) 1 ) s=0 + s f(g exp ty exp sx) s=0 ) t=0 (Leibniz ) = s t f(g exp sx exp ty (exp sx) 1 ) +(Y Xf)(g). s=t=0 ([X, Y ]f)(g) = s t f(g exp sx exp ty (exp sx) 1 ). s=t=0
2.1 Lie Lie 37 2.1.23 Lie G G Lie g X, Y [X, Y ] = 0 f C (G), g G ([X, Y ]f)(g) = s t f(g exp sx exp ty (exp sx) 1 ) = f(g exp ty ) = 0. s t s=t=0 s=t=0 [X, Y ] = 0 2.1.24 Lie g X, Y [X, Y ] = 0 g 2.1.23 Lie Lie 2.1.25 Lie Lie Lie Lie 2.1.26 G, H Lie Lie g, h f : G H Lie 2.1.9α G : g T e (G), α H : h T e (H) df = αh 1 df e α G : g h Lie X g df e (X e ) T e (H) H df(x) g G x G X, Y g df g (X g ) = df g (dl g ) e (X e ) = d(f L g ) e (X e ). (f L g )(x) = f(gx) = f(g)f(x) = (L f(g) f)(x) df g (X g ) = d(f L g ) e (X e ) = d(l f(g) f)(x e ) = (dl f(g) ) e df e (X e ) = df(x) f(g). df g (X g ) = df(x) f(g), df g (Y g ) = df(y ) f(g) (g G) Lie df g ([X, Y ] g ) = [df(x), df(y )] f(g) (g G)
38 2 df e ([X, Y ] e ) = [df(x), df(y )] e 2.1.9 [df(x), df(y )] H df([x, Y ]) = [df(x), df(y )]. df : g h Lie 2.1.27 Lie f : G H df = αh 1 df e α G : g h f 2.1.26 Lie Lie 2.1.28 A, B, C Lie Lie a, b, c A a f : A B, g : B C Lie d(g f) = dg df : a c d(id A ) = αa 1 d(g f) = αc 1 = αc 1 d(id A ) e α A = α 1 A d(g f) e α A = α 1 C dg e α B α 1 B id Te (A) α A = id dg e df e α A df e α A = dg df 2.1.29 A, B Lie Lie a, b f : A B Lie df : a b Lie df d(f 1 ) = d(f f 1 ) = d(id B ) = id d(f 1 ) df = id d(f 1 ) = df 1 df Lie 2.1.30 G, H Lie Lie g, h f : G H Lie f(exp X) = exp(df(x)) (X g) exp G exp H Lie Lie ( 2.1.25) t f(exp tx) H 2.1.15 Y h f(exp tx) = exp ty (t R)
2.1 Lie Lie 39 t = 0 () = d dt f(exp tx) t=0 = df e (X e ) () = d dt exp ty t=0 = Y e. df e (X e ) = Y e df(x) = Y f(exp tx) = exp(tdf(x)) t = 1 f(exp X) = exp(df(x)) 2.1.31 Lie G V G GL(V ) Lie G Lie g V g gl(v ) Lie g 2.1.32 Lie g X ad(x)(y ) = [X, Y ], Y g ad(x) gl(g) ad : g gl(g) Lie X, Y, Z g [ad(x), ad(y )](Z) = ad(x)ad(y )(Z) ad(y )ad(x)(z) = [X, [Y, Z]] [Y, [X, Z]] = [X, [Y, Z]] + [Y, [Z, X]] = [Z, [X, Y ]] (Jacobi ) = ad([x, Y ])(Z) [ad(x), ad(y )](Z) = ad([x, Y ]) ad Lie 2.1.33 Lie g ad : g gl(g) g 2.1.34 Lie G g Ad(g) = d(l g R g 1) G Lie g Ad(g) GL(g) g exp(x)g 1 = exp(ad(g)x) (g G, X g) Ad : G GL(g) Lie Ad g L g R g 1 G 2.1.29 Ad(g) g Ad(g) GL(g) 2.1.30 g exp(x)g 1 = exp(ad(g)x) (g G, X g) 2.1.28 g, h G Ad(gh) = d(l gh R (gh) 1) = d(l g L h R g 1 R h 1) = d(l g R g 1 L h R h 1) = d(l g R g 1) d(l h R h 1) = Ad(g) Ad(h) Ad : G GL(g)
40 2 Ad C GL(g) g X 1,..., X n θ 1,..., θ n GL(g) R; u θ i (u(x j )) G R; g θ i (Ad(g)(X j )) C 2.1.9 θ i (Ad(g)(X j )) = θ i (α 1 G dl g dr g 1 α G (X j )) g C Ad g 2.1.22 X, Y g, f C (G), g G ([X, Y ]f)(g) = s t f(g exp sx exp ty (exp sx) 1 ) s=t=0 = ( ) f(g exp(ad(exp sx)ty )) s t t=0 s=0 = s ((Ad(exp sx)y )f(g)) s=0 = s ((Ad(exp sx)y )) s=0 f(g) d Ad(exp sx)y ds Ad ad = [X, Y ] = ad(x)(y ) s=0 d Ad(exp sx) = ad(x) ds s=0 2.1.35 Lie G Ad : G GL(V ) G 2.1.36 V GL(V ) 2.1.18 GL(V ) g GL(V ), X gl(v ) Ad(g)X = d dt (getx g 1 ) = d t=0 dt etgxg 1 = gxg 1. t=0 2.1.37 Lie H Lie G Lie H G H G 2.1.38 G Lie Lie g G Lie H ι : H G dι : h dι(h) Lie ι dι : h g 2.1.26 Lie H G dι : h dι(h) Lie
