I II Morse 1998

I II Morse 1998

1 Morse 1 1.1.............................. 1 1.2......................... 8 1.3......................... 16 1.4 Morse................................. 20 1.5........................................ 26 2 Riemann 31 2.1............................. 31 2.2................................. 34 2.3 Gauss............................ 38 3 Chern-Lashof 40 3.1 Riemann............................ 40 3.2 Γ...................................... 47 3.3...................................... 50 3.4 Chern-Lashof.............................. 57 4 61 4.1................................... 61 4.2...................................... 67 4.3.................................. 75 4.4................................. 81 4.5....................................... 85 4.6..................................... 89 i

1 Morse C Riemann C Morse 1.1 1.1.1 M p T p M C F : M N p df p : T p M T F (p) N f : M R M C f p M df p : T p M T f(p) R = R 0 p f f(p) f a ff p a = f(p) 1.1.2 M C f : M R p M (x 1,..., x n ) n f df p = x i (p)dxi p dx 1 p,..., dx n p T p M (T p M ) p f f f (p) = = x1 x (p) = 0 n 1.1.3 M C f : M R a M a = {x M f(x) a} a f f(x) = a x M df x : T x M R f 1 (a) MM a 1

2 1 Morse 1.1.4 M C f : M R p M X, Y T p M p X, Ỹ ( 2 f) p (X, Y ) = ( XỸ f)(p) ( 2 f) p (X, Y ) X, Y ( 2 f) p : T p M T p M R ( XỸ f)(p) = X p (Ỹ f) = X(Ỹ f) X ( XỸ f)(p) (Ỹ Xf)(p) = ([ X, Ỹ ]f)(p) = df p([ X, Ỹ ]) = 0 ( 2 f) p ( XỸ f)(p) Y 1.1.5 p M M C f : M R 1.1.4 ( 2 f) p f p Hessian ( 2 f) p p f 1.1.6 M C f : M R p M p (x 1,..., x n ) 1.1.2 f f (p) = = x1 x (p) = 0 n X, Y T p M n X = X i x i, Y = p n Y i Y p n Ỹ = Ỹ i x i x i n ( 2 f) p (X, Y ) = X(Ỹ f) = X i n x i Ỹ j f p j=1 x = n X i Y j 2 f j i,j=1 x i x (p). j [ 2 ] f x i x (p) j x 1,..., p x n ( 2 f) p p [ 2 ] f p f x i x (p) j p

1.1 3 1.1.7 V H : V V R H H index(h) H V 0 = {v V H(v, w) = 0 ( w V )} H nullity nullity(h) M C f : M R p f Hessian( 2 f) p nullity f p nullity 1.1.8 V H V v 1,..., v n H V v 1,..., v n v 1,..., v λ 0 v λ+1,..., v n v 1,..., v λ V 1 v λ+1,..., v n V 2 V V 1 V 2 H V 1 Hλ Hλ W dim(w V 2 ) = dim W + dim V 2 n (λ + 1) + (n λ) n 1. 0 v W V 2 v V 2 H(v, v) 0 v W H(v, v) < 0 Hλ 1.1.9 f R n 0 V C V C g i (1 i n) n f(x) = f(0) + x i g i (x) (x = (x 1,..., x n ) V ) g i (0) = f x (0) i g i (x) = x V tx V (t [0, 1]) 1 0 f(x) f(0) = = = 1 0 1 f (tx)dt xi d dt f(tx)dt n 0 ( n 1 x i 0 f x i (tx)xi dt f x i (tx)dt ).

4 1 Morse 1.1.10 (Morse) M C f : M R p M p (x 1,..., x n ) f(x) = f(p) (x 1 ) 2 (x λ ) 2 + (x λ+1 ) 2 + + (x n ) 2 λ f p p (x 1,..., x n ) f(x) = f(p) (x 1 ) 2 (x λ ) 2 + (x λ+1 ) 2 + + (x n ) 2 λ f p 2 f x i x (p) = j 2 (i = j λ) 2 (i = j > λ) 0 () f p Hessian ( 2 f) p x 1,..., p x n p 2... 2 2... 2 1.1.8 R n ( 2 f) p λ p (y 1,..., y n ) y j (p) = 0 (1 j n) 1.1.9 n f(y) = f(p) + y j g j (y), j=1 g j (0) = f y j (0) = 0 C g j g j 1.1.9 n g j (y) = y i h ij (y) C h ij n f(y) = f(p) + y i y j h ij (y) i,j=1

1.1 5 h ij (y) = 1 2 (h ij + h ji ) n f(y) = f(p) + y i y j hij (y), i,j=1 hij (y) = h ji (y) h ij h ij h ij = h ji f(u) = f(p) + i r 1 ±(u i ) 2 + r i,j u i u j H ij (u), [H ij (u)] p (U 1 ; u 1,..., u n ) f(v) = f(p) + ±(v i ) 2 + i r v i v j H ij(u) r+1 i,j [H ij(v)] p (U 2 ; v 1,..., v n ) [ 2 ] f u i u (0) = j ±2... ±2 2H rr (0) 2H rn (0)..... 2H nr (0) 2H nn (0) p f H ij (0) 0 H rr (0) 0 p U 2 U 1 H rr (u) U 2 0 g(u) = H rr (u) (u U 2 ) g U 2 C g v 1,..., v n v i = u i (i r) v r (u) = g(u) u r + r+1 i u i H ir(u). H rr (u) v r (0) = g(0) 0 ur

6 1 Morse [ ] v i det u (0) = g(0) 0. j p U 3 U 2 v 1,..., v n ±(v r ) 2 = H rr (u) u r + f = f(p) + r+1 i = H rr (u) (u r ) 2 + 2u r = (u r ) 2 + 2u r i r 1 = f(p) + i r 1 r+1 i ±(u i ) 2 + = f(p) + ±(v i ) 2 + i r H ij(v) r i,j u i H ir(u) H rr (u) r+1 i u i H ir (u) + u i u j H ij (u) ±(v i ) 2 + (u r ) 2 H rr (u) + 2 r+1 i,j f(v) = f(p) + ±(v i ) 2 + i r 2 u i H ir(u) H rr (u) + r+1 i,j r+1 i r+1 i 2 u i H ir(u) H rr (u) u i u j H ir(u)h jr (u). H rr (u) u i u r H ir (u) + u i u j ( H ij (u) H ir(u)h jr (u) H rr (u) r+1 i,j H ij (u) H ir(u)h jr (u) H rr (u) v i v j H ij(v) r+1 i,j ). u i u j H ij (u) r = 1 r n f(x) = f(p) + ±(x i ) 2 (x U) p (U; x 1,..., x n ) 1.1.11 C 1.1.10 C f p f(x) = f(p) (x 1 ) 2 (x λ ) 2 + (x λ+1 ) 2 + + (x n ) 2 f p

1.1 7 1.1.12 M C M X M C X M φ t dφ t (p) = X φt (p) (t R, p M) dt φ 0 (p) = p (p M) M p p U p ɛ p > 0 dφ t (q) = X φt(q) ( t < ɛ p, q U p ) dt φ 0 (q) = q (q U p ) {U p } p M MX {U p1,..., U pk } X ɛ 0 = min{ɛ p1,..., ɛ pk } X p X p = 0 t R φ t (p) = p M t < ɛ 0 t φ t s, t, s + t < ɛ 0 s, t φ s φ t = φ s+t t R φ t t R k, rk 0 t = k(ɛ 0 /2) + r, k Z, r < ɛ 0 /2 φ t = φ t k < 0 φ t = k {}}{ φ ɛ0 /2 φ ɛ0 /2 φ r k {}}{ φ ɛ0 /2 φ ɛ0 /2 φ r φ t t φ t φ s φ t = φ s+t φ t Mφ t X φ t

8 1 Morse 1.1.13 M Riemann, M C f df fgradf M X gradf, X = df(x) gradf f gradf f f Riemann 2 f M X, Y ( f)(y ) = Y f ( 2 f)(x, Y ) = X(( f)(y )) ( f)( X Y ) = XY f ( X Y )f f 2 f 1.1.5 f Hessian M 2 f f Hessian 1.2 1.2.1 X, Y A Xf, g : X Y f A = g A (1) (3) F : X [0, 1] Y f g rela f, g A (1) F (x, 0) = f(x) (x X), (2) F (x, 1) = g(x) (x X), (3) F (x, t) = f(x) = g(x) (x A, t [0, 1]). A f g 1.2.2 f : X Y f : Y X f f 1 X, f f 1 Y fx Y 1.2.3 X A F : X [0, 1] X (1) F (x, 0) = x (x X)

1.2 9 (2) F (x, t) = x (x A, t [0, 1]) (3) F (x, 1) A (x X) A X 1.2.4 A X i : A X 1.2.5 R 1 R 2 R k R E 0 = (0,..., 0,...) E 1 = (1, 0,..., 0,...) E 2 = (0, 1, 0,..., 0,...). E q = (0,..., 0,. q 1, 0,..., 0,...) E 0, E 1, E 2,..., E q,... E 0,..., E q q q { } q q = (t 1,..., t q, 0,...) t i 0, t i 1. q > 0 F i q : q 1 q F i q(e j ) = { Ej (j < i) E j+1 (j i) 1.2.6 X σ : q X X q F X q F S q (X; F ) S q (X; F ) qx q a σ σ σ a σ σ 0 1.2.7 X X qσ 0 i q X q 1 σ (i) σ (i) = σ F i q : q 1 X q (σ) = ( 1) i σ (i) S q 1 (X; F ) i=0

10 1 Morse : S q (X; F ) S q 1 (X; F ) ( X ) = 0 B q (X; F ) = { (c) S q (X; F ) c S q+1 (X; F )} Z q (X; F ) = {c S q (X; F ) (c) = 0} S q (X; F ) = 0 B q (X; F ) Z q (X; F ) Z q (X; F )/B q (X; F ) X q H q (X; F ) 1.2.8 X { F (q = 0) H q (X; F ) = {0} (q 1) X = {x} q X q q > 0 (σ q ) = Z q (X; F ) = B q (X; F ) = σ q (t) = x (t q ) q (σ q ) = ( 1) i σ q 1 i=0 { σq 1 (q ) 0 (q ). { {0} (q ) S q (X; F ) H q (X; F ) = {0}. (q ). q = 0 Z 0 (X; F ) = S 0 (X; F ) = F σ 0 B 0 (X; F ) = {0} H 0 (X; F ) = Z 0 (X; F )/B 0 (X; F ) = F 1.2.9 X X = k X k q H q (X; F ) = k H q (X k ; F )

1.2 11 q q X qσ : q XX X k S q (X; F ) = k S q (X k ; F ) X X (S q (X; F )) S q (X k ; F ) Z q (X; F ) = k Z q (X k ; F ), Z q (X k ; F ) = Z q (X; F ) S q (X k ; F ) B q (X; F ) = k B q (X k ; F ), B q (X k ; F ) = B q (X; F ) S q (X k ; F ) H q (X; F ) = k H q (X k ; F ) 1.2.10 X 0 H 0 (X; F ) X 1.2.9 XH 0 (X; F ) 1 X 0 XX 1 σ (σ) = σ(1) σ(0) 0 c = x a x x c B 0 (X; F ) x a x = 0 x a x = 0 x 0 X x X X 1 σ x σ x (0) = x 0, σ x (1) = x c = ( ) a x x a x x 0 = ( ) (σ x (1) σ x (0)) = a x σ x x x x x c B 0 (X; F ) { B 0 (X; F ) = a x x x } a x = 0. x # : S 0 (X; F ) F ; x # a x x x a x S 0 (X; F )/ker( # ) = F. ker( # ) B 0 (X; F ) H 0 (X; F ) = F 1

