2004 / D : 0,.,., :,.,.,,.,,,.,.,,.. :,,,,,,,., web page.,,. C-613 e-mail tamaru math.sci.hiroshima-u.ac.jp url http://www.math.sci.hiroshima-u.ac.jp/ tamaru/index-j.html 2004 D - 1 -
1 1.1 [ ].,. 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).,. 1.1.2 (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.,,.. 1.1.3 ( ) f : R n R n C -, p R n., (df) p, f p C -. (df) p, (Jf) p ( n ).. 1.1.4 ( ) 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) ).,,. 2004 D - 2 -
1.2,.,., (, ). 1.2.1 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 ). 1.2.2 S m := {x R m+1 x = 1} R m+1. 1.2.3 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). 1.2.4 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)...,. 1.2.5 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 )..,. 2004 D - 3 -
1.3 C -,.,., M R m+1. 1.3.1 (U, φ, V ) M : 1.2.5 (1) (3).,., ( )., ( ). 1.3.2 f : M R, M R m+1., M C 0 (M). 1.3.3 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 -. 1.3.4 C 0 (M) f C -, : (U, φ, V ) :, f φ : U R : C..,,. 1.3.5 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 -. 1.3.6 F : M N C -, : (U, φ, V ) : M, f φ : U R n+1 : C. 2004 D - 4 -
1.4,,.,,,. 1.4.1 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+1. 1.4.2 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.,.,,. 1.4.3 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+1. 1.4.4 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) ). 2004 D - 5 -
1.5 C - F : R m R n (df ) p,.,,. 1.5.1 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.. 1.5.2 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,. 1.5.3 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 ). 1.5.4 F : M N C - (U, φ, V ) : s.t. F φ : C. 1.3.4.. 1.5.5 (U, φ, V ), φ : U V R m+1 C -., C -. 1.5.6 M p M, (U, φ, V ) (U, φ, V ) p., C -. φ 1 φ : (φ ) 1 (V V ) φ 1 (V V )..,,. 1.5.7 C - F : M N C - ( ), : F : F 1 : C. 1.5.8 ( ) C - F : M N p M, (F ) p : T p M T F (p), F p C -. 2004 D - 6 -
1.6.,,.. 1.6.1 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 ). 1.6.2 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)).,,, ( ). 1.6.3 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 ),. 1.6.4 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). 1.6.5 M X Γ(M R m+1 ), : (i) p M, X(p) = 1, (ii) u p T p M, u, X(p) = 0. ξ. ( ).,. 1.6.6 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. 2004 D - 7 -
1.7,.,. 1.7.1 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.. 1.7.2 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 ). 1.7.3 X, Y Γ(T M) bracket : [ X, Y ] := D X Y D Y X. bracket, Jacobi. 1.7.4 ξ M, X Γ(T M), D X ξ Γ(T M). M (M, ξ). 1.7.5 (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) ( ).. 1.7.6 (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 ).,,.,. 2004 D - 8 -
1.8,..,., 1.8.1 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=1 1.8.2 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. 1.8.3 V W, < V W >.,, V W : a(v w) := (av) w = v (aw). 1.8.4 {e i } V, {h j } W, {e i h j } V W. V n n V (V (V V ) = (V V ) V ). 1.8.5 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 ). 2004 D - 9 -
1.9..,., T p M T p M. 1.9.1 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). 1.9.2 (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 : } ( ). 1.9.3 (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 ( ). 1.9.4 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 ). 2004 D - 10 -
1.10..,. 1.10.1 (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 ). 1.10.2 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 ),,. 1.10.3 ( ),.,,,..,, ( ξ )., ( ),.. 2004 D - 11 -
2 2.1 2.1.1 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 ).,. 2.1.2 R m+1 m.,.,. 2.1.3 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. 2.1.4 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 (, ). 2.1.5 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. 2004 D - 12 -
2.2 T p M T p M.. 2.2.1 M, (tangent bundle) : T M := T p M. 2.2.2 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 ).,,,. 2.2.3 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. 2.2.4 E, F E F := (E p F p )., (, ). 2.2.5 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). 2004 D - 13 -
2.3,., 2 ( ). V, 2 ( V V R ). 2.3.1 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) > 0. 2.3.2 M g (M, g). g g p, 1. 2.3.3 {(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 }. 2.3.4 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 2 + + dx m dx m. R m. 2.3.5. 2.3.6 (M, g), γ : [a, b] M C -. γ : b L(γ) := g γ(t) ( γ(t), γ(t))dt. a R m,.,. 2.3.7 H m := {(x 1,..., x m ) x m > 0} g := (1/x 2 m)(dx 1 dx 1 +... + dx m dx m ) (real hyperbolic space).,,.. 2004 D - 14 -
2.4 (Levi-Civita ), ( ).,.,.,. 2.4.1 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. 2.4.2 (M, ).,, R m,.,. 2.4.3 (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. 2004 D - 15 -
2.5,..,., (M, g), Levi-Civita, c : I M (I ). 2.5.1 c, : ċċ = 0.,. c, ċ : R T M., ( ċ : c(i) T M).. c(i) ċċ = 0 ( ),. 2.5.2 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., 0. 2.5.3 ċċ = 0, t 2.,. 2.5.4 p M, X p T p M, 1 c : s.t. c(0) = p, ċ(0) = X p., 2.,. 2.5.5 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). 2004 D - 16 -
2.6 ( ).,. 2.6.1 (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)). 2.6.2 (M, g), Levi-Civita., (0, 4)- : R(X, Y, Z, W ) := g(r(x, Y )Z, W ). 2.6.3 R m g 0 (R m, g 0 ), R = 0. 2.6.4 T p M 2 σ, K σ := g p (R p (u, v)v, u) σ. {u, v} σ. K σ, σ,. 2.6.5 (H 2, g) p, K TpH 2 = 1. 2.6.6. ( ), ( ),,.,., (, )., ( ). 2.6.7 R 3 M, g., M, (M, g) (i.e., p M, K p = K TpM).,.,, (, )., : 2.6.8 ( (Theorema egregium)) ( ). 2004 D - 17 -