2.1 Lie Lie 41 2.1.39 2.1.38 dι(h) Lie H Lie dι h dι(h) 2.1.40 G Lie H G Lie G, H Lie g, h exp G, exp H X h exp G (X) = exp H (X) ι : H G ι Lie 2.1.30 X h ι(exp H (X)) = exp G (dι(x)) ι dι exp G (X) = exp H (X) 2.1.41 G Lie H G H G H Lie 2.1.42 2.1.41 Lie Lie Lie Lie 2.1.43 Lie G Lie H Lie g, h h = {X g exp tx H(t R), t exp tx H } H Lie h = {X g exp tx H(t R)} h = {X g exp tx H(t R), t exp tx H } h h 2.1.38 2.1.39X h, s R exp sx G (W ; x 1,..., x n ) V = {z W x i (z) = 0 (k + 1 i n)} H exp sxx 1,..., x k H t exp tx H s I exp tx V (t I) I W ; t exp tx C t x i (exp tx) I C t exp tx H C X h h h h = h H Lie H X g exp tx H(t R) t exp tx H h = {X g exp tx H(t R)}
42 2 2.1.44 Lie G Lie H, K Lie h, k H K G Lie Lie h k 2.1.45 Lie Lie 2.1.46 GL(n, R) 2.1.3gl(n, R) T e (GL(n, R)) gl(n, R) Lie GL(n, R) Lie g X gl(n, R) X g X g = (dl g ) e (X) (g GL(n, R)) : gl(n, R) g; X X Lie 2.1.9 GL(n, R) = α 1 Lie (i, j)- 1 0 n E ij {E ij 1 i, j n} gl(n, R) {x ij 1 i, j n} x ij gl(n, R) GL(n, R) gl(n, R) e GL(n, R) X gl(n, R) t R e + tx GL(n, R) g GL(n, R) X g = (dl g ) e (X) = (dl g ) e ( d dt (e + tx) t=0 ) = d dt L g(e + tx) = d (g + tgx) t=0 dt t=0 = x ij (gx) = x ik (g)x kj (X) g i,j=1 x ij i,j=1 k=1 X, Y gl(n, R), g GL(n, R). x ij g = = = [ X, Ỹ ] g ( ) x ik (g)x kj (Y ) k=1 x pr (g)x rq (X) x pq i,j,p,q=1 r=1 ( ) x ik (g)x kj (X) k=1 x pr (g)x rq (Y ) r=1 x pq x ij g x ir (g)x rk (X)x kj (Y ) x ir (g)x rk (Y )x kj (X) i,j=1 k,r=1 k,r=1 x ij (gxy gy X) i,j=1 x ij g x ij g
2.1 Lie Lie 43 = i,j=1 = [X, Y ] g. x ij (g[x, Y ]) x ij g [ X, Ỹ ] = [X, Y ] Lie 2.1.47 V n Lie GL(V ) GL(n, R) Lie gl(v ) gl(n, R) v 1,..., v n V f End(V ) f v 1,..., v n R(f) f[v 1,..., v n ] = [v 1,..., v n ]R(f) R : End(V ) End(R n ) R R Lie R : gl(v ) gl(n, R) R GL(V ) Lie R GL(V ) : GL(V ) GL(n, R) 2.1.48 2.1.46 Lie : gl(n, R) g Lie gl(n, R) Lie GL(n, R) Lie g gl(n, R) GL(n, R) Lie 2.1.47 V gl(v ) GL(V ) Lie 2.1.49 Lie g ρ : g gl(v ) v V h = {X g ρ(x)v = 0} h g Lie a, b R, X, Y h ρ(ax + by )v = aρ(x)v + bρ(y )v = 0, ρ([x, Y ])v = ρ(x)ρ(y )v ρ(y )ρ(x)v = 0 ax + by, [X, Y ] h h g Lie
44 2 2.1.50 Lie G ρ : G GL(V ) v V H = {g G ρ(g)v = v} H G Lie h H Lie h = {X g dρ(x)v = 0} g, h Hρ(gh 1 )v = ρ(g)ρ(h 1 )v = v gh 1 H H G ρ : G GL(V ) C ρ v : G V ; g ρ(g)v ρ v C H = ρ 1 v (v) G H G Lie ( 2.1.41 2.1.42) 2.1.43 X h t R exp tx H ρ(exp tx)v = v t = 0 2.1.30 0 = d dt t=0 ρ(exp tx)v = d dt etdρ(x) v = dρ(x)v. t=0 dρ(x)v = 0 X g t R ρ(exp tx)v = exp(tdρ(x))v = exp tx H 2.1.43 X h n=0 1 n! (tdρ(x))n v = v 2.1.51 V det : GL(V ) GL(R) = R {0} GL(V ) det tr : gl(v ) gl(r) = R det : GL(V ) GL(R) = R {0} GL(V ) X gl(v ) X : V V λ i (1 i k) p i k k det(e tx ) = (e λit ) p i = e p iλ i t. d dt det(etx ) = d t=0 dt det tr i=1 i=1 k e p iλ i t k = p i λ i = tr(x) i=1 t=0 i=1 2.1.52 V SL(V ) = {g GL(V ) detg = 1} 2.1.50 2.1.51 SL(V ) Lie SL(V ) SL(V ) Lie sl(v ) 2.1.50 2.1.51 sl(v ) = {X gl(v ) trx = 0} R n Lie SL(n, R), sl(n, R)