12 1 Morse 1.2.11 S n = {x R n+1 x = 1} S n n n 1 { F (q = 0 n) H q (S n ; F ) = {0} () 1.2.12 f : X Y X qσ f σ Y qσ f σ S q (f) : S q (X; F ) S q (Y ; F ) ; σ a σ σ σ a σ f σ Y S q (f) = S q 1 (f) X S q (f)b q (X; F ) B q (Y ; F ), S q (f)z q (X; F ) Z q (Y ; F ) S q (f) H q (f) : H q (X; F ) H q (Y ; F ) ; [z] [S q (f)z] H q (f) f S q (1 X ) = 1 Sq (X;F ), S q (g f) = S q (g) S q (f) H q (1 X ) = 1 Hq(X;F ), H q (g f) = H q (g) H q (f) f H q (f) 1.2.13 f, g : X Y f, g H q (f), H q (g) : H q (X; F ) H q (Y ; F ) 1.2.14 f : X Y f H q (f) : H q (X; F ) H q (Y ; F ) 1.2.15 X { F (q = 0) H q (X; F ) = {0} (q 1) B n = {x R n x 1} B n { F (q = 0) H q (B n ; F ) = {0} (q 1)

1.2 13 1.2.16 A X : S q (X; F ) S q 1 (X; F ) (S q (A; F )) S q 1 (A; F ) : S q (X; F )/S q (A; F ) S q 1 (X; F )/S q 1 (A; F ) = 0 = 0 Im[ : S q+1 (X; F )/S q+1 (A; F )) S q (X; F )/S q (A; F )] Ker[ : S q (X; F )/S q (A; F ) S q 1 (X; F )/S q 1 (A; F )] Ker[ : S q (X; F )/S q (A; F ) S q 1 (X; F )/S q 1 (A; F )] Im[ : S q+1 (X; F )/S q+1 (A; F )) S q (X; F )/S q (A; F )] X A q H q (X, A; F ) A = H q (X, ; F ) = H q (X; F ) 1.2.17 A X Z q (X, A; F ) = {c S q (X; F ) c S q 1 (A; F )} B q (X, A; F ) = S q (A; F ) + S q+1 (X; F ) H q (X, A; F ) = Z q (X, A; F )/B q (X, A; F ) Ker[ : S q (X; F )/S q (A; F ) S q 1 (X; F )/S q 1 (A; F )] = Z q (X, A; F )/S q (A; F ) Im[ : S q+1 (X; F )/S q+1 (A; F )) S q (X; F )/S q (A; F )] = B q (X, A; F )/S q (A; F ) H q (X, A; F ) = Z q (X, A; F )/B q (X, A; F ) 1.2.18 A XH 0 (X, A; F ) = {0}

14 1 Morse x 0 A c = x a x x S 0 (X; F ) = Z 0 (X, A; F ) XX 1 σ x σ x (0) = x 0, σ x (1) = x ( ) a x σ x = x x a x x x a x x 0 = c x a x x 0. c S 0 (A; F ) + S 1 (X; F ) = B 0 (X, A; F ) Z 0 (X, A; F ) = B 0 (X, A; F ) 1.2.17 H 0 (X, A; F ) = Z 0 (X, A; F )/B 0 (X, A; F ) = {0}. 1.2.19 f : X Y A, B X, Y f f(a) B S q (f) S q (f) S q (f)(s q (A; F )) S q (B; F ) S q (f) : S q (X; F )/S q (A; F ) S q (Y ; F )/S q (B; F ) H q (f) : H q (X, A; F ) H q (Y, B; F ) H q (f) f 1.2.20 Z Y X i : Y X j : X X H q (i) : H q (Y, Z; F ) H q (X, Z; F ) H q (j) : H q (X, Z; F ) H q (X, Y ; F ) δ q : H q (X, Y ; F ) H q 1 (Y, Z; F ) δ q+1 Hq (Y, Z; F ) δ q Hq 1 (Y, Z; F ) δ q 1 δ 1 H0 (Y, Z; F ) {0} H q (i) H q (X, Z; F ) H q 1 (i) H q 1 (X, Z; F ) H 0 (i) H 0 (X, Z; F ) H q (j) H q (X, Y ; F ) H q 1 (j) H q 1 (X, Y ; F ) H 0 (j) H 0 (X, Y ; F )

1.2 15 Z = δ q+1 Hq (Y ; F ) δ q δ 1 H0 (Y ; F ) {0} H q (i) H q (X; F ) H 0 (i) H 0 (X; F ) H q (j) H q (X, Y ; F ) H 0 (j) H 0 (X, Y ; F ) 1.2.21 H q (B n, S n 1 ; F ) B n 1.2.18 H 0 (B n, S n 1 ; F ) = {0}. (B n, S n 1 ) H 1 (B n ; F ) H 1 (B n, S n 1 ; F ) H 0 (S n 1 ; F ) H 0 (B n ; F ) H 0 (B n, S n 1 ; F ) {0} {0} n = 1 1.2.10 n 2 q 2 {0} H 1 (B 1, S 0 ; F ) F F F {0} H 1 (B 1, S 0 ; F ) = F. {0} H 1 (B n, S n 1 ; F ) F F {0} H 1 (B n, S n 1 ; F ) = {0}. H q (B n ; F ) H q (B n, S n 1 ; F ) H q 1 (S n 1 ; F ) H q 1 (B n ; F ) {0} {0} H q (B n, S n 1 ; F ) = H q 1 (S n 1 ; F ) 1.2.11 H q (B n, S n 1 ; F ) = H q (B n, S n 1 ; F ) = { F (q = n) {0} (q n, q 2) { F (q = n) {0} (q n)

16 1 Morse 1.2.22 U A X U A i : X U X H q (i) : H q (X U, A U; F ) = H q (X, A; F ) 1.3 f M C a M a = f 1 (, a] = {x M f(x) a} 1.3.1 f M C a < b a, b f 1 [a, b] fm a M b M a M b M a M b M Riemann, f gradf f 1 [a, b] f 1 [a, b] U ŪU gradff 1 [a, b] f 1 [a, b] 1 U M C ρm X ρ X = gradf gradf 2 X f 1 [a, b] gradf/ gradf 2 U X 1.1.12 X Mφ t p Mφ t (p) f 1 [a, b] 1.1.13 ( ) d d dt f(φ t(p)) = df dt φ t(p) = gradf, d dt φ t(p) = 1. t f(φ t (p)) φ t (p) f 1 [a, b] 1 φ b a : M a M b φ t f b r t : M b M b r t (p) = { p (f(p) a) φ t(a f(p)) (p) (a f(p) b) M a M b

1.3 17 1.3.2 M M φ : S k 1 M x S k 1 φ(x) M M B k M φ B k M B k 1.3.3 f M C p f λ f(p) = c ɛ > 0 f 1 [c ɛ, c + ɛ] p f ɛ > 0 M c+ɛ M c ɛ B λ M c ɛ B λ 1.1.10 p (U; x 1,..., x n ) f(x) = f(p) (x 1 ) 2 (x λ ) 2 + (x λ+1 ) 2 + + (x n ) 2 n (x i (x)) 2 2ɛ (x U) { } λ x U (x i (x)) 2 ɛ, x λ+1 (x) = = x n (x) = 0 B λ B λ B λ S k 1 f 1 (c ɛ) = M c ɛ M c ɛ B λ M c ɛ B λ C µ : R R µ(0) > ɛ µ(r) = 0 (r 2ɛ) 1 < µ (r) 0 (r R). µ { f(x) µ((x 1 ) 2 + + (x λ ) 2 + 2(x λ+1 ) 2 + + 2(x n ) 2 ) (x U) F (x) = f(x) (x / U) M F x U U (x i ) 2 2ɛ µ((x 1 ) 2 + + (x λ ) 2 + 2(x λ+1 ) 2 + + 2(x n ) 2 ) = 0 F M C U C ξ, η : U R ξ(x) = (x 1 (x)) 2 + + (x λ (x)) 2 (x U) η(x) = (x λ+1 (x)) 2 + + (x n (x)) 2 (x U).

18 1 Morse U f = c ξ + ηf U F = c ξ + η + µ(ξ + 2η) 1 F 1 (, c + ɛ] = M c+ɛ = f 1 (, c + ɛ] U F fu µ U ξ + 2η 2ɛF fξ + 2η < 2ɛ F = f µ(ξ + 2η) f = c ξ + η c + 1 2 ξ + η = c + 1 (ξ + 2η) c + ɛ 2 U F 1 (, c + ɛ] F 1 (, c + ɛ] = f 1 (, c + ɛ] 2 F f U F fu F ξ F η = 1 µ (ξ + 2η) < 0 = 1 2µ (ξ + 2η) 1 0 df = F F dξ + ξ η dη dξ dη 0 F p F f 3 F 1 (, c ɛ] M c+ɛ F 1 (, c + ɛ] = f 1 (, c + ɛ] x M F (x) f(x) F 1 [c ɛ, c + ɛ] f 1 [c ɛ, c + ɛ] f 1 [c ɛ, c + ɛ] F 1 [c ɛ, c + ɛ] f 1 [c ɛ, c + ɛ] f p f F F 1 [c ɛ, c + ɛ] F p F (p) = c µ(0) < c ɛ p / F 1 [c ɛ, c + ɛ] F 1 [c ɛ, c + ɛ] F 1.3.1 F 1 (, c ɛ] F 1 (, c + ɛ] = M c+ɛ

1.3 19 H = F 1 (, c ɛ] M c ɛ F 1 (, c ɛ] = M c ɛ H 4 M c ɛ B λ M c ɛ H r t : M c ɛ H M c ɛ H U r t U ξ ɛ R 1 r t (u 1,..., u n ) = (u 1,..., u λ, tu λ+1,..., tu n ) r t r 1 R 1 r 0 R 1 B λ F η 1 r t(f 1 (, c ɛ]) F 1 (, c ɛ] ɛ ξ η + ɛ R 2 r t ɛ ξ η + ɛη 0 r t (u 1,..., u n ) = (u 1,..., u λ, s t u λ+1,..., s t u n ) t + (1 t) ( ) ξ ɛ 1/2 (η 0) s t = η 1 (η = 0). 0 ξ ɛ η 1 s t [0, 1] r t λ + 1 i n lim s tu i = u i η 0 η 0 (u i s t u i ) 2 = (1 s t ) 2 (u i ) 2 ( ) 1/2 2 ξ ɛ = 1 t (1 t) (u i ) 2 η ( ) 1/2 2 ξ ɛ = (1 t) 2 1 (u i ) 2 η (u i ) 2 η.

