1 1.1 [ ]., D R m, f : D R n C -. f p D (df) p : (df) p : R m R n f(p + vt) f(p) : v lim. t 0 t, (df) p., R m {x 1,..., x m }, (df) p (x i ) =
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1 2004 / D : 0,.,., :,.,.,,.,,,.,.,,.. :,,,,,,,., web page.,,. C-613 tamaru math.sci.hiroshima-u.ac.jp url tamaru/index-j.html 2004 D - 1 -
2 1 1.1 [ ]., D R m, f : D R n C -. f p D (df) p : (df) p : R m R n f(p + vt) f(p) : v lim. t 0 t, (df) p., R m {x 1,..., x m }, (df) p (x i ) = f x i (p)., (df) p Jacobi, (Jf) p : (Jf) p = f 1 f 1 x 1 (p) x m (p)..... f n f x 1 (p) n x m (p), Jacobi.,, ( ) f : R n R n C -, p R n., (df) p, f p C -. (df) p, (Jf) p ( n ) ( ) F : R m R n R n C -, (x 0, y 0 ) R m R n., (df {x0 } R n) (x 0,y 0 ), : W : R m x 0, f : W R n : C s.t. x W, F (x, f(x)) = 0., R 2 y = ax., f(x) = ax y = f(x), F (x, y) = y ax F (x, y) = 0.,. (F (x, y) = y f(x) )., (, F (x, y) = x y = f(x) ).,, D - 2 -
3 1.2,.,., (, ) R m+1 M, : p M, (W, F ) s.t. (1) W p R m+1, (2) F : W R : C, (3) q W, (JF ) q (0,..., 0), (4) M W = {x W F (x) = 0}., R m+1 M, (W F (x) = 0 ) S m := {x R m+1 x = 1} R m R m U C - f : U R, graph(f) := {(x 1,..., x m, f(x)) R m+1 x = (x 1,..., x m ) U} R m+1 ( f )., f.,., x m+1 = f(x), x k = f(x) (k = 1,..., m + 1) R m+1 M, : p M, (W, f, U) s.t. (1) W p R m+1, (2) R m U :, (3) f : U R : C, (4) W M = graph(f)..., R m+1 M, : p M, (U, φ, V ) s.t. (1) R m U :, φ : U R m+1 : C, (2) q U, rank(jφ) q = m, (3) V = φ(u), V p M, φ : U V :. (U (x 1,..., x m ), m M ).., D - 3 -
4 1.3 C -,.,., M R m (U, φ, V ) M : (1) (3).,., ( )., ( ) f : M R, M R m+1., M C 0 (M) C 0 (M) f C - : p M, (W, f) s.t. (1) W R m+1 p, (2) f : W R : C, (3) f M W = f., f C 0 (M) C -, R m+1 C -. M R m+1, M., R m+1 C C 0 (M) f C -, : (U, φ, V ) :, f φ : U R : C..,, R m+1 M, R n+1 N. F : M N C -, F = (f 1,..., f n+1 ), i, f i C (M). C -, F C -, C - F : R m+1 R n+1., C - C - C F : M N C -, : (U, φ, V ) : M, f φ : U R n+1 : C D - 4 -
5 1.4,,.,,, p R m+1, T p R m+1 := {p} R m+1 R m+1 p., u p := (p, u) T p R m+1. u p p., T p R m+1 R m M R m+1 p M. p (W, F ), M p : T p M := {p} Ker(dF ) p = { u p = (p, u) T p R m+1 (df ) p (u) = 0}.,., F (x, y) = 0 (a, b) F x (a, b) (x a) + F y (a, b) (y b) = 0. {(a, b) + (u 1, u 2 ) F x (a, b) u 1 + F y (a, b) u 2 = 0}.,, (df ) (a,b) (u 1, u 2 ) = F x (a, b) u 1 + F y (a, b) u 2.,.,, R I, R m+1 M. C - c : I M M C -., t 0 I, c c(t 0 ) : ċ(t 0 ) := (c(t 0 ), dc dt (t 0)) T p R m M R m+1 p T p M, : (1) T p M m, (2) T p M = {ċ(0) c : I M : C, c(0) = p}, (3) (U, φ, V ) p, T p M = {p} Im(dφ) φ 1 (p), (4) T p M. (4) (2) ((2) ) D - 5 -
6 1.5 C - F : R m R n (df ) p,.,, F : M N C -, p M. p F C - (W, F ), F p M (F ) p : (F ) p : T p M T F (p) N : (p, u) (F (p), (d F ) p (u))., C -., (F (p), (d F ) p (u)) T F (p) N F : M N C -, p M, u p T p M., C - c : I M ċ(0) = u p, d(f c) (F ) p ( u p ) = (F (p), (0)). dt F c N,.,, F, F : M N C -, p M, (U, φ, V ) p, (F ) p ( u p ) = (F (p), d(f φ) φ 1 (p)((dφ 1 ) p (u)).,. F., (TEX ) F : M N C - (U, φ, V ) : s.t. F φ : C (U, φ, V ), φ : U V R m+1 C -., C M p M, (U, φ, V ) (U, φ, V ) p., C -. φ 1 φ : (φ ) 1 (V V ) φ 1 (V V )..,, C - F : M N C - ( ), : F : F 1 : C ( ) C - F : M N p M, (F ) p : T p M T F (p), F p C D - 6 -