2.1 Lie Lie 45 2.1.53 V A : V V R G = {g GL(V ) A(gu, gv) = A(u, v) (u, v V )} G Lie g G Lie g = {X gl(v ) A(Xu, v) + A(u, Xv) = 0 (u, v V )} V V R M 2 (V, R) b, c R, B, C M 2 (V, R) (bb + cc)(u, v) = bb(u, v) + cc(u, v) (u, v V ) bb + cc M 2 (V, R) M 2 (V, R) g GL(V ), B M 2 (V, R) (ρ(g)b)(u, v) = B(g 1 u, g 1 v) (u, v V ) ρ(g)b M 2 (V, R) ρ(g) GL(M 2 (V, R)) g, h GL(V ), u, v V (ρ(gh)b)(u, v) = B((gh) 1 u, (gh) 1 v) = B(h 1 g 1 u, h 1 g 1 v) = (ρ(h)b)(g 1 u, g 1 v) = (ρ(g)(ρ(h)b))(u, v) ρ(gh) = ρ(g)ρ(h) ρ : GL(V ) GL(M 2 (V, R)) u, v V, B M 2 (V, R) g (ρ(g)b)(u, v) C ρ C ρ Lie ρ G = {g GL(V ) ρ(g)a = A} 2.1.50 G Lie G Lie ρ X gl(v ), B M 2 (V, R), u, v V (dρ(x)b)(u, v) = d dt (ρ(etx )B)(u, v) = d t=0 dt B(e tx u, e tx v) 2.1.50 = B(Xu, v) B(u, Xv) g = {X gl(v ) dρ(x)a = 0} = {X gl(v ) A(Xu, v) + A(u, Xv) = 0 (u, v V )}. 2.1.54 V A V V A O(V ) = O(V ; A) 2.1.53 O(V ) Lie O(V ) O(V ) Lie o(v ) = o(v ; A) 2.1.53 o(v ) = {X gl(v ) A(Xu, v) + A(u, Xv) = 0 (u, v V )} R n Lie O(n), o(n) t=0
46 2 2.1.55 O(n) n o(n) n 2.1.56 V A V SO(V ) = SO(V ; A) = SL(V ) O(V ; A) 2.1.44 SO(V ) Lie SO(V ) SO(V ) Lie so(v ) = so(v ; A) 2.1.44 so(v ) = sl(v ) o(v ) = o(v ) R n Lie SO(n), so(n) SO(n) O(n) O(n) 2.2 2.2.1 π E : E M M (1) E, Mπ E : E M C (2) k M p p U Φ U : π 1 E (U) U R k 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 Uπ 1 E (x) φ U π 1 E (x) : π 1 E (x) R k E M π E π 1 E (x) x k ranke 2.2.2 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
2.2 47 2.2.3 Lie G Lie g Lie φ : G g T G ; (g, X) X g φφ T G G 2.2.4 M T M = x M T 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 ) π 1 (U) u u = ξ i x i π(u) 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 i y i π(v) π 1 (V ) (y 1,..., y n, η 1,..., η n ) (x 1,..., x n, ξ 1,..., ξ n ) η i = ξ j yi 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 )
48 2 π : T M M C Φ U Φ U (u) U π(u) ( ) φ U ξ i = (ξ 1,..., ξ n ) x i x Uφ U π 1 (x) : π 1 (x) R n π : T M M M 2.2.5 π E : E M M C σ : M E π E σ = 1 M E E Γ(M, E) Γ(E) 2.2.6 M M MΓ(T M) M V p (T M, V ) = x M p (T x M, V ) ω p (T M, V ) ω p (T x M, V ) x M π(ω) = x π : p (T M, V ) M 2.2.4 π : p (T M, V ) M M Γ( p (T M, V )) = Ω p (M; V ) 2.2.7 M p M T p M, p M C X, Y X, Y M C, M Riemann (M,, ) Riemann Riemann Riemann Euclid 2.2.8 ι : M M M Riemann ( M, g) M x ι dι x : T x M T ι(x) M M Riemann g dι g = ι g M Riemann (M, g) ( M, g) Riemann 2.2.9 2.2.8 M Riemann M Riemann M Riemann Riemann (M, g) ( M, g) C ιm x dι x : T x M T ι(x) M ι(m, g) ( M, g) Riemann
2.2 49 2.2.10 ι : M ( M, g) Riemann ( M, g) Riemann 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 M M n ñm p p (U; x 1,..., x n ) ι U : U M U x ι ι(x) M U M x U T x M T x Mp M (Ũ; x1,..., xñ) U = {y Ũ xn+1 (y) = = xñ(y) = 0} x U x 1,..., x x n x T x M ( ) x 1,..., x xñ x Gram-Schmidt (e 1 ) x,..., (eñ) x a 1 [ ] 1(x) a 1 ñ(x). [(e 1 ) x (eñ) x ] = x 1 0... xñ x x........ 0 0 aññ(x) a i j(x) U C ( ) T Ũ U C (e 1 ) x,..., (eñ) x C (e i ) x x 1,..., x x i x