20 1 Morse r 1 R 2 t = 0 ( ξ ɛ s 1 = η ) 1/2 f(u 1,..., u λ, s 1 u λ+1,..., s 1 u n ) = c ((u 1 ) 2 + + (u λ ) 2 ) + s 2 1((u λ+1 ) 2 + + (u n ) 2 ) = c ξ + ξ ɛ η η = c ɛ. r 0 (R 2 ) f 1 (c ɛ) R 1 r t (F 1 (, c ɛ]) F 1 (, c ɛ] R 1 R 2 ξ = ɛ s t = t R 1 r t R 2 r t η + ɛ ξ R 3 r t R 3 R 3 f c ɛ R 3 M c ɛ R 2 R 3 ξ = η + ɛ R 2 s t = 1 r t R 3 r t r t : M c ɛ H M c ɛ H M c ɛ B λ M c ɛ H 1 3 4 M c ɛ B λ M c+ɛ 1.4 Morse 1.4.1 X Y S(X, Y ) S X Y Z S(X, Z) S(X, Y ) + S(Y, Z) S S X Y Z S(X, Z) = S(X, Y ) + S(Y, Z) 1.4.2 F X Y b q (X, Y ; F ) = dim H q (X, Y ; F ), χ(x, Y ; F ) = q 0( 1) q b q (X, Y ; F ) b q (X, Y ; F ) χ(x, Y ; F ) b q (X, Y ; F ) (X, Y ) q Betti χ(x, Y ; F ) (X, Y ) Euler q Betti Euler

1.4 Morse 21 1.2.20 H q (Y, Z; F ) Hq(i) H q (X, Z; F ) Hq(j) H q (X, Y ; F ) dim H q (X, Z; F ) = dim ImH q (j) + dim KerH q (j) = dim ImH q (j) + dim ImH q (i) dim H q (X, Y ; F ) + dim H q (Y, Z; F ). b q (X, Z; F ) b q (X, Y ; F ) + b q (Y, Z; F ) q Betti 1.2.20 δ q+1 Hq (Y, Z; F ) δ q Hq 1 (Y, Z; F ) H q (i) H q (X, Z; F ) H q 1 (i) H q 1 (X, Z; F ) H q (j) H q (X, Y ; F ) H q 1 (j) H q 1 (X, Y ; F ) r q = dim ImH q (i) = dim KerH q (j) s q = dim ImH q (j) = dim Kerδ q t q = dim Imδ q = dim KerH q 1 (i) b q (X, Z; F ) = dim H q (X, Z; F ) = r q + s q b q (X, Y ; F ) = dim H q (X, Y ; F ) = s q + t q b q (Y, Z; F ) = dim H q (Y, Z; F ) = t q+1 + r q. χ(x, Z; F ) = q 0( 1) q (r q + s q ) Euler = ( 1) q (s q + t q ) + q 0 q 0( 1) q (t q+1 + r q ) = χ(x, Y ; F ) + χ(y, Z; F ) 1.4.3 S X 0 X 1 X n S n S(X n, X 0 ) S(X i, X i 1 )

22 1 Morse S n S(X n, X 0 ) = S(X i, X i 1 ) n S n = 1 n 1 S(X n 1, X 0 ) n 1 S(X i, X i 1 ) n S(X n, X 0 ) S(X n, X n 1 ) + S(X n 1, X 0 ) S(X i, X i 1 ) S 1.4.4 (Morse ) Mf M C q C q b q (M; F ) C q, χ(m; F ) = q 0( 1) q C q c 1 <... < c k f a 0 < c 1 < a 1 <... < c i < a i <... < c k < a k a 0 < a 1 <... < a k f 1 (c i ) λ j p j (1 j l) 1.3.3 p j M c i+ɛ M c i ɛ B λ j (1 j l) M c i ɛ B λ 1 B λ l 1.2.22 H q (M a i, M a i 1 ; F ) = H q (M c i+ɛ, M c i ɛ ; F ) = H q (M c i ɛ B λ 1 B λ l, M c i ɛ ; F ) = H q (B λ 1, B λ 1 ; F ) H q (B λ l, B λ l ; F ) = f 1 (c i ) q F b q (M a i, M a i 1 ; F ) = dim H q (M a i, M a i 1 ; F ) = f 1 (c i ) q

1.4 Morse 23 1.4.3 = M a 0 M a k = M S = bq b q (M; F ) = b q (M a k, M a 0 ; F ) k b q (M a i, M a i 1 ; F ) = k = C q (f 1 (c i ) q ) b q (M; F ) C q 1.4.3 = M a 0 M a k = M S = χ χ(m; F ) = χ(m a k, M a 0 ; F ) = k χ(m a i, M a i 1 ; F ) = = k q 0 k q 0 = q 0( 1) q C q ( 1) q b q (M a i, M a i 1 ; F ) ( 1) q (f 1 (c i ) q ) χ(m; F ) = q 0( 1) q C q 1.4.5 X Y S q (X, Y ) = S q q ( 1) i b q i (X, Y ; F ) i=0 1.4.2 S q (Y, Z) S q (X, Z) + S q (X, Y )

24 1 Morse q = ( 1) i (b q i (Y, Z; F ) b q i (X, Z; F ) + b q i (X, Y ; F )) i=0 q = ( 1) i (t q i+1 + r q i r q i s q i + s q i + t q i ) i=0 = t q+1 0. S q (X, Z) S q (Y, Z) + S q (X, Y ) S q 1.4.6 (Morse ( )) 1.4.4 q q ( 1) i b q i (M; F ) ( 1) i C q i i=0 i=0 1.4.4 1.4.4 b q (M a i, M a i 1 ; F ) = f 1 (c i ) q 1.4.5 S q 1.4.3 = M a 0 M a k = M S = Sq S q (M) = S q (M a k, M a 0 ) k S q (M a i, M a i 1 ) = k q ( 1) j b q j ((M a i, M a i 1 ; F ) j=0 = k q ( 1) j (f 1 (c i ) q j ) j=0 = q ( 1) j C q j j=0 q S q (M) ( 1) j C q j j=0

1.4 Morse 25 1.4.7 1.4.6 1.4.4 q b q (M; F ) b q 1 (M; F ) + ± b 0 (M; F ) C q C q 1 + ± C 0 b q 1 (M; F ) b q 2 (M; F ) + b 0 (M; F ) C q 1 C q 2 + C 0 b q (M; F ) C q n = dim M b n (M; F ) b n 1 (M; F ) + ± b 0 (M; F ) C n C n 1 + ± C 0 b n+1 (M; F ) b n (M; F ) + b 0 (M; F ) C n+1 C n + C 0 b n+1 (M; F ) = C n+1 = 0 b n (M; F ) b n 1 (M; F ) + ± b 0 (M; F ) = C n C n 1 + ± C 0 χ(m; F ) = ( 1) i b i (M; F ) = i 0 i 0( 1) i C i 1.4.8 1.4.4 q C q+1 = C q 1 = 0 b q+1 (M; F ) = b q 1 (M; F ) = 0 b q (M; F ) = C q 1.4.4 0 b q+1 (M; F ) C q+1 = 0 0 b q 1 (M; F ) C q 1 = 0 b q+1 (M; F ) = b q 1 (M; F ) = 0 1.4.6 b q (M; F ) b q 1 (M; F ) + ± b 0 (M; F ) C q C q 1 + ± C 0 b q+1 (M; F ) b q (M; F ) + b 0 (M; F ) C q+1 C q + C 0 b q+1 (M; F ) = C q+1 = 0 b q (M; F ) b q 1 (M; F ) + ± b 0 (M; F ) = C q C q 1 + ± C 0 b q 1 (M; F ) = C q 1 = 0 b q 2 (M; F ) b q 3 (M; F ) + ± b 0 (M; F ) = C q 2 C q 3 + ± C 0 b q (M; F ) = b q (M; F ) b q 1 (M; F ) = C q C q 1 = C q b q (M; F ) = C q

26 1 Morse 1.5 1.5.1 (Reeb) M n f : M R C M n f p M q M f(p) = a, f(q) = b p 0 1.1.10 p (U; x 1,..., x n ) f(x) = f(p) + (x 1 ) 2 + + (x n ) 2 ɛ > 0 f 1 [a, a + ɛ] B n q n 1.1.10 p (V ; y 1,..., y n ) f(y) = f(q) (y 1 ) 2 (y n ) 2 ɛ > 0 f 1 [b ɛ, b] B n f 1 [a + ɛ, b ɛ] f 1.3.1 f 1 [a, a + ɛ] f 1 [a, b ɛ] f 1 [a, b ɛ] B n M = f 1 [a, b ɛ] f 1 [b ɛ, b] M n 1.5.2 C Betti n CP n C = C {0} C n+1 {0} C n+1 {0} (C n+1 {0})/C CP n CP n U(1) = {λ C λ = 1} S 2n+1 = {z C n+1 z = 1} S 2n+1 S 2n+1 /U(1) CP n = S 2n+1 /U(1) (z 0, z 1,..., z n ) S 2n+1 CP n [z 0, z 1,..., z n ] 0 i n U i = {[z 0, z 1,..., z n ] CP n z i 0}

1.5 27 [z] = [z 0, z 1,..., z n ] U i λ U(1) w j = z i z j z i λz i λz j λz i = z i z j z i w j w j = z j [z] (w 0,..., ŵ i,..., w n ) C n = R 2n U i ŵ i w i c 0 < c 1 < < c n CP n C f λ U(1) n f[z 0,..., z n ] = c j z j 2 j=0 n n c j λz j 2 = c j z j 2 j=0 j=0 f f (U i ; w 0,..., ŵ i,..., w n ) [z 0,..., z n ] U i z i 2 = 1 z j 2 j i f = c i 1 z j 2 + c j z j 2 j i j i = c i + (c j c i ) z j 2 j i = c i + (c j c i ) w j 2 j i f CP n C U i i p i = [0,..., 1,..., 0] fp i 2#{c j c j c i < 0} = 2i f f k C k { 1 (k0 k 2n) C k = 0 ()

28 1 Morse 1.4.8 { 1 (k0 k 2n) b k (CP n ; F ) = 0 (). H k (CP n ; F ) = { F (k0 k 2n) {0} (). 1.5.3 C Betti n HP n H = H {0} H n+1 {0} H n+1 {0} (H n+1 {0})/H HP n HP n Sp(1) = {λ H λ = 1} S 4n+1 = {z H n+1 z = 1} S 4n+1 S 4n+1 /Sp(1) HP n = S 4n+1 /Sp(1) (z 0, z 1,..., z n ) S 4n+1 HP n [z 0, z 1,..., z n ] 0 i n [z] = [z 0, z 1,..., z n ] U i λ Sp(1) U i = {[z 0, z 1,..., z n ] HP n z i 0} w j = z i z j z 1 i z i λ z j λ(z i λ) 1 = z i z j λλ 1 z i = z i z j z 1 i w j w j = z j [z] (w 0,..., ŵ i,..., w n ) H n = R 4n U i c 0 < c 1 < < c n HP n C f n f[z 0,..., z n ] = c j z j 2 j=0

1.5 29 λ Sp(1) n n c j z j λ 2 = c j z j 2 j=0 j=0 f f (U i ; w 0,..., ŵ i,..., w n ) [z 0,..., z n ] U i z i 2 = 1 z j 2 j i f = c i 1 z j 2 + c j z j 2 j i j i = c i + (c j c i ) z j 2 j i = c i + (c j c i ) w j 2 j i f HP n C U i i p i = [0,..., 1,..., 0] fp i 4#{c j c j c i < 0} = 4i f f k C k { 1 (k 4 0 k 4n) C k = 0 () 1.4.8 { 1 (k 4 0 k 4n) b k (HP n ; F ) = 0 (). H k (HP n ; F ) = { F (k 4 0 k 4n) {0} (). 1.5.4 0 < b < a 2 T 2 T 2 = {((a + b cos u) cos v, (a + b cos u) sin v, b sin u) u, v R} u, v T 2 T 2 f f(u, v) = (a + b cos u) cos v

30 1 Morse f T 2 C f f f u = b sin u cos v, f v = (a + b cos u) sin v. 0 (u, v) = (0, 0), (π, 0), (0, π), (π, π) f f Hessian f 2 f = b cos u cos v, u2 2 f = b sin u sin v. u v 2 f = (a + b cos u) cos v, v2 f Hessian det 2 f = b(a + b cos u) cos u cos 2 v b 2 sin 2 u sin 2 v tr 2 f = (a + 2b cos u) cos v. f (u, v) = (0, 0) T 2 (a + b, 0, 0) det( 2 f) (0,0) = b(a + b) > 0 ( 2 f) (0,0) tr( 2 f) (0,0) = (a + 2b) < 0 ( 2 f) (0,0) 2 (u, v) = (π, 0) T 2 (a b, 0, 0) det( 2 f) (π,0) = b(a b) < 0 ( 2 f) (π,0) 1 (u, v) = (0, π) T 2 ( (a + b), 0, 0) det( 2 f) (0,π) = b(a + b) > 0 ( 2 f) (0,π) tr( 2 f) (0,π) = a + 2b > 0 ( 2 f) (0,π) 0 (u, v) = (π, π) T 2 ( (a b), 0, 0) det( 2 f) (π,π) = b(a b) < 0 ( 2 f) (π,0) 1 f 0 C 0 = 1 f 1 C 1 = 2 f 2 C 2 = 1 1.4.4 χ(t 2 ; F ) = C 0 C 1 + C 2 = 0. T 2 2 b 0 (T 2 ; F ) = b 2 (T 2 ; F ) = 1 0 = χ(t 2 ; F ) = b 0 (T 2 ; F ) b 1 (T 2 ; F ) + b 2 (T 2 ; F ) = 2 b 1 (T 2 ; F ) b 1 (T 2 ; F ) = 2 H 0 (T 2 ; F ) = F, H 1 (T 2 ; F ) = F F, H 2 (T 2 ; F ) = F.

