等質空間の幾何学入門

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1 2006/12/04 08

2 i, 2006/12/ , 4.,,.,,.,.,.,,.,,,.,.,,.,,,.,.

3 ii :

4 1 1,.,,. :,., G/K.,.,., [KO, 7], [M, 4]. 1.1 :,.,. 1.1 M n (R) n n, : (1) GL n (R) := {g M n (R) det(g) 0} general linear group. (2) O(n) := {g GL n (R) t gg = I n } orthogonal group. (3) SL n (R) := {g GL n (R) det(g) = 1} special linear group. (4) SO(n) := SL n (R) O(n) special orthogonal group.,. O(n), R n GL n (R). 1.2 O(n) = {g GL n (R) x, y R n, gx, gy = x, y }., R n.

5 , M n (C) n n, : (1) GL n (C) := {g M n (C) det(g) 0}. (2) U(n) := {g GL n (C) t gg = I n } unitary group. (3) SL n (C) := {g GL n (C) det(g) = 1}. (4) SU(n) := SL n (C) U(n) special unitary group. 1.4 U(n) = {g GL n (C) x, y C n, gx, gy = x, y }., C n., H. classical group.,., 1.5 H 3 Heisenberg : 1 x z H := 0 1 y x, y, z R Heisenberg H. 1.2 G M.,, g, h G, e G, p, q M. 1.7 Φ : G M M : (g, p) Φ(g, p) =: g.p G M action, : (1) (gh).p = g.(h.p), (2) e.p = p.. g.p = Φ(g, p), g p gp, g.p. G M, G M. 1.8 Φ : GL n (R) R n R n : (g, v) g.v := gv GL n (R) R n. GL n (R) G R n., G M, G G M.,,., :

6 RH 2 := {z C Im(z) > 0}. SL 2 (R) RH 2 : [ Φ : SL 2 (R) RH 2 RH 2 a b : ( c d ], z) az + b cz + d. RH 2., , SL 2 (R) RH 2..,,.,, : 1.11 M, Aut(M) := {f : M M : }.., (1) Φ : G M M, : ϕ : G Aut(M) : g Φ(g, ) Φ(g, ) : p Φ(g, p). (2), ϕ : G Aut(M) : g ϕ g, : Φ : G M M : (g, p) ϕ g (p). ϕ ,.,, Aut(M). Aut(M) M G V ϕ : G GL(V ), representation., G M, G.p := {g.p M g G} G p M orbit. M.,,.

7 , G M transitive, : p, q M, g G : g.p = q. : 1.16 R n R n : R n R n R n : (g, p) g+p.,, : 1.17 G M, o M., p M, g G s.t. g.p = o, G M., : 1.18 n 2. O(n) S n 1., R n 2 O(n), R n 2 O(n), O(n) S n 1., M, M G. G G K, g h : g 1 h K G., G/K := G/ G K coset space. G/K = {gk g G}. K, G/K. M G-, M G.

8 M G-. p M, G p := {g G g.p = p}, : G/G p M : [g] g.p. G p isotropy subgroup G := O(n + 1) S n, {[ ] } 1 G e1 = O(n + 1) α O(n) = O(n). α S n : S n = O(n + 1)/O(n)., p G p : {[ α G en+1 = 1 ] } O(n + 1) α O(n) G e1., G p G q, : G/G p = G/Gq M G-, p, q M. G p G q g G : g 1 G p g = G q. M = G/G p, M (G, G p ),., M (G, G p ) G G/K, G G/K : G G/K G/K : (g, [h]) g.[h] := [gh] RP n, G k (R n ), G k1,...,k l (R n ) : (1) RP n := (R n+1 \ {0})/, v w : c 0 : v = cw. (2) G k (R n ) := {V K n V, dim V = k}. (3) G k1,...,k l (R n ) := {(V k1,..., V kl ) V k1 V kl :, dim V ki = k i } : RP n G 1 (R n+1 ) : [v] Rv.