50 2 (e 1 ) x,..., (e n ) x T x M T x M (e n+1 ) x,..., (eñ) x Tx Mπ 1 (U) T M u ñ u = ξ i (e i ) π(u) i=n+1 Φ U (u) = (π(u), ξ n+1,..., ξñ) (u π 1 (U)) Φ U : π 1 (U) U Rñ n π 1 (U) (x 1,..., x n, ξ n+1,..., ξñ) (V ; y 1,..., y n ) Gram-Schmidt y V (f 1 ) y,..., (fñ) y b 1 [(f 1 ) y (fñ) y ] = 1(y) b 1 ñ(y). y 1 0... yñ y y........ 0 0 bññ(y) v π 1 (V ) v = ñ i=n+1 η i (f i ) π(v) π 1 (V ) (y 1,..., y n, η n+1,..., ηñ) U V π 1 (U) π 1 (V ) x U V x i = x ñ j=1 y j x i y j x [ x 1 x ] [ = xñ x y 1 x yñ x ] y 1 x 1 (x). yñ (x) x 1 y 1 xñ (x). yñ (x) xñ. [ ] y i J(x) = x (x) j A(x) = [a i j(x)], B(x) = [b i j(x)]
2.2 51 [ x 1 [e 1 eñ] = [f 1 fñ] = xñ ] = [ ] x A 1 xñ [ ] y B 1 yñ [ ] y J. 1 yñ [e 1 eñ] = = [ ] x A 1 xñ [ ] y JA 1 yñ = [f 1 fñ] B 1 JA [e 1 eñ] = [f 1 fñ] B 1 JA u π 1 (U V ) u = [f 1 fñ] 0. 0 η n+1. ηñ = [e 1 eñ] 0. 0 ξ n+1. ξñ = [f 1 fñ]b 1 JA 0. 0 ξ n+1. ξñ. 0. 0 η n+1. ηñ = B 1 JA 0. 0 ξ n+1. ξñ (x 1,..., x n, ξ n+1,..., ξñ) (y 1,..., y n, η n+1,..., ηñ) C T M
52 2 π π(x 1,..., x n, ξ n+1,..., ξñ) = (x 1,..., x n ) π : T M M C Φ U Φ U (u) U π(u) ñ φ U ξ i e i = (ξ n+1,..., ξñ) i=n+1 x Uφ U π 1 (x) : π 1 (x) Rñ n π : T M M π : T M MRiemann M T M M 2.3 1.4 1.4.8 3 Euclid 2.3.1 π : E M M r x M E x = π 1 (x) r P x = {(x; ẽ 1,..., ẽ r ) ẽ 1,..., ẽ r E x } P = P x x M P p M p U Φ U : π 1 (U) U R r P x R r R r GL(r, R) Lie Φ r U : x U P x U GL(r, R) ; (x; ẽ 1,..., ẽ r ) (x, Φ U (ẽ 1 ),..., Φ U (ẽ r )) P R r e 1,..., e r R r E x u (u(e 1 ),..., u(e r )) P x = {u u R r E x }
2.3 53 R r a GL(r, R) R r E x u ua GL(r, R) P x GL(r, R) P u P x x π : P M π C x Mπ 1 (x) GL(r, R) P E 2.3.2 G Lie π P : P M M G (1) P, Mπ P : P M C (2) Lie G P u P a G π P (u a) = π P (u) x M G πp 1 (x) (3) M p p U Φ U : πp 1 (U) U G u πp 1 (U) Φ U (u) U π P (u) Φ U (u) = (π P (u), φ U (u)) (u πp 1 (U)) a G φ U (u a) = φ U (u)a P M π P πp 1 (x) x a G P R a R a (u) = u a 2.3.3 1.4.8 M O(M) M O(2) 2.3.1 M r M GL(r, R) 2.3.4 π P : P M G M G ρ : G GL(V ) M E = P ρ V G P V (u, v) g = (ug, ρ(g) 1 v) (g G, (u, v) P V ) P V G P ρ V P ρ V (u, v) P V [u, v] π E [u, v] = π P (u) ((u, v) P V )
54 2 π E : E M g G π P (ug) = π P (u) π E 2.3.2 (3) Φ U : πp 1 (U) U G Ψ U : π 1 E (U) U V ; [u, v] (π P (u), ρ(φ U (u))v) Ψ U g G [ug, ρ(g) 1 v] (π P (ug), ρ(φ U (ug))ρ(g) 1 v) = (π P (u), ρ(φ U (u))ρ(g)ρ(g) 1 v) = (π P (u), ρ(φ U (u))v) Ψ U Ψ U E π E : E M M E = P ρ V ρ u P V E π(u) ; v [u, v] u uv = [u, v] P E 2.3.5 E M r P EP GL(r, R) 1 : GL(r, R) GL(r, R) P 1 R r P 1 V E ; [u, v] uv E 2.3.6 P M P M G G f : P P f : G G f (u a ) = f (u )f (a ) (u P, a G ) f = (f, f ) P P 2.3.7 2.3.6 f : P P M M f (u a ) = f (u )f (a ) (u P, a G ) f (π 1 (π(u ))) = f (u G ) = f (u )f (G ) f (u )G = π 1 (π(f (u ))) f : P P M M
2.3 55 2.3.8 2.3.6 2.3.7 2.3.9 2.3.6f : P P f : G G f f f : M M P f(p ) G G M M P P 2.3.10 P M G G Lie g X Xu = d u exp tx (u P ) dt t=0 P X X X u P dπ u X u = 0 X 2.3.11 P M G G Lie g X g G (Ad(g)X) = dr g 1X u P (Ad(g)X) u = d u exp tad(g)x = d ug exp txg 1 dt t=0 dt ( t=0 ) d = (dr g 1) ug ug exp tx = (dr dt g 1) ug Xug t=0 = (dr g 1X ) u (Ad(g)X) = dr g 1X 2.3.12 E M, E E s, t s, t (x) = s(x), t(x) (x M) M s, t C, E (E,, ) 2.3.13 2.2.7 Riemann Riemann Riemann Riemann