2 Riemann 2.1 2.1.1 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 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 MRiemann M T M C M 2.1.2 2.1.1 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 ι : M ( M, g) Riemann ( M, g) Riemann 2.1.3 M Levi-Civita M X, Y C (T M) X Y T M M C X Y = X Y + h(x, Y ) ( X Y T M, h(x, Y ) T M) Riemann Levi-Civita h L 2 (T M, T M) C h 31

32 2 Riemann x M c(0) = x c (0) = X x M c c c(t) c(0) τ 0 t ( 1 X Y )(x) = lim t 0 t (τ 0Y t c(t) Y x ) X Y T M M C a, b M C ax (by ) = a((xb)y + b X Y ) = (a(xb)y + ab X Y ) + abh(x, Y ) ax (by ) = a(xb)y + ab X Y h(ax, by ) = abh(x, Y ) T Mh L 2 (T M, T M) C M Levi-Civita 0 = X Y Y X [X, Y ] = X Y + h(x, Y ) Y X h(y, X) [X, Y ] = ( X Y Y X [X, Y ]) + (h(x, Y ) h(y, X)). T M T M X Y Y X [X, Y ] = 0 h(x, Y ) h(y, X) = 0 h M Levi-Civita M Riemann gm X, Y, Z C (T M) X(g(Y, Z)) = X( g(y, Z)) = g( X Y, Z) + g(y, X Z) = g( X Y, Z) + g(y, X Z) = g( X Y, Z) + g(y, X Z). M Riemann gm Levi-Civita M Levi-Civita

2.1 33 2.1.4 2.1.3 h M X, Y C (T M) X Y Gauss X Y = X Y + h(x, Y ) ( X Y T M, h(x, Y ) T M) 2.1.5 M Levi-Civita M X C (T M) ξ C (T M) X ξ T M M C X ξ = A ξ X + Xξ (A ξ X T M, Xξ T M) T M Riemann T M T M A L(T M, Sym(T M)) C Sym(T M) T M X ξ T M M C 2.1.3 a, b M C ax (bξ) = a((xb)ξ + b X ξ) = aba ξ X + (a(xb)ξ + ab Xξ) ax(bξ) = a(xb)ξ + ab Xξ A bξ (ax) = aba ξ X T MA L(T M, End(T M)) C M Riemann gmξ, η C (T M) X( g(ξ, η)) = g( X ξ, η) + g(ξ, X η) = g( Xξ, η) + g(ξ, Xη). T M g A ξ Sym(T M) 2.1.6 M X, Y C (T M) ξ C (T M) g(h(x, Y ), ξ) = g(a ξ X, Y )

34 2 Riemann Y Mξ M g(y, ξ) = 0 0 = X( g(y, ξ)) = g( X Y, ξ) + g(y, X ξ) = g(h(x, Y ), ξ) + g(y, A ξ X). g(h(x, Y ), ξ) = g(a ξ X, Y ) 2.1.5 2.1.6 2.1.3 g(a ξ X, Y ) = g(h(x, Y ), ξ) = g(h(y, X), ξ) = g(a ξ Y, X). A ξ 2.1.7 2.1.5 MA M X C (T M) ξ C (T M) X ξ X ξ = A ξ X + Xξ (A ξ X T M, Xξ T M) Weingarten R R ξ C (T M) ξ = 0 X C (T M) Xξ = 0 ξ 2.1.8 h = 0 Riemann M M M M n H = tr(h) = h(e i, e i ) ({e i } T M ) HH T M C H M H = 0 Riemann M 2.2 Riemann Gauss Codazzi Ricci ι : M M Riemann M Riemann M M Riemann,

2.2 35 2.2.1 (Gauss ) M M R R M X, Y, Z, W R(X, Y )Z, W = R(X, Y )Z, W + h(x, Z), h(y, W ) h(x, W ), h(y, Z) Gauss Weingarten R(X, Y )Z = X Y Z Y X Z [X,Y ] Z = X ( Y Z + h(y, Z)) Y ( X Z + h(x, Z)) ( [X,Y ] Z + h([x, Y ], Z)) = X Y Z + h(x, Y Z) A h(y,z) X + X(h(Y, Z)) Y X Z h(y, X Z) + A h(x,z) Y Y (h(x, Z)) [X,Y ] Z h([x, Y ], Z) = R(X, Y )Z A h(y,z) X + A h(x,z) Y +h(x, Y Z) h(y, X Z) h([x, Y ], Z) + X(h(Y, Z)) Y (h(x, Z)). 2.1.6 R(X, Y )Z, W = R(X, Y )Z, W A h(y,z) X, W + A h(x,z) Y, W = R(X, Y )Z, W h(x, W ), h(y, Z) + h(y, W ), h(x, Z). Codazzi L 2 (T M, T M) 2.2.2 a C (L 2 (T M, T M)) X, Y, Z C (T M) ( X a)(y, Z) = X(a(Y, Z)) a( X Y, Z) a(y, X Z) : C (T M) C (L 2 (T M, T M)) C (L 2 (T M, T M)) L 2 (T M, T M) f, g C (M) ( X a)(fy, gz) = X(a(fY, gz)) a( X (fy ), gz) a(fy, X (gz)) = X(fg)a(Y, Z) + fg X(a(Y, Z))

36 2 Riemann (Xf)ga(Y, Z) fga( X Y, Z) f(xg)a(y, Z) fga(y, X Z) = fg( X(a(Y, Z)) a( X Y, Z) a(y, X Z)) = fg( X a)(y, Z) X a C (L 2 (T M, T M)) T M T M ( X (fa))(y, Z) fx a = f X a = X(fa(Y, Z)) fa( X Y, Z) fa(y, X Z) = (Xf)a(Y, Z) + f X(a(Y, Z)) fa( X Y, Z) fa(y, X Z) = (Xf)a(Y, Z) + ( X a)(y, Z) X (fa) = (Xf)a + f X a L 2 (T M, T M) 2.2.3 (Codazzi ) M X, Y, Z C (T M) R(X, Y )Z ( R(X, Y )Z) = ( X h)(y, Z) ( Y h)(x, Z) 2.2.1 R(X, Y )Z = R(X, Y )Z A h(y,z) X + A h(x,z) Y +h(x, Y Z) h(y, X Z) h([x, Y ], Z) + X(h(Y, Z)) Y (h(x, Z)). ( R(X, Y )Z) = h(x, Y Z) h(y, X Z) h([x, Y ], Z) + X(h(Y, Z)) Y (h(x, Z)) = h(x, Y Z) h(y, X Z) h( X Y, Z) + h( Y X, Z) + X(h(Y, Z)) Y (h(x, Z)) = ( X h)(y, Z) ( Y h)(x, Z).

2.2 37 2.2.4 (Ricci ) M X, Y C (T M) C (T M) ξ, η R(X, Y )ξ, η = R (X, Y )ξ, η [A ξ, A η ]X, Y Weingarten R(X, Y )ξ, η = X Y ξ Y X ξ [X,Y ] ξ, η = X ( A ξ Y + Y ξ) Y ( A ξ X + Xξ) ( A ξ [X, Y ] + [X,Y ]ξ), η = h(x, A ξ Y ) + X Y ξ h(y, A ξ X) Y Xξ [X,Y ]ξ, η = R (X, Y )ξ, η h(x, A ξ Y ), η + h(y, A ξ X), η = R (X, Y )ξ, η A η X, A ξ Y + A η Y, A ξ X = R (X, Y )ξ, η A ξ A η X, Y + Y, A η A ξ X = R (X, Y )ξ, η [A ξ, A η ]X, Y. 2.2.5 M K Riemann M Gauss, Codazzi, Ricci M X, Y, Z, W C (T M) ξ, η C (T M) K( Y, Z X, W X, Z Y, W ) = R(X, Y )Z, W + h(x, Z), h(y, W ) h(x, W ), h(y, Z), ( X h)(y, Z) = ( Y h)(x, Z), R (X, Y )ξ, η = [A ξ, A η ]X, Y. M R M S, T, U R(S, T )U = K( T, U S S, U T ) 2.2.1(Gauss ) K( Y, Z X, W X, Z Y, W ) = R(X, Y )Z, W + h(x, Z), h(y, W ) h(x, W ), h(y, Z) 2.2.3(Codazzi ) ( X h)(y, Z) ( Y h)(x, Z) = ( R(X, Y )Z) = K( Y, Z X X, Z Y ) = 0.

38 2 Riemann ( X h)(y, Z) = ( Y h)(x, Z) 2.2.4(Ricci ) R (X, Y )ξ, η [A ξ, A η ]X, Y = R(X, Y )ξ, η = K( Y, ξ X, η X, ξ Y, η ) = 0. R (X, Y )ξ, η = [A ξ, A η ]X, Y 2.3 Gauss R 3 2 Riemann () Gauss R n+r n Riemann R 2 1 Riemann () c(s) R 2 s c (s) 1 e 1 (s) = c (s) e 1 (s) π/2 e 2 (s) c(s) κ(s) e 1(s) = κ(s)e 2 (s) Frenet-Serret c(s) A e 2(s) = κ(s)e 1 (s) e 2(s) = e1 (s)e 2 (s) = A e2 (s)e s (s) A e2 (s)e 1 (s) = κ(s)e 1 (s). 2.3.1 ι : M R 3 2 Riemann x M ξ T x M A ξ : T x M T x M M x ξ M x Gauss G(x) G(x) = det A ξ

2.3 Gauss 39 2.3.2 2.3.1 T x M 1 ξ ξ ξ A ξ = A ξ Gauss ξ M ι Gauss M Riemann Gauss ( 2.2.1 2.2.5) M X, Y, Z, W R(X, Y )Z, W = R(X, Y )Z, W + h(x, Z), h(y, W ) h(x, W ), h(y, Z) R 3 R 0 2.1.6 0 = R(X, Y )Z, W + h(x, Z), h(y, W ) h(x, W ), h(y, Z). R(X, Y )Y, X = h(x, X), h(y, Y ) h(x, Y ), h(x, Y ) = h(x, X), ξ h(y, Y ), ξ h(x, Y ), ξ h(x, Y ), ξ = A ξ X, X A ξ Y, Y A ξ X, Y A ξ X, Y. X, Y MM K K = R(X, Y )Y, X = det A ξ Gauss K M Riemann 2.3.3 ι : M R n+1 n Riemann x M ξ T x M A ξ : T x M T x M M x ξ n M x ξ Gauss-Kronecker G(x, ξ) G(x, ξ) = det A ξ 2.3.4 2.3.3 T x M 1 ξ ξ ξ A ξ = A ξ G(x, ξ) = det A ξ = det( A ξ ) = ( 1) n det A ξ = ( 1) n G(x, ξ). M Gauss-Kronecker G(x, ξ) ξ M 1 Gauss-Kronecker 1 M ι M Gauss-Kronecker M Riemann det A ξ 2 det A ξ 2 2.3.2 M M Riemann 2.3.5 ι : M R n+r n Riemann x M ξ T x M A ξ : T x M T x M M x ξ n M x ξ Lipschitz-Killing G(x, ξ) G(x, ξ) = det A ξ