9 6 1,, GL n (R) G k (R n ) : g.v := {gv v V }. (1) G k (R n ) GL n (R)-, G k (R n ) = GL n (R)/B. {[ ] } B = GL 0 n (R). (2) G k (R n ) O(n)-, G k (R n ) = O(n)/O(k) O(n k). 1.27,.,, RH 2 SL 2 (R) 1.9.., : 1.29 G k (R n ) = G n k (R n ). G k (R n ) = O(n)/O(k) O(n k) G n k (R n ) = O(n)/O(n k) O(k).,., : G k (R n ) G n k (R n ) : V V.

10 7 2 M G/K.., M, G/K, M = G/K,. G/K, G.,. :, +.,.,.,.,.,,., [KO, 5 ], [O2], [W, Chapter 3]. 2.1,,.,. C G Lie group, : (1) G G G : (g, h) gh C -. (2) G G : g g 1 C -. (1), (2) : G G G : (g, h) gh 1 C -.

11 : (1) R n. (2) GL n (R). (3) 3 Heisenberg. R n. GL n (R), M n (R) = R n Heisenberg H, R 3, H.,,., 2.4 SO(2) = S 1., 4, C -., SO(n),.,. 2.5 F : GL n (R) R m C -, G := {g GL n (R) F (g) = 0}., dim Ker(dF ) g = k g G, G k-., rank(jf ) g = n 2 dim Ker(dF ) g = n 2 k., G k. C -,.,. p G, T p G := {ċ(0) c : I M n (R) : C, c(i) G, c(0) = p}. T p M = Ker(dF ) g., dim T p G p, G. 2.6 F g (df ) g, : (df ) g (X) := lim t 0 (1/t)(F (g + tx) F (g)). 2.7 O(n), dim O(n) = n(n 1)/2. T e O(n) = {X M n (R) t X + X = 0}., : O(n).

12 : SL n (R), dim SL n (R) = n : GL n (C), dim GL n (C) = 2n 2., F (g) = 0 F. SL n (R). GL n (C) : GL n (C) GL 2n (R). n = , g R, [, ] : g g g. (g, [, ]) Lie algebra, : (1) [X, Y ] = [Y, X]. (2) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0..,. (g, [, ]), [, ] bracket product. (ii) Jacobi identity. 2.11, [X, Y ] := 0 abelian Lie algebra gl n (R) := M n (R), [X, Y ] := XY Y X.,,. 2.13, [X, Y ] := XY Y X : gl n (R) := M n (R) general linear Lie algebra, o(n) := {X gl n (R) X + t X = 0} orthogonal Lie algebra, sl n (R) := {X gl n (R) tr(x) = 0} special linear Lie algebra., SO(n) := O(n) SL n (R). so(n) := sl n (R) o(n), so(n) = o(n), O(n) SO(n), [X, Y ] := XY Y X : gl n (C) := M n (C), u(n) := {X gl n (C) X + t X = 0} unitary Lie algebra,

13 10 2 sl n (C) := {X gl n (C) tr(x) = 0}, su(n) := sl n (C) u(n). 2.15, 3 Heisenberg : h := 0 x z 0 0 y x, y, z R.,., R n C n,, (1) o(n) = {X gl n (R) v, w R n, Xv, w + v, Xw = 0}. (2) u(n) = {X gl n (C) v, w C n, Xv, w + v, Xw = 0}. 2.3,, G a G, : L a : G G : g ag., L a (dl a )g : T g G T ag G : (dl a )g(v) : C (M) R : ϕ v(ϕ L a )., T G := g G T g G G, dl a : T G T G X : G T G, : L a G G X X T G dl a R n : T G 2.19 R n X := f i. x i, f i

14 2.4 11, R n : (dl a ) g ( x i ) g = ( x i ) a+g G, g, G. X, Y, [X, Y ], G = R n, g = span R { x 1,..., x n }, n : α : g T e G : X X e., g T e G.,. 2.4,.,,, g, g. ϕ : g g, : X, Y g, ϕ([x, Y ]) = [ϕ(x), ϕ(y )]., GL n (R) g, gl n (R). G = GL n (R), M n (R) = R n2, T e G M n (R) = R n2., G g, T e G., : g = T e G = M n (R) = gl n (R)., : ϕ : g gl n (R) : X (X e x ij ) ij., x ij : G R, (i, j)- G. ϕ gl n (R) [X, Y ] = XY Y X G, G H, : (1) G H. (2) G H. (3) (1), (2) H.