56 2 2.3.14 π : E M M r x M E x = π 1 (x) r P x = {(x; ẽ 1,..., ẽ r ) ẽ 1,..., ẽ r E x } P = P x x M P p M p U Φ U : π 1 (U) U R r P x R r R r O(r) Lie Φ r U : x U P x U O(r) ; (x; ẽ 1,..., ẽ r ) (x, Φ U (ẽ 1 ),..., Φ U (ẽ r )) P R r e 1,..., e r R r E x u (u(e 1 ),..., u(e r )) P x = {u u R r E x } R r a O(r) R r E x u ua O(r) P x O(r) P u P x x π : P M π C x Mπ 1 (x) GL(r, R) P E 2.3.15 E M r P E P O(r) 1 : O(r) GL(r, R) P 1 R r E 2.3.16 π : P M M G ρ : G GL(V ) P ρ E = P ρ V M E Γ(E) {Φ Ω 0 (P ; V ) Φ(ua) = ρ(a) 1 Φ(u) (u P, a G)} M E φ Φ(u) = u 1 (φ(π(u))) (u P ) P V Φ
2.3 57 M E φ Φ Φ(ua) = (ua) 1 (φ(π(ua))) = ρ(a) 1 u 1 (φ(π(u))) = ρ(a) 1 Φ(u) Φ(ua) = ρ(a) 1 Φ(u) P V Φx M π(u) = x u P φ(x) = uφ(u) E φ u P a G ua (ua)φ(ua) = uρ(a)ρ(a) 1 Φ(u) = uφ(u) π 1 (x) M E φ Φ(u) = u 1 (φ(π(u))) (u P ) P V Φ x Mπ(u) = x u P uφ(u) = uu 1 (φ(π(u))) = φ(x) φ Φ(ua) = ρ(a) 1 Φ(u) P V Φx M π(u) = x u P φ(x) = uφ(u) E φ Φ u 1 φ(π(u)) = u 1 uφ(u) = Φ(u)
3 3.1 1.3Euclid 3.1.1 M E M : Γ(T M) Γ(E) Γ(E); (X, φ) X φ (1) (4) E (1) X+Y φ = X φ + Y φ, (X, Y Γ(T M), φ Γ(E)) (2) X (φ + ψ) = X φ + X ψ, (X Γ(T M), φ, ψ Γ(E)) (3) fx φ = f X φ, (X Γ(T M), φ Γ(E), f C (M)) (4) X (fφ) = f X φ + (Xf)φ. (X Γ(T M), φ Γ(E), f C (M)) X Γ(T M) X φ = 0 φ Γ(E) 3.1.2, M EE X φ, ψ = X φ, ψ + φ, X ψ (X Γ(T M), φ, ψ Γ(E)), 3.1.3 1.3 Euclid 3.1.1 1.3.1 M Eφ Γ(E) 3.1.1 x M X T x M X φ E x M x φ T x M E x 1.2 58
3.1 59 3.1.4 E n M M x p (T x (M), E x ) ω x ωω E M p ()M (U; x 1,..., x n ) ( ) x ω x x i,, 1 x x ip i 1,..., i p U E C M E p Ω p (M; E) ( 3.1.1) 3.1.5 M E M : Ω 0 (M; E) Ω 1 (M; E); φ φ (1) (2) E (1) (φ + ψ) = φ + ψ, (φ, ψ Ω 0 (M; E)) (2) (fφ) = f φ + dfφ. (φ Ω 0 (M; E), f C (M)) 3.1.6, M EE x d φ, ψ = φ, ψ + φ, ψ (φ, ψ Γ(E)), 1.4 3 Euclid E M r P E EM U e 1,..., e r Ω 0 (U; E) e i Ω 1 (U; E) U 1 ω j i Ω 1 (U) e 1,..., e r r e i = e j ω j i j=1 ω = (ω j i ) Ω 1 (U; gl(r, R)) E Uφ φ = e i φ i φ = e i (dφ i ) + e j φ j = e i ( dφ i + ω i jφ j) E ωω 1.4 1.4.1
60 3 3.1.7 M E M U σ = [e 1,..., e r ] σ = [ē 1,..., ē r ] ω ω σ = σω σ = σ ω σ = σf f Ω 0 (U; M r (R)) f r GL(r, R) ω = f 1 df + f 1 ωf. σ = σf σ = σf + σdf = σωf + σdf = σ(ωf + df). σ = σ ω = σf ω. σf ω = σ(df + ωf) f ω = df + ωf ω = f 1 df + f 1 ωf 3.1.7 π : E M M r M {U α α A} α A x U α Φ α : π 1 (U α ) U α R r Φ α Ex : E x R r φ α (x) = Φ α Ex : E x R r U β x U α U β ψ αβ (x) = φ α (x) φ β (x) 1 : R r R r C ψ αβ : U α U β GL(r, R) {ψ αβ α, β A} {U α } E 3.1.8 M E {ψ αβ } ψ αα (x) = 1 r (x U α ) ψ βα (x) = ψ αβ (x) 1 (x U α U β ) ψ αβ (x) ψ βγ (x) ψ γα (x) = 1 r. (x U α U β U γ ) M {U α } C ψ αβ : U α U β GL(r, R) {ψ αβ }
