A A 1 A 5 A 6 1 2 3 4 5 6 7 1 1.1 1.1 (). Hausdorff M R m M M {U α } U α R m E α ϕ α : U α E α U α U β = ϕ α (ϕ β ϕβ (U α U β )) 1 : ϕ β (U α U β ) ϕ α (U α U β ) C M a m dim M a U α ϕ α {x i, 1 i m} {U, x 1, x 2,..., x m } 1. R m {x x 1} 2. R 3 g xy = 0 yz xz
A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2. R m C f a a f 1 (a) m 1 ( ) a f 1 ({a}) f 4. 1 n S L(n, R) n 2 1 1.2 M f M (U, ϕ) f ϕ 1 ϕ(u) C 1.3. C M N M N C C (M, N) R (diffeomorphism) 1956 J.Milnor 7 28 3 R n, n 4 R 4 1982 S.Donaldson 2 2.1 R m R m p 2.1 ( ). C (R m, R) R v : C (R m, R) R R m p f, g C (R m, R), λ, µ R v v(λ f + µg) = λv( f ) + µv(g) v( f g) = v( f )g(p) + f (p)v(g) v( f ) = v(x k ) f (p) x k k
A 3 v v = v(x k ) x k k 2.2. R m p m T p R m x 1, C(t) C(0) = p x 2,..., x m Ċ = DC dt d( f C) ( f ) = (0) dt Ċ C C(0) = p 2. Ċ 2.2 v M 2.3. v : C (M, R) R M p v( f g) = f (p)v(g) + v( f )g(p) p m T p M M U ϕ = (x 1, x 2,, x n ) x 1, U p T p M x 2,..., x m 2.3 M, N m, n Φ M N Φ p : T p M T Φ(p) N N C f Φ p (v)( f ) = v( f Φ) 3. T p M T Φ(p) N Φ p
A 4 2.4 Φ M N M p Φ p Φ (immersion) Φ (embedding) M N (submanifold) Whitney 2.4. 2 n R 2n+1 5. S m R m+1 R m T m R T 2 f (t) = (e it, e ait ) a R 3 3 3.1 M C C (M) C (M) X : C (M) C (M) X( f g) = X( f )g + f X(g) p X p : C (M) R, X p ( f ) = X( f )(p) X p T p M U f (x) = g(x), x U U f = g X( f ) f (U, {x 1, x 2,..., x n }) X = X k x k X k = X(x k ) X k U C M X(M) X(M) C (M) 4. Φ : M N X X(M) Φ X : C (N, R) C (N, R) Φ X( f ) = X( f Φ) Φ 1 N Φ 3.2 C : I M C(t) t C (t) T C(t) M, C (t)( f ) = C (t) = X C(t) d( f C) (t) dt
A 5 C X C(0) 3.1. (1) X X(M) p M p (2) p (, ) M 5. X M Φ : M N C(t) X p Φ C(t) Φ X Φ(p) 6. c X [ c, c] X 3.3 1 X M x X t ϕ t (x) 3.2. ϕ t : M M ϕ t ϕ t ϕ s = ϕ t+s, ϕ 1 t = ϕ t, ϕ 0 = Id M {ϕ t t R} X 1 7. X M Φ : M N {ϕ t t R} X 1 Φ ϕ t Φ 1 Φ X 1 3.4 Lie X *1 {ϕ t } 1 Y ϕ t Y 1 L X Y = lim t 0 t (ϕ t Y Y) Y X Lie C f L X Y( f ) = X(Y f ) Y(X f ) L X Y [X, Y] = XY YX 8. L X Y (L X Y) p X, Y p 9. X, Y X(M) [X, Y] X(M) (1) [X, Y] = [Y, X] (2) [αx + βy, Z] = α[x, Z] + β[y, Z] *1 Lie p t ϕ t