15 12 2., i : H G,, i i : H G, di : h g., h di(h)., h g., GL n (R), gl n (R), O(n) o(n). 2.26, O(n). O(n) 2.7, : (1) SL n (R) sl n (R). (2) GL n (C) gl n (C). (3) U(n) u(n). 2.5,.,., G, g X e T e G, 1 c X : R G : s.t. ċ X (0) = X e. c X G, g. exp : g G : X c X (1). : C -,., 0 g : d exp 0 : g T e G : X X e., : 2.31 exp : g G, 0 g e G.,.,,.

16 exp : gl n (R) GL n (R), : exp(a) := e A := k=0 2.33, : A k k!. (1) Be A B 1 = e BAB 1, (2) det e A = e tra, (3) e A+B = e A e B if AB = BA exp : gl n (R) GL n (R),. : X gl n (R), c X (t) := (tx) k /k! c X (0) = X., 1 exp, R R > X o(2), e X SO(2)., G GL n (R), g = {X gl n (R) t R, e tx G}., O(n) o(n) , : (1) SL n (R) sl n (R). (2) GL n (C) gl n (C). (3) U(n) u(n) Heisenberg H, h : 1 x z 0 x z H := 0 1 y x, y, z R, h := 0 0 y x, y, z R :.

17 14 3, G M, G/G p M.,, G/G p M. : G M, G p. G, H, G/H. dim G/H = dim G dim H. M = G/H, H T p M., reductive g = h p.,., [KO, 6 ], [W, Chapter 3] G M C -, G M M : (g, p) g.p C -.,,. 3.2 : (1) GL n (R) R n, (2) O(n + 1) SO(n + 1) S n.

18 ,. 3.3 G a G. G, : (1) L a : G G : g ag. (2) I a : G G : g aga 1 I a. (3) Ad a : g g : X (di a ) e (X)., G g T e G. I Ad,., ϕ : G M : g g.p,. {p}, : 3.4 G M, p M, G p := {g G g.p = p} G. 3.2 G/G p,., G H, G/H. H : 3.5 G, H, π : G G/H. π G/H, H G. G G/H : a G, a.[g] := [ag]. 3.6 G H, G/H : G G/H., G/H., exp, : g = h p p. π exp : p G/H, 0 p U [e] G/H N., (N, (π exp) 1 ) [e]. G : {(gn, (π exp) 1 g 1 )} g G, G/H.

19 G M. ϕ : G/G p M : [g] g.p, G/G p C., M, G G p. 3.8 S 2 = O(3)/O(2), ψ p := (1, 0, 0) : ψ(a, b) := (cos a cos b, sin a cos b, sin b)., : 3.9 dim G/H = dim G dim H., S n = O(n + 1)/O(n) dim O(n + 1) = dim O(n) + n. : dim O(n) = n(n 1)/2. dim O(n) = dim o(n) G k (R n ) = O(n)/O(k) O(n k), : dim G k (R n ) = dim O(n) (dim O(k) + dim O(n k)) = k(n k) G 1,2,...,n 1 (R n ). 3.3 M = G/G p, G p T p M., ϕ : G Aut(M). p M, G p T p M isotropy representation : (dϕ) : G p GL(T p M) : a (dϕ a ) p.,., 1.23., : 3.14 G M, p, p q.

20 3.3 17, 3.15 α : G 1 GL(V ) β : G 2 GL(W ) equivalent, : : ϕ : G 1 G 2 :, F : V W : s.t. α g V V F F W β ϕ(g), : 3.16 O(3) S 2, O(2) R 2.,., 3.3 Ad : G GL(g). Ad H : H GL(g) G/H reductive, : p g : (Ad H )- s.t. g = h p., reductive.,, G/H reductive g = h p., Ad H : H GL(p). : 3.19 O(n + 1) S n, O(n) R n SL 2 (R) H 2, : SO(2) R 2. S 2 = SO(3)/SO(2), SO(2) R 2.,., S 2 H O(n) G k (R n ), : O(k) O(n k) M n k,k (R), (a, b).x := bx t a. M n k,k (R) (k, n k)-. p = T p G k (R n ), dim G k (R n ) = k(n k). W

21 18 4 G M., p M G.p M.,.,,., R 3 SL 3 (R)/SO(3),,. 4.1.,. 4.1 G N. N M G, : G G : s.t. M G -., : G N, p N M := G.p N., G.p := {g.p N g G }. 4.2, : M = G /G p.