3.1 61 M {U α } C ψ αβ : U α U β GL(r, R) {ψ αβ } α (U α R r ) (x, u) U α R r (y, v) U β R r x = y u = ψ αβ (x)v (x, u) (y, v) U α R r {ψ αβ } 3.1.9 E M r M {U α } {ψ αβ } R r e 1,..., e r EU α E e i (x) = φ α (x) 1 e i (x U α ) {e i } ω α U α U β ω β = ψ 1 αβ dψ αβ + ψ 1 αβ ω αψ αβ U α gl(r, R) {ω α } {ω α } E {e i } U β E ē i (x) = φ β (x) 1 e i (x U β ) {ē i } ω β x U α U β ē i (x) = φ β (x) 1 e i = φ α (x) 1 φ α (x)φ β (x) 1 e i = r φ α (x) 1 ψ αβ (x)e i = φ α (x) 1 e j (ψ αβ (x)) j i r = e j (x)(ψ αβ (x)) j i. j=1 j=1 σ = [e 1,..., e r ], σ = [ē 1,..., ē r ] σ = σψ αβ 3.1.7 U α gl(r, R) {ω α } ω α U α U E ξ = σφ = [e 1... e r ] φ 1. φ r
62 3 ξ = ( dφ i + φ j (ω α ) i j) ei = σ(dφ + ω α φ) U α M U α U β U α U β f = ψ αβ E U α U β ξ = σφ = σ φ σφ = σ φ = σf φ φ = f φ φ = f 1 φ ff 1 = 1 r σ(d φ + ω β φ) dff 1 + fd(f 1 ) = 0 = σf(d(f 1 φ) + (f 1 df + f 1 ω α f)f 1 φ) = σf(df 1 φ + f 1 dφ + f 1 dff 1 φ + f 1 ω α φ) = σ(fdf 1 φ + dφ + dff 1 φ + ω α φ) = σ(dφ + ω α φ). U α U β M 3.2 1.2 3.2.1 E n M E ω Ω p (M; E) d ω Ω p+1 (M; E) () M (U; x 1,..., x n ) ω ω x = a i1 i p (x)(dx i 1 ) x (dx i p ) x i 1 < <i p ( a i1 i p Ω 0 (U; E) ) (d ω) x = ( ) a i1 i p i 1 < <i p i=1 x i (dx i ) x (dx i 1 ) x (dx i p ) x x
3.2 63 1.2.10 ( ) a i1 i p i 1 < <i p i=1 x i (dx i ) x (dx i 1 ) x (dx ip ) x x p+1 (T x (M), E x ) 1.1.12 a i1 i p (x) ω x ( a i1 i p (x) = ω x x i,, 1 x ) x ip x (1 i 1 < < i p n) i 1,..., i p a i1 i p (x) (V ; y 1,..., y n ) M 1 ( ) b i1 i p (x) = ω x y i,, 1 x y i p x ω x = b i1 i p (x)(dy i 1 ) x (dy ip ) x i 1 < <i p x U V T x (M) 2 x i x y i x y i = x j=1 ( ) b i1 i p (x) = (dy i ) x = j 1,,j p =1 i=1 j=1 x j y i (x) x j x y i x j (x)(dxj ) x x j 1 xjp (x) y i 1 y (x)a ip j 1 j p (x) x j yi (x) yi x (x) = δ k jk ( ) b i1 i p i 1 < <i p i=1 y i (dy i ) x (dy i 1 ) x (dy ip ) x x = 1 /y p! i x b i1 i p (dy i ) x (dy i 1 ) x (dy ip ) x i,i 1,...,i p =1
64 3 = 1 p! = 1 p! i,i 1,...,i p=1 i 1,...,i p =1 j,j 1,...,jp=1 k,k 1,...,kp=1 x j { } x j y (x) 1 xjp i /x j x (x) y i 1 y (x)a i j p 1 j p (x) y i x k (x)(dxk ) x yi1 x k 1 (x)(dxk 1 ) x yip x k p (x)(dxk p ) x j 1,...,jp=1 k,k 1,...,kp=1 { x j 1 xjp /x k x (x) y i 1 y (x)a ip j 1 j p (x) (dx k ) x yi 1 x k 1 (x)(dxk 1 ) x yip x k p (x)(dxkp ) x. } i r =1 i r =1 x j r yir (x) ir y x (x) = δ kr j r k r ( ) x j r y i r (x) x k ir y x (x) = n x j ( ) r y i r (x) (x) kr ir i r =1 y x k kr x x jr = y (x) 2 y ir i r x k x (x) k r i r =1 k k r (dx k ) x (dx k 1 ) x (dx k p ) x k k r ( ) = 1 p! i 1,...,i p=1 j 1,...,jp=1 k,k 1,...,kp=1 x j 1 xjp (x) y i 1 y (x) i /x p k x a j1 j p (dx k ) x yi 1 x k 1 (x)(dxk 1 ) x yip x k p (x)(dxk p ) x = 1 p! /x k x a j1 j p (dx k ) x (dx k 1 ) x (dx k p ) x k,k 1,...,k p =1 ( ) = a k1 k p k 1 < <k p k=1 x k (dx k ) x (dx k 1 ) x (dx k p ) x. x d ω d ω d ω E p + 1 d ω 3.2.2 3.2.1 p = 0 ω Ω 0 (M; E) E d ω ( ) (d ω) x = ω x i (dx i ) x i=1 x
3.2 65 M X d ω = ω ( ) (d ω) x (X) = ω i=1 x i (dx i ) x (X) = ( ω) x (X). x 3.2.3 E M E 3.2.1 d : Ω p (M; E) Ω p+1 (M; E) E 3.2.4 E M E d : Ω p (M; E) Ω p+1 (M; E) 3.2.1 d 3.2.5 E M E φ Ω p (M; E) ψ Ω q (M) d (φ ψ) = d φ ψ + ( 1) p φ dψ 1.2.16(U; x 1,..., x n ) M Uφ ψ φ x = ψ x = (a i1 i p Ω p (U; E), b j1 j q (φ ψ) x = a i1 i p (x)(dx i 1 ) x (dx i p ) x i 1 < <i p b j1 j q (x)(dx j 1 ) x (dx j q ) x j 1 < <j q i 1 < <i p Ω q (U) ) j 1 < <j q a i1 i p (x)b j1 j q (x) (dx i 1 ) x (dx i p ) x (dx j 1 ) x (dx j q ) x k 1 < < k p+q {i 1,..., i p, j 1,..., j q } = {k 1,..., k p+q } sgn ( i1 i p j 1 j q k 1 k p k p+1 k p+q ) = 0