3 Chern-Lashof 3.1 Riemann 3.1.1 V V V A V V α A(x, y) = (α(x))(y) (x, y V ) V V V V Hom(V, V ) α Hom(V, V ) V V A 0 x V (α(x))(x) > 0 A V α Hom(V, V ) A(x, y) = (α(x))(y) (x, y V ) A V V A V V α Hom(V, V ) V V Hom(V, V ) A 0 x V (α(x))(x) > 0 A(x, x) > 0 A V 3.1.2 V,, 3.1.1 Hom(V, V ) α p V ( p V ) α : V V α (p,0) : p V p V = ( p V ) ( p V ) ( 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 ) 40

3.1 Riemann 41 α (p,0) ( p V ) ( 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 ) = σ S p sgn(σ) u σ(1), v 1 u σ(p), v p = det( u i, v j ) 1 i,j p. u 1,..., u n V u i1 u ip (1 i 1 < < i p n) p V 1 i 1 < < i p n 1 j 1 < < j p n A(u i1 u ip, u j1 u jp ) = δ i1 j 1 δ ip j p A p V 3.1.3, V p V 3.1.2 A A, u = u, u 3.1.4 3.1.2V u 1,..., u p v 1,..., v p u 1 u p, v 1 v p = det( u i, v j ) 1 i,j p V u 1,..., u n u i1 u ip (1 i 1 < < i p n) p V 3.1.5 V u 1, u 2 θ u 1, u 2 = u 1 u 2 cos θ

42 3 Chern-Lashof 3.1.4 u 1 u 2 2 = u 1 2 u 2 2 u 1, u 2 2 = u 1 2 u 2 2 u 1 2 u 2 2 cos 2 θ = u 1 2 u 2 2 sin 2 θ u 1 u 2 = u 1 u 2 sin θu 1 u 2 2 V V 3 3.1.6 V W n F : V W V v 1,..., v n JF = F (v 1 v n ) v 1 v n JF v 1,..., v n = F (v 1) F (v n ) v 1 v n v 1,..., v n V v 1,..., v n (a ij ) n v j = a ij v i v 1 v n = det(a ij )v 1 v n F (v 1 v n) = det(a ij )F (v 1 v n ). F (v 1 v n) v 1 v n = det(a ij) F (v 1 v n ) det(a ij ) v 1 v n = F (v 1 v n ). v 1 v n 3.1.7 X 2 X [0, ] µ µ X (1) µ( ) = 0 (2) X {A i } A A i A 2 X µ(a) µ(a i ). 3.1.8 µ XX A µ(t ) = µ(t A) + µ(t A) T 2 X A X µ

3.1 Riemann 43 3.1.9 f µ X S [, ] µ(x S) = 0 [, ] O f 1 (O) X µf µ 3.1.10 R n Lebesgue Lebesgue Fubini 3.1.11 X σ Borel µ XX Borel µ A X Borel B A B µ(a) = µ(b) µ Borel 3.1.12 X Hausdorff X Borel µ K Xµ(K) < µ Radon 3.1.13 (Riesz ) X Hausdorff X K(X) K(X) L : K(X) R (1) f 0 f K(X) L(f) 0 (2) K X sup{l(f) f K(X), f 1, suppf K} < Radon µ X L(f) = X fdµ (f K(X)) 3.1.14 C 3.1.13 3.1.15 (M, g) Riemann M (U; x 1,..., x n ) x U n T x (M) Riemann suppf U f K(M) L(f) = U f(x 1,..., x n ) x 1 x n dx 1 dx n L(f) Euclid Lebesgue Riemann L(f)

44 3 Chern-Lashof K(M) L : K(M) R L 3.1.13Radon µ M L(f) = M fdµ (f K(M)) µ M Riemann Riemann Riemann µ µ (M,g) Riemann µ M vol(m) = µ M (M) vol(m) M M 1 M 2 3.1.16 n n Riemann 3.1.17 M Riemann (U; x 1,..., x n ) M U µ M φµ M φ 0 U φdµ M = U φ(x 1,..., x n ) x 1 x n dx 1 dx n 3.1.18 f : M N M N C x M df x : T x (M) T f(x) (N) x fm y N f(x) = y f x y fn 3.1.19 (Sard ) U R n f : U R p C f CR p Lebesgue µ µ(f(c)) = 0 3.1.20 f : M N Riemann M Riemann N C f C µ N (f(c)) = 0 M NM N {U i } {V i } i (1) U i M (2) V i N (3) f(u i ) V i

3.1 Riemann 45 C i = C U i V i N (V i ; x 1,..., x p ) V i R p R p Lebesgue µ 3.1.19 µ(f(c i )) = 0 x 1 x p V i V i ( V i ) A = sup V i 3.1.17 µ N (f(c i )) = 0 x 1 x p 0 µ N (f(c i )) Aµ(f(C i )) = 0 0 µ N (f(c)) i µ N (f(c i )) = 0, µ N (f(c)) = 0 3.1.21 f : M N n Riemann M n Riemann N C x M 3.1.6 J Jf(x) = Jdf x 3.1.22 () f : M N n Riemann M n Riemann N C φ Mµ M N y φ(x) Nµ N φjf M µ M x f 1 (y) φ 0 N x f 1 (y) φ(x) dµ N (y) = M φ(x)jf(x)dµ M (x) Jf M O = {x M Jf(x) 0} MO µ M φ(x)jf(x)dµ M (x) = φ(x)jf(x)dµ M (x) M O

46 3 Chern-Lashof x O x U x f(u x ) f(x) f : U x f(u x ) f Euclid Euclid M N R n f : M N x 1,..., x n M R n y 1,..., y n N R n φ M φ f y 1 y n N Mf y φ(f 1 (y)) y 1 y n N 3.1.17 df ( ) ( ) x φjfdµ M = φ 1 df x n M M x 1 x 1 x n dx 1 dx n x n ( ) ( ) = φ df df x M 1 x n dx 1 dx n ( ) = φ det yi f x M j y 1 y n dx 1 dx n = φ f 1 y N 1 y n dy 1 dy n = φ f 1 dµ N N f Euclid Euclid x O x U x f(u x ) f(x) f : U x f(u x ) Euclid Euclid O{U x } x O OM O O {U x } x O {U k } {U k } O{U k } {ψ k } f U k f k : U k V k = f(u k ) ψ k φ y (ψ k φ)(fk 1 (y)) N

3.2 Γ 47 V k µ N V k (ψ k φ)(f 1 k (y))dµ V k (y) = y k (ψ k φ)(fk 1 (y)) Nµ N {ψ k } U k ψ k φjfdµ Uk. (ψ k φ)(fk 1 (y)) = φ(x) k x f 1 (y) y x f 1 (y) Nµ N φjf M µ M Lebesgue φ 0 Lebesgue M φjfdµ M = k = k = = φ(x) ψ k φjfdµ Uk U k (ψ k φ)(fk 1 (y))dµ V k (y) V k ( ) (ψ k φ)(fk 1 (y)) dµ N (y) N k N x f 1 (y) φ(x) dµ N (y) φjfdµ M = M φ(x) dµ N (y) N x f 1 (y) 3.1.23 3.1.22 f 3.2 Γ Γ Euclid Riemann ( 3.3.2) Lipschitz-Killing ( 2.3.5) ΓΓ

48 3 Chern-Lashof 3.2.1 Γ(z) Γ(z) = Γ(z) Γ 3.2.2 Γ(z + 1) = zγ(z) (z > 0) 0 e t t z 1 dt (z > 0) z > 0 Γ(z + 1) = 0 e t t z dt = [ e t t z ] o + Γ(z + 1) = zγ(z) 0 e t zt z 1 dt = zγ(z) 3.2.3 n Γ(n) = (n 1)! 0! = 1 Γ(1) = 0 e 1 dt = [ e t ] 0 = 1 Γ(1) = 1 3.2.2 z > 0 n Γ(z + n) = (z + n 1)(z + n 2) zγ(z) Γ(1) = 1 Γ(n) = (n 1)! 3.2.4 Γ(z) = 2 e t2 t 2z 1 dt (z > 0) 0 Γ(z) t = s 2 Γ(z) = 0 e z2 s 2z 2 2sds = 2 e s2 s 2s 1 ds. 0 3.2.5 Γ(p)Γ(q) π/2 Γ(p + q) = 2 cos 2p 1 θ sin 2q 1 θdθ (p, q > 0) 0

3.2 Γ 49 3.2.4 Γ(p) = 2 Γ(q) = 2 0 0 e u2 u 2p 1 du e v2 v 2q 1 dv Γ(p)Γ(q) = 4 0 0 u = r cos θ, v = r sin θ Γ(p)Γ(q) = 4 = 2 π/2 0 0 0 = Γ(p + q)2 e u2 v 2 u 2p 1 v 2q 1 dudv. e r2 r 2p 1 cos 2p 1 θr 2q 1 sin 2q 1 θrdrdθ e r2 r 2p+2q 1 dr2 π/2 0 π/2 0 cos 2p 1 θ sin 2q 1 θdθ. cos 2p 1 θ sin 2q 1 θdθ Γ(p)Γ(q) π/2 Γ(p + q) = 2 cos 2p 1 θ sin 2q 1 θdθ 0 3.2.6 Γ ( ) 1 2 = π. 3.2.5 p = q = 1/2 Γ(1) = 1 ( ) 1 2 π/2 Γ = 2 dθ = π. 2 0 3.2.7 3.2.4 ( 1 Γ = 2) π. vol(s n 1 ) = 2Γ( 1 2 )n Γ ( ) n 2 Γ(p i ) = 2 0 e u2 i u 2p i 1 i du i Γ(p 1 ) Γ(p n ) = 2 n e (u2 1 + +u2 n ) u 2p 1 1 1 u 2p n 1 n du 1 du n. 0 0

50 3 Chern-Lashof φ : (0, ) (S n 1 (0, ) n ) (0, ) n ; (r, ξ) rξ φφ Jφ(r, ξ) = r n 1 Γ(p 1 ) Γ(p n ) = 2 n e r2 r 2(p 1+ +p n) n ξ 2p 1 1 0 S n 1 (0, ) n 1 ξn 2pn 1 r n 1 dµ S n 1(ξ)dr = 2 e r2 r 2(p 1+ +p n ) 1 dr2 n 1 ξ 2p 1 1 0 S n 1 (0, ) n 1 ξ 2p n 1 n dµ S n 1(ξ) = Γ(p 1 + + p n )2 n 1 1 ξ 2p n 1 dµ S n 1(ξ). S n 1 (0, ) n ξ 2p 1 1 Γ(p 1 ) Γ(p n ) Γ(p 1 + + p n ) = 2n 1 ξ 2p 1 1 S n 1 (0, ) n 1 ξ 2p n 1 n dµ S n 1(ξ). p 1 = = p n = 1/2 Γ ( ) n 1 2 Γ ( n 2 ) = 2 n 1 vol(sn 1 ) = vol(sn 1 ). 2 n 2 n ) n vol(s n 1 ) = 2Γ ( 1 2 Γ ( ) n 2 3.3 3.3.1 ι : M M M Riemann Mx M ξ Tx MX T x M c(0) = x, c (0) = X M c(t) ξ(0) = ξ ξ(t) ξ(t) T M ξ (0) c(t) ξ (0) T ξ (T M) X T x M X dπ ξ X = X T x M T ξ (T M) ; X X H ξ = {X X T x M} H ξ ξdπ ξ : H ξ T x M V ξ = Ker(dπ ξ ) T ξ (T M)

