No.004 [1] J. ( ) ( ) (1968) [2] Morse (1997) [3] (1988) 1

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Transcription:

No.004 [1] J. ( ) ( ) (1968) [2] Morse (1997) [3] (1988) 1

1 (1) 1.1 X Y f, g : X Y { F (x, 0) = f(x) F (x, 1) = g(x) F : X I Y f g f g F f g 1.2 X Y X Y gf id X, fg id Y f : X Y, g : Y X X Y X Y (2) 1.3 X A X A X r t : X X r 0 = id, r 1 = A, r t A = id 1.4 X A X A X A X (3)CW D n = {(x 1,, x n ) R n x 2 1 + + x 2 n 1} 2

1.5 X e n X e n ϕ : D n X ϕ intd n : intd n e n e n X n ϕ e n 0 X 1.6 X X {e λ ; λ Λ} X (X ) X = λ Λ (1) X = λ Λ e λ (2) λ µ e λ e µ = (3) e n ϕ : D n X ϕ( D n ) X n 1 = k<n 1 CW e k 1.7 X = λ Λ e λ Λ Λ A = λ Λ e λ A X 1.8 X = λ Λ e λ X CW (1) X e e X (2) X F F X X e F ē ē (0) X = D 1 X = e 0 e 1 (1) X = D 2 X = e 0 e 1 e 2 (2) X = D 3 X = e 0 e 2 3

2 M, f, M a p q r s M : R 3 V f : M R, f(x) = d(x, V ) : d x V M a = {x M f(x) a } (1) a < f(q) = M a. (2) f(p) < a < f(q) = M a e 2. (3) f(q) < a < f(r) = M a (4) f(r) < a < f(s) = M a 1 (5) f(s) < a = M a = M. (1) (2) = 0 (2) (3) = 1 (3) (4) = 1 4

(4) (5) = 2 M CW (M e 0 e 1 e 1 e 2 ) 5

3 M m f : M R 3.1 (f ) M p 0 f : M R p 0 (x 1,, x m ) f x 1 (p 0 ) = 0,, f x m (p 0 ) = 0 3.2 (Hesse ) 2 f p 0 f : M R m m H f (p 0 ) = ( ) x i x j p 0 f Hesse 3.3 ( ) deth f (p 0 ) 0 p 0 deth f (p 0 ) = 0 p 0 3.4 p 0 (x 1,, x m ), (y 1,, y m ) f : M R Hesse H f (p 0 ), H f (p 0) H f (p 0 ) = t J(p 0 )H f (p 0 )J(p 0 ) J(p 0 ) (y 1,, y m ) (x 1,, x m ) Jacobi p 0 3.5 f : M R p 0 p 0 6

3.6 ( ) f : M R f 3.7 f f(0) = 0 R n 0 V C f(x 1,, x n ) = Σ n i=1x i g i (x 1,, x n ) g i g i (0) = f x i (0) V C 3.8 M f : V R p 0 M f(p 0 ) = 0 p 0 (U, ϕ; x 1,, x n ) U V x 1 (p 0 ) = = x n (p 0 ) = 0 ɛ > 0 p 0 W W = {(x 1,, x n ) R n x i < ɛ, i = 1, n} ϕ(u), W = ϕ 1 (W ) f = f W : W R g i : W R f(x 1,, x n ) = Σ n i=1 x ig i (x 1,, x n ) g i (p 0 ) = f x i (p 0 ) 3.9 ( ) p 0 f : M R p 0 (X 1,, X m ) f f = X1 2 Xλ 2 + Xλ+1 2 + + Xm 2 + f(p 0 ) X 1 (p 0 ) = = X m (p 0 ) = 0 7

3.10 λ p 0 3.11 3.12 f : M R Morse f 8

4 Morse 4.1 M p 0 M M f : M R C (M) R L : C (M) R L(f + g) = L(f) + L(g) L(af) = al(f) L(f g) = f(p 0 )L(g) + L(f)g(p 0 ) L p 0 p 0 T p0 (M) = {L L p 0 } L 1 + L 2 al (L 1 + L 2 )(f) = L 1 (f) + L 2 (f) (al)(f) = al(f) a R R T p0 (M) p 0 4.2 M O M X O X O p p M X p T p (M) X : O T p (M) p O U p (U; x i,, x n ) X(p) = Σ n i=1ξ i (p)( x i ) p ξ i : U R, i = 1,, n 9

4.3 M X M f : M R Xf : M R (Xf)(p) = X p (f) X : C (M) C (M) X X(f + g) = X(f) + X(g) X(af) = ax(f) X(f g) = f(p 0 )X(g) + X(f)g(p 0 ) X : C (M) C (M) X p (f) = (Xf)(p) M 4.4 M X M f : M R (fx) p = f(p)x p fx M 4.5 f : R n R grad f = ( f x 1,, f x n ) R (grad f) p = Σ n f i=1 (p)( ) p x i x i, p U f 4.6 M X M c : (α, β) M X dc dt (t) = X c(t), t (α, β) 10

4.7 M X M c, c 0 J, J X c(0) = c(0) c, c M φ : R M M (1) φ(0, p) = p p M (2) φ(s, φ(t, p)) = φ(s + t, p) s, t R, p M φ 1 φ : R M M φ(t, p) φ t (p) t R φ t : M M (1),(2) (1) φ 0 = 1 (2) φ s φ t = φ s+t s, t R φ t φ t φ t 4.8 p 0 c : R M, c(t) = φ t (p 0 ) t = 0 p 0 φ p 0 4.9 M φr M M M X X p (f) = lim t 0 f(φ t (p)) f(p) t = df(φ t(p)) dt t=0 f C (M) 11

X φ X φ 4.10 M φ : R M M X φ M p 0 M c : R M, c(t) = φ t (p 0 ) t = 0 p 0 X : dc dt (t) = X c(t) t R 4.11 M φ, φ X φ φ 4.12 M M X φ : R M M 12

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