66 3 = (φ ψ) x sgn k 1 < <k p+q i 1 < <i p j 1 < <j q a i1 i p (x)b j1 j q (x)(dx k 1 ) x (dx k p+q ) x ( i1 i p j 1 j q k 1 k p k p+1 k p+q ) = = d (φ ψ) x sgn k 1 < <k p+q i 1 < <i p j 1 < <j q ( i1 i p j 1 j q k 1 k p k p+1 k p+q /x i x (a i1 i p b j1 j q )(dx i ) x (dx k 1 ) x (dx k p+q ) x i=1 sgn k 1 < <k p+q i 1 < <i p j 1 < <j q { i=1 ( i1 i p j 1 j q k 1 k p k p+1 k p+q } ( /x i x a i1 i p )b j1 j q (x) + a i1 i p (x) b j 1 j q (x) x i ) ) = (dx i ) x (dx k 1 ) x (dx k p+q ) x { ( /x i x a i1 i p )b j1 j q (x) + a i1 i p (x) b } j 1 j q (x) x i i 1 < <i p j 1 < <j q i=1 (dx i ) x (dx i 1 ) x (dx i p ) x (dx j 1 ) x (dx j q ) x = (d φ ψ) x + ( 1) p (φ dψ) x. d (φ ψ) = d φ ψ + ( 1) p φ dψ 0 1.2.17 0 0 3.2.6 E M E φ Ω 0 (M; E) ψ Ω p (M) d d (φψ) = (d d φ) ψ p = 0 d d (φψ) = (d d φ)ψ d d : Ω 0 (M; E) Ω 2 (M; E) x M d d : E x 2 (T x M, E x )
3.2 67 R = d d Ω 2 (M; EndE) d d φ = R φ. (φ Ω p (M; E)) 3.2.5φ Ω 0 (M; E) ψ Ω p (M) d d (φψ) = d ((d φ) ψ + φdψ) = (d d φ) ψ (d φ) dψ + (d φ) dψ + φddψ ( 1.2.16 ddψ = 0) = (d d φ) ψ. p = 0 d d (φψ) = (d d φ)ψ ψ Ω 0 (M) = C (M) x M (d d φ) x 2 (T x M, E x ) φ(x) d d : E x 2 (T x M, E x ) R = d d Ω 2 (M; EndE) φ Ω p (M; E) φ 1 Ω 0 (M; E) φ 2 Ω p (M) φ 1 φ 2 φ = φ 1 φ 2 d d φ = R φ d d φ = d d (φ 1 φ 2 ) = (d d φ 1 ) φ 2 = (R φ 1 ) φ 2 = R φ 1 φ 2 = R φ. 3.2.7 3.2.6R 3.2.8 E M E ω Ω p (M; E) p = 0 X X(M) p = 1 X, Y X(M) d ω(x) = X ω. d ω(x, Y ) = X (ω(y )) Y (ω(x)) ω([x, Y ]). 1.2.18p = 0 d ω = ω X X(M) d ω(x) = X ω. p = 1 ω ω = i a i dx i ω d ω d ω = /x ja i dx j dx i i,j
68 3 X, Y X = i ξ i x, i Y = j η j x j d ω(x, Y ) = i,j /x ja i (ξ j η i η j ξ i ) ω(y ) = i a i η i X (ω(y )) = i,j ξ j /x j(a i η i ) = i,j ξ j ( ( /x ja i )η i + a i η i x j ). X Y Y (ω(x)) = i,j η j ( ( /x ja i )ξ i + a i ξ i x j ) [X, Y ] = ξ i ηj i,j x i x η j ξi j i,j x j x i = ( ) ξ j ηi ξi ηj i,j xj x j x i ω([x, Y ]) = i,j ( ) a i ξ j ηi ξi ηj. xj x j d ω(x, Y ) = X (ω(y )) Y (ω(x)) ω([x, Y ]) 3.2.9 3.2.8 p > 0 X 1,... X p+1 X(M) = d ω(x 1,..., X p+1 ) p+1 i=1 ( 1) i 1 Xi (ω(x 1,..., ˆX i,..., X p+1 )) + i<j( 1) i+j ω([x i, X j ], X 1,..., ˆX i,..., ˆX j,..., X p+1 ).
3.2 69 3.2.10 E M E R R (X, Y )φ = ( X Y Y X [X,Y ] )φ. (X, Y X(M), φ Ω 0 (M; E)) Ricci 3.2.8 R(X, Y )φ = (d d φ)(x, Y ) = X (d φ(y )) Y (d φ(x)) d φ([x, Y ]) = X ( Y φ) Y ( X φ) [X,Y ] φ = ( X Y Y X [X,Y ] )φ. 3.2.11 E M E Φ Ω 0 (M; EndE) ( X Φ)φ = X (Φφ) Φ( X φ) (X X(M), φ Ω 0 (M; E)) X Φ Ω 0 (M; EndE) EndE X Φ Ω 0 (M; EndE) f C (M) ( X Φ)(fφ) = X (Φ(fφ)) Φ( X (fφ)) = X (fφφ) Φ(f X φ + (Xf)φ) = f X (Φφ) + (Xf)Φφ fφ( X φ) (Xf)Φφ = f( X (Φφ)) Φ( X φ)) = f( X Φ)φ. x M ( X Φ)φ x φ(x) X Φ : Ω 0 (M; E) Ω 0 (M; E) x M X Φ : E x E x X Φ Ω 0 (M; EndE) EndE 3.1.1 (1) (3) E f C (M) ( X (fφ))φ = X (fφφ) fφ( X φ) = f X (Φφ) + (Xf)Φφ fφ( X φ) = f( X Φ)φ + (Xf)Φφ.