3.3 51 V ξ ξv ξ T x M T ξ (T M) = H ξ + V ξ H ξ dπ ξ : H ξ T x M T x M V ξ T x MH ξ V ξ T ξ (T M) T M Riemann x M U x M = {u T x M u = 1} U M = x M U x M U M π : T M M U M ππ : U M M T M Riemann U M Riemann U M Riemann 3.3.2 ι : M R n+r n M Euclid M Lipschitz-Killing U M M τ(m, ι) = 1 det A vol(s n+r 1 ξ dµ ) U M(ξ) U M 3.3.3 R 2 c(s) κ(s) 2.3 e 2 (s) κ(s) e 2 (s) e 2 (s) e 2 (s) π : U c c τ(c) = 2 κ(s) ds = 1 κ(s) ds. vol(s 1 ) c π c π 3.3.4 c(s) R 1+r s e 1 (s) = c (s) c(s) κ(s) e 1(s) c h e 1(s) = e1 (s)e 1 (s) = h(e 1 (s), e 1 (s)). κ(s) = h(e 1 (s), e 1 (s)) ξ U c h(e 1 (s), e 1 (s)), ξ = A ξ (e 1 ), e 1 det A ξ = h(e 1 (s), e 1 (s)), ξ

52 3 Chern-Lashof τ(c) = = = 1 vol(s r ) 1 vol(s r ) 1 vol(s r ) U c c c det A ξ dµ U c(ξ) det A ξ dµ U U c(s) c c(s) c (ξ)ds U c(s) c h(e 1 (s), e 1 (s)), ξ dµ U c(s) c (ξ)ds. h(e 1 (s), e 1 (s)), ξ dµ U U c(s) c c(s) c (ξ) ξ r dµ S r 1(ξ) p = (0,..., 0, 1) S r 1 S r 1 φ : S r 2 ( π/2, π/2) S r 1 {±p} ; (u, θ) (cos θu, sin θ) φφ Jφ(u, θ) = cos r 2 θ π/2 ξ r dµ S r 1(ξ) = vol(s r 2 ) sin θ cos r 2 θdθ S r 1 π/2 3.2.7 = vol(s r 2 )2 π/2 vol(s r 2 ) = 2Γ( 1 2 )r 1 Γ( r 1 2 ) Γ ( 3.2.5) p = r 1 2, q = 1 2 π/2 0 π/2 vol(s r 2 )2 0 Γ(p)Γ(q) π/2 Γ(p + q) = 2 cos 2p 1 θ sin 2q 1 θdθ 0 cos r 2 θ sin θdθ = 0 cos r 2 θ sin θdθ. r 1 Γ( )Γ(1) 2 Γ( r+1) = Γ( r 1 ) 2 Γ( r+1). 2 2 cos r 2 θ sin θdθ. = 2Γ( 1 2 )r 1 Γ( r 1 = 2Γ( 1 2 )r 1 Γ( r+1 = ) 2 ) 2 ) Γ( r 1 Γ( r+1 2 2 ) 1 2Γ( 1 Γ( 1 2 )r+1 2 )2 Γ( r+1 = 1 π vol(sr ). 2 )

3.3 53 U c(s) c h(e 1 (s), e 1 (s)), ξ dµ U c(s) c (ξ) = 1 π vol(sr ) h(e 1 (s), e 1 (s)) = 1 π vol(sr )κ(s). τ(c) = 1 vol(s r ) c 1 = vol(s r ) c = 1 κ(s)ds. π c U c(s) c h(e 1 (s), e 1 (s)), ξ dµ U c(s) c (ξ)ds 1 π vol(sr )κ(s)ds 3.3.5 ι : M R n+r n M Euclid x M T x M R n+r ξ U x M ξ S n+r 1 ν : U M S n+r 1 ; ξ ξ ν C ν ι : M R n+r Gauss Ker(dν) ξ = {X X KerA ξ } dim Ker(dν) ξ = dim KerA ξ Jν(ξ) = det A ξ τ(m, ι) = = 1 #(ν 1 (u))dµ vol(s n+r 1 S n+r 1(u) ) S n+r 1 1 #{x M u T vol(s n+r 1 x M}dµ S n+r 1(u) ) S n+r 1 C ν : U M S n+r 1 ( 3.1.22) ν x M, ξ Ux Me 1,..., e n T x Mξ 1,..., ξ r = ξ Tx M 3.3.1 e 1,..., e n, ξ 1,..., ξ r 1 T ξ (U M) 1 s r 1 dν ξ (ξ s ) = d ν(cos tξ + sin tξ dt s ) t=0 = d (cos tξ + sin tξ dt s ) t=0 = ξ s.

54 3 Chern-Lashof 1 i n c(0) = x, c (0) = X T x M M c(t) c(t) ξ(0) = ξ ξ(t) 3.3.1 X = ξ (0) dν ξ (X ) = d ν(ξ(t)) dt t=0 = d ξ(t) dt t=0 = X ξ(t) = A ξ (X). (Weingarten ( 2.1.7)) Ker(dν) ξ = {X X KerA ξ } dim Ker(dν) ξ = dim KerA ξ 3.1.21 3.1.6 Jν(ξ) = dν ξ(e 1) dν ξ (e n) dν ξ (ξ 1 ) dν ξ (ξ r 1 ) e 1 e n ξ 1 ξ r 1 = A ξ (e 1 ) A ξ (e n ) ξ 1 ξ r 1 = A ξ (e 1 ) A ξ (e n ) = (det A ξ )e 1 e n = det A ξ. C ν : U M S n+r 1 S n+r 1 φ = 1 #(ν 1 (u))dµ S n+r 1(u) = det A ξ dµ U M(ξ) S n+r 1 U M u S n+r 1 τ(m, ι) = τ(m, ι) = 1 #(ν 1 (u))dµ vol(s n+r 1 S n+r 1(u) ) S n+r 1 #(ν 1 (u)) = #{x M u T x M} 1 #{x M u T vol(s n+r 1 x M}dµ S n+r 1(u) ) S n+r 1 3.3.6 ι : M R n+r n M Euclid u S n+r 1 f u (x) = x, u (x M)

3.3 55 M C f u : M R x M f u u T x M x M f u f u Hessian ( 2 f u ) x ( 2 f u ) x (X, Y ) = A u X, Y (X, Y T x M) x MX T x M c(0) = x, c (0) = X M c(t) (df u ) x (X) = (df u ) x (c (0)) = d f dt u (c(t)) t=0 = d c(t), u dt t=0 = X, u. x M f u u T x M f u Hessian x M f u u Tx MY T x MY x Ỹ Ỹ f u = df u (Ỹ ) = Ỹ, u. X T x M Weingarten ( 2.1.7) u T x M 2.1.6 XỸ f u = X Ỹ, u = X Ỹ, u = h(x, Y ), u = A u X, Y. f u x M ( 2 f u ) x (X, Y ) = A u X, Y (X, Y T x M) 3.3.7 ι : M R n+r n M Euclid τ(m, ι) 2 M u S n+r 1 f u 3.3.6 x M f u u T x M u S n+r 1 #{x M u T x M} = #{x M x f u } 2 τ(m, ι) = 1 #{x M u T vol(s n+r 1 x M}dµ S n+r 1(u) 2 ) S n+r 1

56 3 Chern-Lashof 3.3.8 3.3.7 M R 1+r τ(c) 2 3.3.4 τ(c) = 1 κ(s)ds π c c κ(s)ds 2π Fenchel 3.3.9 n M Euclid ι : M R n+r ι(x) = (ι 1 (x),..., ι n+r (x)) (x M) ι : M R n+r (r < r ) ι(x) = (ι 1 (x),..., ι n+r (x), 0,..., 0) (x M) τ(m, ι) = τ(m, ι) r = r + 1 r = r + 1 ι : M R n+r 3.3.5 Gauss ν : U M S n+r 1 ι : M R n+r+1 Gauss ν : Ũ M S n+r u S n+r 1 π/2 < θ < π/2 (cos θu, sin θ) S n+r (cos θu, sin θ) T x M u T x M #{x M (cos θu, sin θ) T x M} = #{x M u T x M} p = (0,..., 0, 1) S n+r φ : S n+r 1 ( π/2, π/2) S n+r {±p} ; (u, θ) (cos θu, sin θ) φφ Jφ(u, θ) = cos n+r 1 θ 3.3.5 τ(m, ι) 1 = #{x M (cos θu, sin θ) T vol(s n+r x M}dµ S n+r(cos θu, sin θ) ) S n+r 1 π/2 = #{x M u T vol(s n+r x M}dµ S n+r 1(u) cos n+r 1 θdθ ) π/2 S n+r 1 = vol(sn+r 1 ) π/2 cos n+r 1 θdθτ(m, ι). vol(s n+r ) π/2

3.4 Chern-Lashof 57 3.2.7 vol(s n+r 1 ) = 2Γ( 1 2 )n+r Γ( n+r ( 3.2.5) p = n + r, q = 1 2 2 Γ( n+r 2 )Γ( 1 2 ) Γ( n+r+1 2 ) 2 ), vol(sn+r ) = 2Γ( 1 Γ(p)Γ(q) π/2 Γ(p + q) = 2 cos 2p 1 θ sin 2q 1 θdθ 0 2 )n+r+1 Γ( n+r+1 2 ) π/2 π/2 = 2 cos n+r 1 θdθ = cos n+r 1 θdθ. 0 π/2 vol(s n+r 1 ) vol(s n+r ) τ(m, ι) = τ(m, ι) π/2 π/2 cos n+r 1 θdθ = 2Γ( 1 2 )n+r Γ( n+r+1 ) 2 Γ( n+r)2γ( 1 Γ( n+r )Γ( 1) 2 2 2 2 )n+r+1 Γ( n+r+1 ) 2 = 1. 3.4 Chern-Lashof S. S. Chern and R. K. Lashof, On th total curvature of immersed manifolds, Amer. J. Math. 79 (1957), 306 318. S. S. Chern and R. K. Lashof, On th total curvature of immersed manifolds, II, Mich. Math. J. 5 (1958), 5 12. 3.4.1 ι : M R n+r n M Euclid F n τ(m, ι) b q (M; F ) q=0 3.3.7 u S n+r 1 f u : M R 1.4.4 n n #{x M x f u } = C q b q (M; F ) q=0 q=0

58 3 Chern-Lashof τ(m, ι) = 1 #{x M x f vol(s n+r 1 u }dµ S n+r 1(u) ) S n+r 1 n b q (M; F ) q=0 3.4.2 ι : M R n+r n M Euclid 0 u S n+r 1 f u : M R x M f u 3.3.6 u T x M ( 2 f u ) x (X, Y ) = A u X, Y (X, Y T x M) f u x A u : T x M T x M 3.3.5 0 det A u = Jν(u). u U x M νu S n+r 1 ν f u Sard ( 3.1.20) 0 u S n+r 1 u νf u : M R 3.4.3 ι : M R n+r n M Euclid τ(m, ι) < 3 M n 3.3.5 3.3.7 τ(m, ι) = 1 #{x M x f vol(s n+r 1 u }dµ S n+r 1(u) ) U M S n+r 1 2 #{x M x f u } {u S n+r 1 2 = #{x M x f u }} 0 τ(m, ι) < 3 3.4.2 u S n+r 1 f u Reeb ( 1.5.1) M n