A 6 (3) Φ Φ [X, Y] = [Φ X, Φ Y] (4) [[X, Y], Z] + [[Y, z], X] + [[Z, X], Y] = 0 S L(n, R) Lie Lie G v T e G X g = L g v X X(G) L g : G G L g (h) = gh X L g X = X g T e G Lie Lie 4 4.1 V n V R V V {e 1, e 2,..., e n } V ω j : V R, ω j (e k ) = δ jk ω j V V dim V = dim V = n {ω 1, ω 2,..., ω n } {e 1, e 2,..., e n } α V p α : V } {{ V } R, p V p p V *2 0 V = R, 1 V = V α p V, β q V α β p+q V α β(v 1, v 2,... v p+q ) = 1 sgn(σ)α(v σ(1),..., v σ(p) )β(v σ(p+1),..., v σ(p+q) ) p!q! σ p + q sgn(σ) 4.1. (1) (α β) γ(v 1, v 2,..., v p+q+r ) = 1 sgn(σ)α(v σ(1),..., v σ(p) )β(v σ(p+1),..., v σ(p+q) )γ(v σ(p+q+1),..., v σ(p+q+r) ) p!q!r! (2) p V n C p V {ω 1,... ω n } ω j1 ω j2 ω jp, 1 j 1 < j 2 < < j p n p > n p V = {0} V = n p=0 p V 2 n V *2
A 7 10. (1) α p V, β q V α β = ( 1) pq β α (2) α i V α 1 α 2 α n (v 1, v 2,..., v n ) = det(α i (v j )) 4.2 M p T p M T pm f d f p : T p M R, d f p (v) = v( f ) () f 4.2. (U, {x 1, x 2,..., x m }) a d(x 1 ) p, d(x 2 ) p,..., d(x m ) p T p M x 1, x 2,..., x m a x j U d(x j ) x j 11. d f p d f p = m k=1 f x k d(x k ) p 4.3 M n ω : X(M) X(M) C } {{ } (M), p C (M) p C (M) C ω ω(x σ(1), X σ(2),..., X σ(p) ) = sgn(σ)ω(x 1, X 2,..., X p ) ω( f X 1 + gx 1, X 2,..., X p ) = f ω(x 1, X 2,..., X p ) + gω(x 1, X 2,..., X p ) f, g C (M). ω x M ω(x 1, X 2,..., X p )(x) X j (x) T x M ω x : T x M T x M R } {{ } p
A 8 ω x p Tx M {U, (x 1, x 2,..., x n )} ω = ω j1 j 2... j p (x)dx j1 dx j2 dx jp j 1 < j 2 < < j p ω j1 j 2... j p ( = ω,,..., x j1 x j2 x jp ) C M p A p (M) A 0 (M) = C (M) α A p (M), β A q (M) α β A p+q (M) α β(x 1, X 2,..., X p+q ) = 1 sgn(σ)α(x σ(1),..., X σ(p) )β(x σ(p+1),..., X σ(p+q) ) p!q! p q 0 4.4 M, N Φ : M N M N C *3 N p β Φ β(x)(v 1,..., v p ) = β(φ(x))(φ (v 1 ),..., Φ (v p )) β Φ (pull back) Φ : A p (N) = A p (M) p = 0 Φ ( f ) = f Φ 12. (1) Φ (α β) = Φ α Φ β (2) p {x 1, x 2,..., x m } Φ(p) {y 1, y 2,..., y n } Φ (dy j1 dy j2 dy jp ) = Ξ j 1 j 2... j p i 1 i 2...i p dx i1 dx i2 dx ip Ξ j 1 j 2... j p i 1 i 2...i p 1 i 1 <i 2 <...i p <m 4.5 p α dα dα(x 1, X 2,..., X p+1 ) = ( 1) j+1 X j α(x 1,..., ˆX j,..., X p+1 ) + ( 1) i+ j α([x i, X j ], X 1,..., ˆX i,..., ˆX j,..., X p+1 ) j dα = d α j1 j 2... j p (x)dx j1 dx j2 dx jp = j 1 < j 2 < < j p = p+1 j 1 < j 2 < < j p+1 k=1 ( 1) k 1 α j 1... ĵ k... j p x jk i< j j 1 < j 2 < < j p dx j1 dx j2 dx jp+1 α j1 j 2... j p dx k dx j1 dx j2 dx jp x k k *3