70 3 X (fφ) = f( X Φ) + (Xf)Φ EndE 3.2.12 E M E Φ Ω p (M; EndE) ψ Ω q (M; E) d (Φ ψ) = d Φ ψ + ( 1) p Φ d ψ 3.2.5(U; x 1,..., x n ) M UΦ ψ Φ x = ψ x = (A i1 i p Ω p (U; EndE), b j1 j q (Φ ψ) x = A i1 i p (x)(dx i 1 ) x (dx ip ) x i 1 < <i p b j1 j q (x)(dx j 1 ) x (dx jq ) x j 1 < <j q i 1 < <i p Ω q (U; E) ) j 1 < <j q A i1 i p (x)b j1 j q (x) (dx i 1 ) x (dx ip ) x (dx j 1 ) x (dx jq ) x k 1 < < k p+q {i 1,..., i p, j 1,..., j q } {k 1,..., k p+q } sgn ( i1 i p j 1 j q k 1 k p k p+1 k p+q ) = 0 = (Φ ψ) x sgn k 1 < <k p+q i 1 < <i p j 1 < <j q A i1 i p (x)b j1 j q (x)(dx k 1 ) x (dx k p+q ) x ( i1 i p j 1 j q k 1 k p k p+1 k p+q ) = d (Φ ψ) x ( i1 i p j 1 j q sgn k 1 < <k p+q i 1 < <i p j 1 < <j q k 1 k p k p+1 k p+q )
3.2 71 = /x i x (A i1 i p b j1 j q )(dx i ) x (dx k 1 ) x (dx k p+q ) x i=1 ( i1 i p j 1 j q sgn k 1 < <k p+q i 1 < <i p j 1 < <j q k 1 k p k p+1 k p+q { } ( /x i x A i1 i p )b j1 j q (x) + A i1 i p (x) /x i x b j1 j q ) i=1 = (dx i ) x (dx k 1 ) x (dx k p+q ) x { } ( /x i x A i1 i p )b j1 j q (x) + A i1 i p (x) /x i x b j1 j q i 1 < <i p j 1 < <j q i=1 (dx i ) x (dx i 1 ) x (dx i p ) x (dx j 1 ) x (dx j q ) x = (d Φ ψ) x + ( 1) p (Φ d ψ) x. d (Φ ψ) = d Φ ψ + ( 1) p Φ d ψ 3.2.13 E M E R d R = 0. Bianchi φ Ω 0 (M; E) 3.2.6 d φ Ω 1 (M; E) d d d φ = d d (d φ) = R d φ 3.2.12 R Ω 2 (M; EndE) d d d φ = d (d d φ) = d (R φ) = (d R )φ + R d φ (d R )φ = 0. φ Ω 0 (M; E) d R = 0 3.2.14 E M E M U E σ = [e 1,..., e r ] ω Ω Ω 2 (U; gl(r, R)) R σ = σω Ω = dω + ω ω (d R )σ = σ(dω Ω ω + ω Ω)
72 3 Bianchi dω Ω ω + ω Ω = 0 Bianchi 3.2.5 Ω R σ = d d σ = d (σω) = (d σ) ω + σdω = σω ω + σdω = σ(dω + ω ω). Ω = dω + ω ω R σ = σω 3.2.12 d (R σ) = (d R )σ + R d σ = (d R )σ + R σω 3.2.5 = (d R )σ + R σ ω = (d R )σ + σω ω. d (σω) = (d σ) Ω + σdω = σω Ω + σdω. (d R )σ = σ(dω Ω ω + ω Ω) 3.2.15 3.2.14 Bianchi Ω = dω + ω ω ddω = 0 dω = dω ω ω dω = (Ω ω ω) ω ω (Ω ω ω) = Ω ω ω ω ω ω Ω + ω ω ω = Ω ω ω Ω. dω Ω ω + ω Ω = 0
3.3 3.3 73 1.4 3 Euclid 3.1.9α σ α (x) = φ α (x) 1 [e 1,..., e r ] (x U α ) E P U α σ α {ω α } P ω σ α ω = ω α 2.3.1 Φ r α : πp 1 (U α ) U α GL(r, R) ; u (π P (u), φ α (π P (u))u) φ α (x) : E x R r u E x φ α (π P (u))u R r GL(r, R) U α GL(r, R) gl(r, R) 1 ω α ( ω α ) (x,g) = g 1 dg + g 1 (ω α ) x g ((x, g) U α GL(r, R)) π P 1 (U α ) (Φ r α) ω α P πp 1 (U α U β ) (Φ r α) ω α (Φ r β) ω β ω β = ((Φ r β) 1 ) (Φ r α) ω α = (Φ r α (Φβ) r 1 ) ω α Φ r α (Φ r β) 1 (x, g) = (x, φ α (x) φ β (x) 1 g) = (x, ψ αβ (x)g) (Φ r α (Φ r β) 1 ) ω α = (ψ αβ g) 1 d(ψ αβ g) + (ψ αβ g) 1 ω α (ψ αβ g) = g 1 ψαβ 1 ((dψ αβ)g + ψ αβ dg) + g 1 ψαβ 1 ω αψ αβ g = g 1 dg + g 1 (ψαβ 1 dψ αβ + ψαβ 1 ω αψ αβ )g = g 1 dg + g 1 ω β g = ω β. (Φ r α) ω α P gl(r, R) 1 ω σ α ω σ α ω = σ α(φ r α) ω α = (Φ r ασ α ) ω α