7 1.6.,, R m+1 M. X : M M R m+1 M, : X : M R m+1 : C s.t. X(p) = (p, X(p)). M Γ(M R m+1 ) Γ(M R m+1 ) C (M)- : f, g C (M) X, Y Γ(M R m+1 ), (f X + g Y )(p) := fx + gy (p) := (p, f(p)x(p) + g(p)y (p)).,,, ( ) M X : M M R m+1, : p M, X(p) = (p, X(p)) Tp M. M Γ(T M). Γ(T M) Γ(M R m+1 ), Γ(T M) Γ(M R m+1 ) C (M)-., Γ(T M),., f, g C (M), X, Y Γ(T M), f X + g Y Γ(T M) M X Γ(M R m+1 ), : (i) p M, X(p) = 1, (ii) u p T p M, u, X(p) = 0. ξ. ( )., r S m (r) := {x R m+1 x = r}, ξ : ξ : S m (r) S m (r) R m+1 : p (p, p/r)., ξ ξ D - 7 -
8 1.7,., f C (M) u p T p M : u p f := df(p + tu) dt t=0., u p f = (df) p (u)., p u X Γ(T M) Y Γ(M R m+1 ) D X Y Γ(M R m+1 ) : D X Y : M T M : p (p, X(p)Y ) X, Y Γ(T M) bracket : [ X, Y ] := D X Y D Y X. bracket, Jacobi ξ M, X Γ(T M), D X ξ Γ(T M). M (M, ξ) (M, ξ) (shape operator) :, p M A : Γ(T M) Γ(T M) : X D X ξ. A p : T p M T p M : X(p) (D X ξ)(p) ( ) (M, ξ) K, H : K : M R : p ( 1) m det(a p ), H : M R : p (1/m)tr(A p ). ξ ξ, ( 1 ).,,., D - 8 -
9 1.8,..,., V : V := {f : V R f : }. V., {e 1,..., e n } V, f 1,..., f n V f i (e j ) := δ ij, {f 1,..., f n } V ( ). X, < X > X., N < X >:= { x k x k X}. k= V, W, < V W > I V W, V W : (v 1 + v 2, w) (v 1, w) (v 2, w) I := (v, w 1 + w 2 ) (v, w 1 ) (v, w 2 ) a R, v, v 1, v 2 V, w, w 1, w 2 W. (av, w) (v, aw),. (v, w) < V W > v w V W, < V W >.,, V W : a(v w) := (av) w = v (aw) {e i } V, {h j } W, {e i h j } V W. V n n V (V (V V ) = (V V ) V ) S n n. x n V, σ S n, σ(x) = x, σ S n, σ(x) = sgn(σ)x., S n n V : σ(v 1 v n ) := v σ(1) v σ(n). n S n V, n n V., v 1... v n (, v 1 v n A : n V n V v 1... v n ) D - 9 -
10 1.9..,., T p M T p M R m+1 M. X : M M ( r R m+1 ) ( s (R m+1 ) ) M, : X : M ( r R m+1 ) ( s (R m+1 ) ) : C s.t. X(p) = (p, X(p)), X(p) ( r T p M) ( s T p M)., (r, s) (0, 2) g, M : T p M T p M g(p) : T p M T p M R : ( X(p), Y (p)) X(p), Y (p). V, V V = {f : V V R : } ( ) (0, 2) h, M : T p M T p M h(p) : T p M T p M R : ( X(p), Y (p)) D X Y, ξ.. M (U, φ, V ), φ 1 (p) = (x 1 (p),..., x m (p)) C - x 1,..., x m ( ) p M x 1,..., x m, {(dx 1 ) p,..., (dx m ) p } T p M. m = 2,. T p M {(dx 1 ) p, (dx 2 ) p }. T p M T p M 4, {(dx 1 ) p (dx 1 ) p, (dx 1 ) p (dx 2 ) p, (dx 2 ) p (dx 1 ) p, (dx 2 ) p (dx 2 ) p }.,, (dx i dx j ) p := (1/2)(dx i ) p (dx j ) p + (1/2)(dx j ) p (dx i ) p, {(dx 1 dx 1 ) p, (dx 1 dx 2 ) p, (dx 2 dx 2 ) p }., 1 (g = Edx 1 dx 1 + 2F dx 1 dx 2 + Gdx 2 dx 2 ) D
11 1.10.., (M, ξ) g, h, A, h( X, Y ) = g(a( X), Y ) for X, Y Γ(T M).,. ( m = 2),. M, p M (U, φ, V ), φ 1 = (x 1, x 2 ) M (p ) g = Edx 1 dx 1 + 2F dx 1 dx 2 + Gdx 2 dx 2, h = Ldx 1 dx 1 + 2Mdx 1 dx 2 + Ndx 2 dx 2, : A p = [ E F ] 1 [ F G L M ] M. N, E(p), F (p), G(p),. trace, : K = LN M 2 EG F 2, EN 2F M + GL H =. 2(EG F 2 ),, ( ),.,,,..,, ( ξ )., ( ), D