3.4 Chern-Lashof 59 3.4.4 ι : M R n+r n M Euclid τ(m, ι) = 2 M n + 1 R n+1 r = 1 M R n+1 r 2 M R n+r M R n+r 3.3.7τ(M, ι) 2 ξ U M det A ξ = 0 det A ξ 0 ξ U M p = π( ξ) M ξ U p M T p M p ξ 1,..., ξ r ξ r ( p) = ξξ U p M r ξ = ξ i ξ i ( p) A ξ ξ r A ξ = ξ i A ξ( p) r 2 ( r ) det A ξ = det ξ i A ξ( p) ξ(θ) = cos θξ r ( p) + sin θξ r 1 ( p) = cos θ ξ + sin θξ r 1 ( p) ξ(θ) U p M A ξ(θ) = det ( cos θa ξ + sin θa ξr 1 ( p)). f(θ) f(θ) cos θ sin θ θ f(0) = det A ξ 0 f(θ) 0 H θ pν(θ) R n+r θ H θ pξ r 1 ( p) ξ r ( p) 2 ι(m) θ H θ p 1 M ι(p 1 ) / θ H θ R n+r = θ H θ θ 1 ι(p 1 ) H θ1 p 1 H θ H θ1 H θ ι(p 1 ) ι(m) H θ1 p 2 M ι(p 2 ) / H θ1 R n+r = θ H θ θ 2 ι(p 2 ) H θ2 ι(p 2 ) / H θ1 H θ1 H θ2 ι(p 1 ) / H θ2 ι(p 1 ) ι(p 2 ) H θ θ f(θ) 0 θθ 3 f ξ(θ3 )(p 1 ) < f ξ(θ3 )( p) < f ξ(θ3 )(p 2 ) f ξ(θ3 )(p 2 ) < f ξ(θ3 )( p) < f ξ(θ3 )(p 1 )

60 3 Chern-Lashof f ξ(θ3 )(p 1 ) < f ξ(θ3 )( p) < f ξ(θ3 )(p 2 ) 3.3.5 ι : M R n+r Gauss Jν(ξ) = A θ ν : U M S n+r 1 ; ξ ξ Jν(ξ(θ 3 )) = A ξ(θ3 ) = f(θ 3 ) = 0. U Mξ(θ 3 ) W Gauss ν W W (p, ξ) W ι(p 1 ) ι(p 2 ) ι(p) ξ (p, ξ) W f ξ (p 1 ) < f ξ (p) < f ξ (p 2 ) f ξ M f ξ (p 2 ) p f ξ M p ξ U p Mf ξ S n+r 1 ν(w ) ξ f ξ M τ(m, ι) = 2 3.3.5 ξ S n+r 1 f ξ ξ S n+r 1 f ξ M R n+r M R n+r 1 R n+r ι 3.3.9 τ(m, ι ) = τ(m, ι) = 2 M R n+r n + 1 3.4.5 3.4.4 τ(m, ι) = 2 M n + 1 R n+1 3.4.6 ι : M R n+1 n M Euclid n M x M Gauss-Kronecker G(x) G(x) 0 M Betti 0 3.4.2 u S n f u : M R 3.3.6 f u x M ( 2 f u ) x (X, Y ) = A u X, Y (X, Y T x M) A u : T x M T x M G(x) = det A u 0 det A u > 0 A u f u x f u 1.4.8 M Betti 0

4 M. Berger, Geometry I, Springer 1987. T. E. Cecil and P. J. Ryan, Tight and taut immersions of manifolds, Pitman 1985. H. G. Eggleston, Convexity, Cambridge University 1958. 4.1 4.1.1 M M C f f Morse 1.1.11 f Morse f C(f) M Morse γ(m) γ(m) = min{c(f) f M Morse } 4.1.2 M F b(m; F ) = q 0 b q (M; F ) b(m; F ) γ(m) M Morse f f q C q (f) Morse ( 1.4.4) b(m; F ) = b q (M; F ) C q (f) = C(f) q 0 q 0 b(m; F ) γ(m) 4.1.3 M γ(m) = inf{τ(m, ι) ι : M R m } 61

62 4 Whitney M Euclid 3.4.1 ι : M R m 1 τ(m, ι) = C(f vol(s m 1 u )dµ S m 1(u) ) S m 1 1 γ(m)dµ vol(s m 1 S m 1(u) ) S m 1 = γ(m) γ(m) τ(m, ι). M Euclid h λ (λ > 0) γ(m) = lim λ τ(m, h λ ) ι : M R m C(φ) = γ(m) M Morse φλ > 0 h λ (x) = (ι(x), λφ(x)) R m R = R m+1 (x M) C h λ : M R m+1 x M rank(dh λ ) x rankdι x = dim M rank(dh λ ) x = dim Mh λ p = (0, 1) R m R = R m+1 x M f p (h λ (x)) = h λ (x), p = λφ(x) φ M Morse f p h λ M Morse 3.3.5 3.3.6 p h λ Gauss q f q f αq = αf q f αq h λ M Morse f q h λ M Morse w R m q = w + p ε > 0 w 2 < εf q h 1 M Morse C(f q h 1 ) = γ(m) f q (h λ (x)) = h λ (x), q = ι(x), w + λφ(x) w = λu R m f q (h λ (x)) = ι(x), λu + λφ(x) = λ( ι(x), u + φ(x)). u 2 < ε w 2 < λ 2 εf q h λ M Morse C(f q h λ ) = γ(m)

4.1 63 q = w + βp q = w/β + p f q h λ f q h λ ( w /β) 2 < λ 2 εf q h λ M Morse C(f q h λ ) = γ(m) q 1 = q 2 = w + βp 2 = w 2 + β 2 w 2 = 1 β 2 ( w /β) 2 < λ 2 ε λ 2 ε > 1 β2 β 2 = 1 β 2 1, α = (1 + λ 2 ε) 1/2 1 + λ 2 ε > 1 β 2 β 2 > (1 + λ 2 ε) 1 β > (1 + λ 2 ε) 1/2 W λ = (R m [ α, α]) S m q S m W λ f q h λ M Morse C(f q h λ ) = γ(m) τ(m, h λ ) = = = 1 C(f vol(s m u h λ )dµ S m(u) ) S m 1 1 C(f vol(s m u h λ )dµ S m(u) + C(f ) S m W λ vol(s m u h λ )dµ S m(u) ) W λ 1 1 vol(s m ) vol(sm W λ )γ(m) + C(f vol(s m u h λ )dµ S m(u) ) W λ α lim λ α = 0 W λ = (R m [ α, α]) S m W 0 = (R m {0}) S m = S m 1 lim λ vol(sm W λ ) = vol(s m S m 1 ) = vol(s m ) lim λ 1 C(f vol(s m u h λ )dµ S m(u) = γ(m) ) S m W λ

64 4 I λ = C(f u h λ )dµ S m(u) W λ lim λ I λ = 0 u S m C(f u h 1 ) > 0 0 I 1 = C(f u h 1 )dµ S m(u) C(f u h 1 )dµ S m(u) = vol(s m )τ(m, h 1 ) <. W 1 S m 0 I 1 < I λ W λ V = S m 1 [ ε 1/2, ε 1/2 ] S m q = w + βp (w R m ) q W λ β αα = (1 + λ 2 ε) 1/2 w 2 = 1 β 2 β α β 2 1 1 + λ 2 ε 1 + λ 2 ε 1 β 2 λ 2 ε 1 β 1 = 1 β 2 2 β 2 λ 2 β 2 w 2 ε 1 λβ w ε 1/2 = w 2 β 2 w = w w, q = (w, ηp) η = λβ w q W λ q V C F : W λ V ; q = w + βp q = (w, ηp) F (w + βp) = ( w w, λβ ) w p. F : W λ V (w, η) V w + η p λ w + η λ p = w w + η λ p + η λ w + η λ p p

4.1 65 2 η λ w + η λ p = 1 λ 2 w + 1 η λ p 2 = 1 λ 2 ε + 1 = α2. 1 ( ) λ 2 1 = 1 + 1 λ 2 + 1 η 2 λ 2 η 2 V W λ C F F F W λ V JF q = w + βp W λ S m 1 c(t) c(0) = w/ w S m 1 w c(t) + βp q W λ F ( w c(t) + βp) = c(t) + λβ w p. t = 0 w df (c (0)) = c (0) df (c (0)) = c (0) w cos θ = w, sin θ = β θw λ q d d(t) = sin θ w dt t=0 w F d(t) = cos(θ + t) w + sin(θ + t)p w ( cos(θ + t) w ) + sin(θ + t)p w df ( β w ) w + w p JF = 1 w m 1 + cos θp = β w w + w p. = w λ sin(θ + t) + w cos(θ + t) p = λ cos 2 θ p = λ w 2 = J(F 1 ) = w m+1. λ λ w m+1. λ w 2 p

66 4 q = w + βp W λ q = w + ηp V q = w + ηp = w w + λβ w p w λ (q ) = w w + η λ p J(F 1 )(q ) = w λ(q ) m+1 λ f q h λ (x) = w, ι(x) + λβφ(x) ( = w w, ι(x) + λβ ) w φ(x) = w ( w, ι(x) + ηφ(x)) = w f q h 1 (x) C(f q h λ ) = C(f q h 1 ) I λ = C(f q h λ )dµ S m(q) W λ = C(f q h 1 )J(F 1 )dµ S m 1dη(q ) = V = 1 λ C(f q h 1 ) w λ(q ) m+1 dµ S m 1dη(q ) λ C(f q h 1 ) w 1 (q ) m+1 w λ(q ) m+1 V w 1 (q ) dµ m+1 S m 1dη(q ). V w λ (q ) w 1 (q ) = w + ηp w + η p w + ηp = ( w + ηp 2 ) 1/2 = (1 + η 2 ) 1/2 λ ( 1 + 1 ) 1/2 ε I λ 1 λ = 1 λ ( 1 + 1 ) (m+1)/2 ε V ( 1 + 1 ) (m+1)/2 I 1 ε C(f q h 1 ) w 1 (q ) m+1 dµ S m 1dη(q )

4.2 67 lim I λ = 0 λ 4.1.4 M ι : M R m τ(m, ι) = γ(m) 4.1.5 M ι : M R m Morse f u (u S m 1 ) C(f u ) = γ(m) Morse f u (u S m 1 ) C(f u ) = γ(m) τ(m, ι) = 1 C(f vol(s m 1 u )dµ S m 1(u) = γ(m) ) S m 1 ι : M R m Morse f v (v S m 1 ) C(f v ) > γ(m) v U u U C(f u ) > γ(m) Morse f u (u S m 1 ) C(f u ) γ(m) τ(m, ι) > γ(m) ι : M R m 4.2 R n A A c y R n r > 0 U(y; r) = {x R n x y < r} B(y; r) = {x R n x y r} R n A d(x, A) = inf{ x y y A} (x R n ) x A d(x, A) r > 0 A U(A; r) U(A; r) = {x R n d(x, A) < r} 4.2.1 R n x, yxy xy = {λx + µy λ 0, µ 0, λ + µ = 1}. R n CC x, y C xy C

68 4 4.2.2 R n y R n r > 0 U(y; r) B(y; r) 4.2.3 C i R n i C 1 C 2 R n 1 R n 2 = R n 1+n 2 (x 1, x 2 ), (y 1, y 2 ) C 1 C 2 λ 0, µ 0, λ + µ = 1 C 1, C 2 λ(x 1, x 2 ) + µ(y 1, y 2 ) = (λx 1 + µy 1, λx 2 + µy 2 ) C 1 C 2 C 1 C 2 4.2.4 k 2 R n C x 1,..., x k C C λ i 0, λ 1 + + λ k = 1 λ 1 x 1 + + λ k x k k k = 2 k = m m + 1 x 1,..., x m+1 C λ i 0, λ 1 + + λ m+1 = 1 λ 1 x 1 + + λ m+1 x m+1 C λ 3 = = λ m+1 = 0 C C x 1,..., x m+1 C λ m+1 = 1 λ i 0, λ 1 + + λ m+1 = 1 λ 1 x 1 + + λ m+1 x m+1 = x m+1 C.