22 4.1 19, M = G/H., M N,. G-., : 4.3 α : G M, β : G N. f : M N G-, : α g M M f f N β g g G, : 4.4 S n = O(n + 1)/O(n) R n+1 O(n + 1)-. e 1 S n, S n = O(n + 1).e 1 R n+1.,,,., G- N, G G := O(3) R 3 R 3, : N (a, v) G w R 3, (a, v).w := aw + v., G R 3 : (1) S 2 (r) := {(x, y, z) x 2 + y 2 + z 2 = r 2 }. (2) S 1 (r) R := {(x, y, z) x 2 + y 2 = r 2 }. (3) R 2 := {(x, y, 0)}.. (1) G 1 := O(3), v := (r, 0, 0). (2) G 2 := O(2) {(0, 0, z)}, v := (r, 0, 0). (3) G 3 := {(x, y, 0)} R 3, v := (0, 0, 0). G O(3) R 3,. (a, v) (b, u) = (ab, au + v) R 3,,. G, R 3. 2,. R 3,, G 1, G 2, G 3., G N., G G, G.,.,.

23 S n, O(n + 1) G S n S n 1 (r) (r 0): {( ) } 1 G := α O(n) = O(n). α, r > 0 1. r = 0, G S n S k 1 (r 1 ) S n k 1 (r 2 ) (r1 2 + r2 2 = 1): {( ) } α G := α O(k), β O(n k) = O(k) O(n k). β, r 1 > 0 r 2 > 0 1. r 1 = 0 S n k 1 (1), r 2 = 0 S k 1 (1)., S k S n k = (O(k + 1) O(n k))/(o(k) O(n k 1)).., SL 3 (R)/SO(3)., s-,. 4.9 SL 3 (R)/SO(3), (1) reductive: sl 3 (R) = o(3) p, p := {X sl 3 (R) t X = X}. (2) : a.x := axa 1 (a SO(3), X p). O(3) p. p X, Y := tr( t XY ), p S 4.

24 p a : λ 1 a := λ 2 λ 1, λ 2, λ 3 R.. : 4.11 p a : a := λ 1 λ 2 λ 1 λ 2 λ 3. λ 3 λ 3, X a X X p. O(3) X h X : h X = {Y o(3) [Y, X] = 0}. bracket, : 4.13 X a, (1) λ 1 = λ 2 > λ 3, h X = o(2). (2) λ 1 > λ 2 = λ 3, h X = o(2). (3) λ 1 > λ 2 > λ 3, h X = 0., : 4.14 X a, (1) λ 1 = λ 2 > λ 3, H.X = O(3)/O(2) O(1). (2) λ 1 > λ 2 = λ 3, H.X = O(3)/O(1) O(2). (3) λ 1 > λ 2 > λ 3, H.X = O(3)/O(1) O(1) O(1). (1) (2), G 1 (R 3 ) = RP 2, S 4. Veronese surface. 4.15, G 2 (R 5 ).,,., a, a. h X,.

25 22 5,,.,. :..,. [A], [B]. 5.1 : 5.1 M = G/H reductive, g = h p reductive. p Ad H -, M G-. G/H G- G/H,,. p T o M, G- T o M., : 5.2, M G-., p M, T p M g p : a G, g a.o (X, Y ) = (da) p (X), (da) p (Y ). g p, well-defined, a.o = b.o (da) p (X), (da) p (Y ) = (db) p (X), (db) p (Y ). Ad H -, well-defined.

26 M = G/H reductive. M G-, Ad H. Ad H p., G/H. Shur : 5.4 M = G/H reductive, Ad H., Ad H p, G-. X(M) M. (M, g) : 5.5 : X(M) X(M) X(M) Levi-Civita : 2g( X Y, Z) = Xg(Y, Z) + Y g(z, X) Zg(X, Y ) +g([x, Y ], Z) + g([z, X], Y ) + g(x, [Z, Y ]).,. 5.2, G/{e}. G, reductive g = {0} g. 5.6 g, G.,. 5.7 : g g g Levi-Civita : 2 X Y, Z = [X, Y ], Z + [Z, X], Y + X, [Z, Y ]. Levi-Civita,., U : g g g : 2 U(X, Y ), Z = [Z, X], Y + X, [Z, Y ]. U. Levi-Civita X Y = (1/2)[X, Y ] + U(X, Y ).