12 M m, : {(U α, φ α )} s.t. (1) {U α } M, (2) φ α : U α φ α (U) R m, (3) φ β φ 1 α : φ α (U α U β ) φ β (U α U β ) C -. {(U α, φ α )} M, (U α, φ α )., R m+1 m.,., M, N. F : M N C -, : p M, (U, φ) : p, (V, ψ) : F (p), ψ F φ 1 φ(p) C -., C C., C C (M) M C -. u : C (M) R p p, : f, g C (M), u(fg) = u(f)g(p) + f(p)u(g). p T p M, p.,,.,,., T p M (, ) M, (U, φ) p. φ = (x 1,..., x m ), ( ) p : C (M) R : f (f φ 1 ) (φ(p)). x i x i p., {( x 1 ) p,..., ( x m ) p } T p M D
13 2.2 T p M T p M M, (tangent bundle) : T M := T p M m M T M 2m. π : T M M, {(U α, φ α )} M., p M Φ α : π 1 (U α ) φ α (U α ) R m : a i ( x i ) p (φ α (p), (a 1,..., a m ) T M, {(π 1 (U α ), Φ α )} T M. T p M T p M T p M T p M, T p M T p M, ( T M := T p M ).,,, E, M, π : E M, : (1) π C, (2) p M, E p := π 1 (p) n, (3) (i.e., p M, U : p s.t. π 1 (U) = U R n ). (E, M, π), E. E E p p E, F E F := (E p F p )., (, ) π : E M, C - s : M E (section), π s = id M. T M s : M T M. s(p) π 1 (p) = T p M., M. T M 1 (1-form), k T M k (k-form) D
14 2.3,., 2 ( ). V, 2 ( V V R ) S 2 (T M) := S 2 (T p M) g : M S 2 (T M), : p M, g p :., S 2 (T p M) = {f : T p M T p M R :, }., g p, : X T p M, X 0 g p (X, X) > M g (M, g). g g p, {(U α, φ α )} M, φ α = (x 1,..., x m ), (1) p U α, {(dx i ) p } T p M, (2) dx i : U α T M : p (dx i ) p., (dx i ) p x i., (dx i ) p ( a j ( x j ) p ) := a j ( x j ) p (x i ) = a i., {(dx i ) p } {( x i ) p } S 2 (T p M) = span R {(dx i dx j ) p i j}. ( (U, φ) ) 1. R m ( ), g := dx 1 dx 1 + dx 2 dx dx m dx m. R m (M, g), γ : [a, b] M C -. γ : b L(γ) := g γ(t) ( γ(t), γ(t))dt. a R m,., H m := {(x 1,..., x m ) x m > 0} g := (1/x 2 m)(dx 1 dx dx m dx m ) (real hyperbolic space).,, D
15 2.4 (Levi-Civita ), ( ).,.,., Γ(T M) M., : Γ(T M) Γ(T M) Γ(T M) : (X, Y ) X Y M, : (1), (2) f C (M), X, Y Γ(T M), fx Y = f X Y, (3) f C (M), X, Y Γ(T M), X (fy ) = (Xf)Y + f X Y., Xf., T M, X : M T M : p X p ( )., X p T p M X p : C (M) R : f X p f. Xf X : C (M) C (M) : f Xf. Xf : M R : p X p f (M, ).,, R m,., (M, g), : (4) T (X, Y ) := X Y Y X [X, Y ] = 0, (5) X, Y, Y Γ(T M), Xg(Y, Z) = g( X Y, Z) + g(y, X Z). (4) T (torsion), (4) torsion free., Levi-Civita. bracket [X, Y ], : [X, Y ] : C (M) C (M) : ϕ [X, Y ]ϕ := X(Y ϕ) Y (Xϕ).,, X Y : g( X Y, Z) = (1/2){Xg(Y, Z) + Y g(z, X) Zg(X, Y ) +g([x, Y ], Z) g([y, Z], X) + g([z, X], Y )}., Levi-Civita D
16 2.5,..,., (M, g), Levi-Civita, c : I M (I ) c, : ċċ = 0.,. c, ċ : R T M., ( ċ : c(i) T M).. c(i) ċċ = 0 ( ), R m g 0. (R m, g 0 )., (R m, g 0 ) c = (c 1,..., c m ), : d 2 dt 2 c i = 0 for i = 1,..., m., ċċ = 0, t 2., p M, X p T p M, 1 c : s.t. c(0) = p, ċ(0) = X p., 2., (H 2, g), : c(t) := (cos s(t), sin s(t))., s(t) ṡ(t) = sin s(t)., ( ). Levi-Civita (, ).,., c(t) := (cos t, sin t) (H 2, g) D