4.2 69 λ m+1 < 1 λ 1 + + λ m = 1 λ m+1 > 0. λ 1 x 1 + + λ m+1 x m+1 ( = (λ 1 + + λ m ) λ 1 λ 1 + + λ m x 1 + + C λ 1 λ m x 1 + + x m C λ 1 + + λ m λ 1 + + λ m λ 1 x 1 + + λ m+1 x m+1 C ) λ m x m + λ m+1 x m+1 λ 1 + + λ m 4.2.5 f R m R n x, y R m f(xy) = f(x)f(y)r m f R n f f L a R n f(x) = Lx + a (x R m ) x, y R m λ 0, µ 0, λ + µ = 1 λf(x) + µf(y) = λ(lx + a) + µ(ly + a) = L(λx + µy) + a = f(λx + µy) f(xy) = f(x)f(y) R m C x, y f(x)f(y) = f(xy) f(c) f(c) R n C x, y f 1 (C) f(x), f(y) C f(xy) = f(x)f(y) C. xy f 1 (C) f 1 (C)

70 4 4.2.6 C 1, C 2 R n λ, µ λc 1 + µc 2 = {λx + µy x C 1, y C 2 } 4.2.3 C 1 C 2 R n R n = R 2n f : R 2n R n ; (x, y) λx + µy ff 4.2.5 f(c 1 C 2 ) = λc 1 + µc 2 4.2.7 C 1, C 2 R n λ, µ 4.2.6λC 1 + µc 2 C 1 C 2 Minkowski 4.2.8 {C i } i I R n i I C i R n x, y i I C i i I x, y C i xy C i xy i I C i i I C i 4.2.9 C R n r > 0 U(C; r) x, y U(C; r) d(x, C) < r, d(y, C) < r x, y C x x < r, y y < r λ + µ = 1 λ 0, µ 0 Cλx + µy C (λx + µy) (λx + µy ) λx λx + µy µy = λ x y + µ y y < r. λx+µy U(C; r) xy U(C; r) U(C; r) 4.2.10 R n

4.2 71 C R n C C C = U(C; r) r>0 4.2.8 4.2.9 C 4.2.11 C R n C x C y C xy {x} c C Cxy C x xy C y C C r > 0 U(y; r) Cz xy {x} c λ 0, µ > 0 λ + µ = 1 z = λx + µy U(z; µr) C z U(z; µr) z z < µr z (λx + µy) < µr. µ > 0 z λx µ y < r. (z λx)/µ U(y; r) C(z λx)/µ C z = λx + µ z λx µ C. U(z; µr) Cz C xy {x} c C 4.2.12 R n C C 4.2.13 4.2.11 x C x C z xy {x, y} c C C x z 1 z 1 z y z 2 1 z = ( y z x + x z y) x y

72 4 z z 1 z 2 z 2 x y z 2 = z 1 + x z (z z 1) = z 1 + x y x z ( 1 x y ( y z x + x z y) z 1 = z 1 + 1 x y ( y z x + x z y) x z x z z 1 y z = x z z y z 1 + x z x + y y z = y x z (z 1 x). y C r > 0 U(y; r) C z 1 ) z 1 x < r x z y z z 1 C x C z 1 z 2 y = y z x z z 1 x < y z r x z x z y z = r z 2 U(y; r) C 4.2.11 z z 1 z 2 {z 1 } c C z C 4.2.14 R n C(C ) = C ( C) = C C C(C ) C x C x 1 C 4.2.13xx 1 {x} c C x (C ) C (C )(C ) = C C CC ( C) x / C x 1 C x 1 x y x yx 1 {y} c y C 4.2.13 x C x y / C x C x / ( C) ( C) C C = ( C)

4.2 73 4.2.15 C R n max{r x 1,..., x r+1 C, x 2 x 1,..., x r+1 x 1 } C dim C 4.2.16 R n x 1,..., x n+1 x 2 x 1,..., x r+1 x 1, x r+2 x 1,..., x n+1 x 1 x R n x = n+1 χ i x i, n+1 χ i = 1 χ 1,..., χ n+1 n+1 λ i x i = 0, n+1 λ i = 0 λ 1 = = λ n+1 = 0 ( ) R n+1 0 = n+1 χ i x i = n+1 ( n+1 χ i x i = 0, n+1 i=2 [ ] x1,..., 1 [ ] xi χ i = 0 1 n+1 χ 1 = χ i ) x 1 + n+1 χ i i=2 n+1 i=2 [ ] xn+1 1 χ i = 0. χ i x i = n+1 i=2 χ i (x i x 1 ). χ 2 = = χ n+1 = 0 χ 1 = 0( ) R n+1 x R n χ 1,..., χ n+1 [ ] x = 1 n+1 [ ] xi χ i 1

74 4 x = n+1 n+1 χ i x i, λ i x i = 0, n+1 n+1 n+1 [ ] xi λ i = 0 1 χ i = 1 λ i = 0 ( ) λ 1 = = λ n+1 = 0 4.2.17 C R n rc r A(C) A(C) C dim C = r x 1,..., x r+1 C x 2 x 1,..., x r+1 x 1 x 2 x 1,..., x r+1 x 1, x r+2 x 1,..., x n+1 x 1 R n x r+2,..., x n+1 4.2.16 x C x = n+1 χ i x i, n+1 χ i = 1 χ 1,..., χ n+1 χ r+2 = = χ n+1 = 0 x 1,..., x r+1, x C dim C = r x 2 x 1,..., x r+1 x 1, x x 1 0 λ 1,..., λ r+1, λ λ 2 (x 2 x 1 ) + + λ r+1 (x r+1 x 1 ) + λ(x x 1 ) = 0 x 2 x 1,..., x r+1 x 1 λ 0 x x 1,..., x n+1 0 = λ 2 (x 2 x 1 ) + λ r+1 (x r+1 x 1 ) + λ(x x 1 ) = (λ(χ 1 1) λ 2 λ r+1 )x 1 +(λχ 2 + λ 2 )x 2 + + (λχ r+1 + λ r+1 )x r+1 +λχ r+2 x r+2 + + λχ n+1 x n+1.

4.3 75 (λ(χ 1 1) λ 2 λ r+1 ) + (λχ 2 + λ 2 ) + + (λχ r+1 + λ r+1 ) +λχ r+2 + + λχ n+1 = λ(χ 1 + + χ n+1 1) = 0. 4.2.16 λχ r+2 = = λχ n+1 = 0 λ 0 χ r+2 = = χ n+1 = 0 x C r+1 x = χ i x i, r+1 χ i = 1 χ 1,..., χ r+1 A(C) = { r+1 } r+1 χ i x i χ i = 1 A(C) r C C ra 1, A 2 C ra 1 A 2 C r C r C r A(C) C 4.2.4 { r+1 χ i x i χ i > 0, χ 1 + + χ r+1 = 1 C A(C) A(C) C 4.2.18 R n C A(C) C C r A(C) C r C } 4.3 4.3.1 C R n C p A(C) R n S y S p y Cσ(y) σ : S (0, ]

76 4 dim C = n p = 0 A(C) = R n y S 0 y l(y) l(y) C0 C r > 0 B(0; r) C σ σ(y) < y σ(y) = y σ(y) < l(y) C f(y) H(f(y), B(0; r)) = {f(y)x x B(0; r)} H(f(y), B(0; r)) f(y) C 4.2.13 H(f(y), B(0; r)) {f(y)} c Cz S l(z) H(f(y), B(0; r)) ρ(z) ρ : S (0, ) ρ σρ ε > 0 δ 1 > 0 z y < δ 1 ρ(z) ρ(y) < ε. ρ(y) = σ(y) z y < δ 1 ε < ρ(z) ρ(y) σ(z) σ(y). H(f(y), B(0; r)) f(y) f(y) D D C = z D C z 1 B(0; r) f(y) zz 1 4.2.11 f(y) C f(y) Cδ 2 > 0 z y < δ 2 l(z) D l(z) D τ(z) τ σ ττ δ 3 > 0 z y < δ 3 τ(z) τ(y) < ε. τ(y) = σ(y) z y < δ 3 δ = max{δ 1, δ 3 } ε > τ(z) τ(y) σ(z) σ(y). z y < δ σ(z) σ(y) < ε σ y σ(y) = l(y) C E = {z 1 z 2 z 1 U(0; r), z 2 l(y)} C E CE E = U(l(y); r) z S l(z) Eρ(z) ρ : S (0, ]

4.3 77 ρ σρ(y) = σ(y) = ρ R > 0 δ > 0 z y < δ R < ρ(z). z y < δ R < ρ(z) σ(z) σ y 4.3.2 C R n C r A(C) C C A(C) dim C = n 0 C A(C) = R n 4.3.1 σ : S (0, ] f : C R n f(x) = 0 (x = 0) x (x 0, σ(x/ x ) = ) σ(x/ x ) x (x 0, σ(x/ x ) < ) σ(x/ x ) x f : C R n {x C x 0, σ(x/ x ) < } R n f 0 f ε > 0 B(0; ε) C σ(z/ z ) = z U(0; ε) {0} c f(z) f(0) = z < ε. σ(z/ z ) < z U(0; ε) {0} c B(0; ε) C z < σ(z/ z ) f(z) f(0) = σ(z/ z ) z < z < ε. σ(z/ z ) z f 0 σ(x/ x ) = x 0 fσ x/ x ε > 0 S x/ x U y U 2 ε ( x + ε ) 2 ε + x + 2 2 < σ(y) V = {ty y U, t > 0}

78 4 V R n x U(x; δ) V, { } ε 0 < δ < min 2, x δ z U(x; δ) z 0 σ(z/ z ) = σ(z/ z ) < f(z) f(x) = z x < ε 2 < ε. f(z) f(x) = f(z) z + z x U(x; δ) V z/ z U σ(z/ z ) z > 2 ε f(z) z + z x σ(z/ z ) = σ(z/ z ) z z z + z x σ(z/ z ) (σ(z/ z ) z ) = z + z x σ(z/ z ) z z 2 = + z x. σ(z/ z ) z ( x + ε ) 2 ε + x + 2 2 z = 2 ( x + ε 2. ε 2) f(z) f(z) < ε 2 + ε 2 = ε. f x f C f f g : R n C g(x) = x 0, σ(x/ x ) < 0 (x = 0) x (x 0, σ(x/ x ) = ) σ(x/ x ) x (x 0, σ(x/ x ) < ) σ(x/ x ) + x g(x) = σ(x/ x ) x < σ(x/ x ) σ(x/ x ) + x g C g f gf(0) = 0 x 0, σ(x/ x ) = gf(x) = g(x) = x.

4.3 79 x 0, σ(x/ x ) < σ(f(x)/ f(x) ) = σ(x/ x ) gf(x) = ( ) σ(x/ x ) g σ(x/ x ) x x = σ(x/ x ) σ(x/ x ) σ(x/ x ) + f(x) σ(x/ x ) x x σ(x/ x ) σ(x/ x ) = σ(x/ x ) + σ(x/ x ) x σ(x/ x ) x x σ(x/ x ) x = σ(x/ x ) 2 σ(x/ x )(σ(x/ x ) x ) + σ(x/ x ) x x = x. gf C fg(0) = 0 x 0, σ(x/ x ) = fg(x) = f(x) = x. x 0, σ(x/ x ) < σ(g(x)/ g(x) ) = σ(x/ x ) fg(x) = ( ) σ(x/ x ) f σ(x/ x ) + x x = σ(x/ x ) σ(x/ x ) σ(x/ x ) g(x) σ(x/ x ) + x x σ(x/ x ) σ(x/ x ) = σ(x/ x ) σ(x/ x ) x σ(x/ x ) + x x σ(x/ x ) + x = σ(x/ x ) 2 σ(x/ x )(σ(x/ x ) + x ) σ(x/ x ) x x = x. fg R n g f {x R n x 0, σ(x/ x ) < } R n g 0 g ε > 0 σ(z/ z ) = z U(0; ε) {0} c g(z) g(0) = z < ε. σ(z/ z ) < z U(0; ε) {0} c g(z) g(0) = σ(z/ z ) z < z < ε. σ(z/ z ) + z