A 9 4.3. d : A p (M) A p+1 (M), ddα = 0 R d(α β) = dα β + ( 1) p α dβ dφ β = Φ dβ X X p α Lie 1 L X α = lim t 0 t (ϕ t α α) i X : A p (M) A p 1 (M) (i X α)(x 1, X 2,..., X p 1 ) = α(x, X 1, X 2,..., X p 1 ) Lie L X α = i X (dα) + d(i X α) 5 5.1 n V V {e 1, e 2,..., e n } { f 1, f 2,..., f n } f j = n p i j e i, det(p i j ) > 0 i=1 V n M n ω T p M {v 1, v 2,..., v n } ω (v 1, v 2,..., v n ) > 0 5.1. n n 1 1 M {U α } C ϕ α : M R, 0 ϕ α (x) 1, Suppϕ α U α, ϕ α 1 α {U α } 1 Suppϕ = {x ϕ(x) 0} ϕ
A 10 5.2 M n ω ω U M ω = U ω(x 1, x 2,..., x n )dx 1 dx 2 dx n (x 1, x 2,..., x n ) M U R n ω = ω(x 1, x 2,..., x n )dx 1 dx 2 dx n n {U α } 1 {ϕ α } M ω = α M ϕ α ω 5.3 Stokes R n R n x n 0 M R n M M M n 1 M = M M n M M ω A n 1 (M) 5.2 (Stokes ). M dω = M k k M ω 5.4 de Rham p α dα = 0 (closed form) α = dβ (p 1) (exact form) M p Z p (M) p B p (M) A p (M) Z p (M) B p (M) 5.3. Z p (M)/B p (M) p de Rham H p DR (M) de Rham de Rham
A 11 6 6.1 M p T p M g p (U, {x 1,..., x m }) ( ) p p M g i j (p) = g p, x i x j U g i j C g = {g p p M} M (M, g) g g = g i j dx i dx j 6. R n g 0 = (dx 1 ) 2 + (dx 2 ) 2 + + (dx n ) 2 7. R 3 S (U, {u, v}) U x U 2 x(u, v) E = x u x u, F = x u x v, F = x v x v g = Edu du + 2Fdu dv + Gdv dv ds 2 (N, h) Φ : M N g p (u, v) = h(φ p v, Φ p v) g M g = Φ h Φ (induced metric) 8. S n = {x R n+1 x = 1} R n+1 (R n+1, g 0 ) i : S n R n+1 i g 0 S n Φ : (M, g) (N, h) g = Φ h Φ (isometric immersion) Φ (isometry) 6.2 (M, g) C : [a, b] M b ( dc L(C) = g dt, dc ) dt dt C M 2 p, q a d(p, q) = inf { L(C) C p q C } 13. d M
A 12 6.3 0 6.1. D : X(M) X(M) X(M) D X Y D M D X1 +X 2 Y = D X1 Y + D X2 Y, D X (Y 1 + Y 2 ) = D X Y 1 + D X Y 2, X, Y X(M) D f X Y = f D X Y, D X ( f Y) = X( f )Y + f D X Y, X, Y X(M), f C (M, R) 3.1 4.3 D X Y p (p ) X(p) T p M p Y 14. (U, {x 1, x 2,... x m }) D X = X i, Y = Y i x i x i Y i D X Y = X j + X j Y k D i jk x j x i i j D i jk X, Y U n D = D i jk x x i k i=1 x j C : ( a, a) M X = X i x i dx i DĊX = + D i jk dt X dc k j dt x i i 0 X C C T C(t) M T C(0) M 6.2. D (M, g) jk jk X(g(Y, Z)) = g(d X Y, Z) + g(y, D X Z), D X Y D Y X = [X, Y], X, Y, Z X(M) X, Y X(M) 6.3. (M, g) Γ k i j = 1 ( g kh gih + g jh g ) i j 2 x j x i x h h g kh (g i j ) Γ k i j
A 13 6.4. (1) R m Γ k i j = 0 (2) (M, g) R m v T p M v X R m v X T p M 6.4 (M, g) p, q 2 C C pq = { C : [0, 1] M C(0) = p, C(1) = q, C } 2 E(C) = 1 0 g ( Ċ, Ċ ) dt, L(C) = 1 0 g ( Ċ, Ċ ) dt Cauchy-Schwarz E(C) L(C) 2, 6.5. (1) inf{e(c) C C pq } = d(p, q) 2 Ċ (2) C E(C) C p, q E(C) C C C pq {C s C pq ϵ < s < ϵ} C 0 = C C : ( ϵ, ϵ) [0, 1] M, C(s, t) = C s (t) C C C *4 C d ds E(C s) = 0 s=0 C *5 6.6. (1) C : [a, b] M ĊĊ = 0. (2) v T p M Ċ(0) = v (1) (2) 9. S m = {x R m+1 x = 1} (2 S m ) π 2 *4 C pq C C *5 [0, 1]