27 R(X, Y )Z := X Y Z + Y X Z + [X,Y ] Z. 5.9 Ric(X, Y ) := R(X, E i )Y, E i Ricci. {E i } g. Ricci σ g 2, {X, Y } σ. K σ := R(X, Y )X, Y σ. σ. 5.3 RH RH 2 := {z C Im(z) > 0} SL 2 (R). G, RH 2 = G/{e} : {( ) } e x y G := 0 e x x, y R. G, : {( ) x y g := 0 x {A, X} : A := 1 2 ( } x, y R. ) ( 0 1, X := 0 0, bracket [A, X] = X. g, c > 0, ; ). A, A 1/c := 1/c 2, A, X 1/c := 0, X, X 1/c := g bracket : [A, X] c := cx., f : (g, [, ],, 1/c ) (g, [, ] c,, ) f(a) = ca, f(x) = X, f.,, bracket.,.

28 (g, [, ] c,, ), : (1) U(A, A) = 0, U(A, X) = (c/2)x, U(X, X) = ca. (2) A A = 0, A X = 0, X A = cx, X X = ca. U, (g, [, ] c,, ), : (1) R(A, X)A = c 2 X. (2) R(A, X)X = c 2 A. R(X, Y )Z = R(Y, X)Z, (g, [, ] c,, ), : (1) Ric(X, Y ) = c 2 X, Y. (2) σ := g, K σ = c 2. (1), Ricci Einstein. (2), g := span R {A, X 1,..., X n 1 }, {A, X 1,..., X n 1 }, bracket : [A, X i ] := cx i, [X i, X j ] := 0. (g, [, ],, ).,, (g, [, ],, ), : (1) U(A, A) = 0, U(A, X i ) = (c/2)x i, U(X i, X j ) = δ ij ca. (2) A A = 0, A X i = 0, Xi A = cx i, Xi X j = δ ij ca (g, [, ],, ), : (1) R(A, X i )A = c 2 X i.

29 26 5 (2) R(A, X i )X j = δ ij c 2 A. (3) R(X i, X j )A = 0. (4) R(X i, X j )X k = δ jk c 2 X i δ ik c 2 X j.,, : 5.19 (g, [, ],, ), : (1) Ric(X, Y ) = c 2 (n 1) X, Y. (2) σ, K σ = c 2., 2 σ., σ : 5.20 f : g g., : X Y = f(x) f(y ), R(X, Y )Z = R(f(X), f(y ))f(z). Aut(g) O(g,, ),., O(n 1) g := span R {A, X, Y, X}, {A, X, Y, Z}, bracket : [A, X] := (1/2)X, [A, Y ] := (1/2)Y, [A, Z] = Z, [X, Y ] := Z bracket 0. Ricci,.., X, Y, Z, 3 Heisenberg. Heisenberg,.

30 27 [A] A. Arvanitoyeorgos, An introduction to Lie groups and the geometry of homogeneous spaces. Translated from the 1999 Greek original and revised by the author. Student Mathematical Library, 22. American Mathematical Society, Providence, RI, xvi+141 pp. [BCO] J. Berndt, S. Console, C. Olmos, Submanifolds and holonomy. Chapman & Hall/CRC Research Notes in Mathematics 434, Chapman & Hall/CRC, Boca Raton, FL, 2003, x+336 pp. [B] A.L. Besse, Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10. Springer-Verlag, Berlin, xii+510 pp. [H] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34. American Mathematical Society, Providence, RI, xxvi+641 pp. [KN1] S. Kobayashi, K. Nomizu, Foundations of differential geometry. Vol. I. Reprint of the 1963 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, xii+329 pp. [KN2] S. Kobayashi, K. Nomizu, Foundations of differential geometry. Vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, xvi+468 pp. [KO],,., [M],., [O1], ( )., [O2], ( )., [W] F. Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, ix+272 pp.

( ),.,,., 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,, (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),,