17 2.6 ( )., (M, ), (1, 3)- : R(X, Y )Z := X Y Z Y X Z [X,Y ] Z. (1, 3)-, R Γ(T M T M T M T M). R Γ(End(T M T M T M, T M)) (M, g), Levi-Civita., (0, 4)- : R(X, Y, Z, W ) := g(r(x, Y )Z, W ) R m g 0 (R m, g 0 ), R = T p M 2 σ, K σ := g p (R p (u, v)v, u) σ. {u, v} σ. K σ, σ, (H 2, g) p, K TpH 2 = ( ), ( ),,.,., (, )., ( ) R 3 M, g., M, (M, g) (i.e., p M, K p = K TpM).,.,, (, )., : ( (Theorema egregium)) ( ) D
2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =
1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,
,.,. 2, R 2, ( )., I R. c : I R 2, : (1) c C -, (2) t I, c (t) (0, 0). c(i). c (t)., c(t) = (x(t), y(t)) c (t) = (x (t), y (t)) : (1)
( ) 1., : ;, ;, ; =. ( ).,.,,,., 2.,.,,.,.,,., y = f(x), f ( ).,,.,.,., U R m, F : U R n, M, f : M R p M, p,, R m,,, R m. 2009 A tamaru math.sci.hiroshima-u.ac.jp 1 ,.,. 2, R 2, ( ).,. 2.1 2.1. I R. c
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.
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,
No.004 [1] J. ( ) ( ) (1968) [2] Morse (1997) [3] (1988) 1
![No.004 [1] J. ( ) ( ) (1968) [2] Morse (1997) [3] (1988) 1](/thumbs/94/119152523.jpg "No.004 [1] J. ( ) ( ) (1968) [2] Morse (1997) [3] (1988) 1")
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
9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin
1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1
![1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1](/thumbs/96/129392387.jpg "1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1")
1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1 f, g C P, λ R (1) v(f + g) = v(f) + v(g) (2) v(λf) = λv(f) (3) v(fg)
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
等質空間の幾何学入門
2006/12/04 08 tamaru@math.sci.hiroshima-u.ac.jp i, 2006/12/04 08. 2006, 4.,,.,,.,.,.,,.,,,.,.,,.,,,.,. ii 1 1 1.1 :................................... 1 1.2........................................ 2 1.3......................................
II 1 II 2012 II Gauss-Bonnet II
II 1 II 212 II Gauss-Bonnet II 1 1 1.1......................................... 1 1.2............................................ 2 1.3.................................. 3 1.4.............................................
1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x
![24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x](/thumbs/94/118744578.jpg "24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x")
24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),
Morse ( ) 2014
Morse ( ) 2014 1 1 Morse 1 1.1 Morse................................ 1 1.2 Morse.............................. 7 2 12 2.1....................... 12 2.2.................. 13 2.3 Smale..............................
( ),.,,., C A (2008, ). 1,, (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,, (M, g) p M, s p : M M p, : (1) p s p, (
( ),.,,., C A (2008, ). 1,,. 1.1. (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,,. 1.2. (M, g) p M, s p : M M p, : (1) p s p, (2) s 2 p = id ( id ), (3) s p ( )., p ( s p (p) = p),,
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
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