A 14 6.5 (M, g) X, Y, Z X(M) R(X, Y)Z = X Y Z Y X Z [X,Y] Z (1 3) *6 6.7. (1) R M R : T p M T p M T p M T p M (2) R 15. R R(u, v)w = R(v, u)w R(u, v)w + R(v, w)u + R(w, u)v = 0 g(r(u, v)w, t) = g(w, R(u, v)t) ( R, x j g(r(u, v)w, t) = g(r(w, t)u, v) x k ) x i = l R l i jk x l R l i jk Γa bc 6.8. (1) T p M 2 H K(H) K(H) = g(r(u, v)v, u), u, v H, g(u, u) = g(v, v) = 1, g(u, v) = 0. (2) T p M v Ric(v) v Ric(v) = m g(r(v, e j )e j, v), g(e i, e j ) = δ i j, e 1 = v. j=2 (3) p M S (p) p S (p) = m Ric(e j ), e j T p M, g(e i, e j ) = δ i j j=1 2 16. ( c, c) (0, a) g = dx 2 + f 2 (x)dy 2 f (x) > 0 *6
A 15 7 7.1 (M, g) M x M v T x M v γ [0, 1] exp x (v) = γ(1) exp x x 7.1 (Hopf-Rinow). (M, g) (1) (2) M v v R 2 Hopf-Rinow 17. γ : [0, a) M lim t a γ(t) 10. G C Lie T e G g e x G g x (u, v) = g e ((L x 1) u, (L x 1) v) L a (x) = ax L a G = O(n) T e G = {A t A = A} g e (A, B) = Tr( t AB) 7.2 Jacobi exp p : T p M M v T p M v T p M {v + sw s R} exp p s = 0 C : ( ε, ε) [0, 1] M, C(s, t) = exp p (t(v + sw)) exp p (v) = q, γ(t) = C(0, t), Y(t) = C (0, t) s Y p q γ γ γ Y + R(Y, γ) γ = 0 2 Jacobi γ Jacobi Y Y(0) = 0, γ Y(0) = w
A 16 (exp p ) v (w) = Y(1) exp p v T p M γ(t) = exp p (tv) Jacobi p q = exp p (v) q p γ q p 7.2. 0 Y(0) = 0, γ Y 0 Jacobi Y 2 2 T p M R n exp p 7.3 (Cartan-Hadamard). (M g) M R m M 2 0 18. 1 S 2 = {(x, y, z) x 2 + y 2 + z 2 = 1} (0, 0, 1) 7.3 (M, g) pq 2 γ : [0, l] M, γ(0) = p, γ(l) = q 2 l γ 0 V V V V(t) T γ(t) M, V(0) = 0, V(l) = 0 V C V I(V, W) = I 7.4. l 0 γ V, γ W + R(V, γ)w, γ dt (1) p q γ W V I(V, W) = 0 V V, V 0 (2) I(V, V) < 0 V V γ. (1) Jacobi V I(V, W) = 0 (2) C(s, t) = exp γ(t) sv(t), γ s (t) = C(s, t) E(γ s ) s = 0 2 * 7 *7
A 17 19. Myers 7.5 (Myers). (M, g) Ric (n 1)H > 0 π/ H M 2 π/ H M M. γ l {E i } γ E 1 = γ W i (t) = sin(πt/l)e i (t), 2 i n I(W i, W i ) = = l 0 l 0 Wi, γ γ W i + R(W i, γ) γ dt (sin πt/l) 2 (π 2 /l 2 R(E i, γ) γ, E i )dt n I(W i, W i ) = i=2 l 0 (sin πt/l) 2 ((n 1)π 2 /l 2 Ric( γ, γ))dt l > π/ H W i I(W i, W i ) γ M T p M π/ H D = {v T p M v π/ H} exp p (D) = M D M M 20. Myers R 3