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...................................... 4 1.4........................................ 4 1.5.................................... 5 2 7 2.1........................................ 7 2.2........................................ 9 2.3................................... 10 2.4................................. 11 2.5...................................... 12 3 14 3.1................................... 14 3.2................................. 15 3.3.................................. 16 4 18 4.1...................................... 18 4.2................................. 20 4.3.................................. 20 5 22 5.1.................................... 22 5.2................................. 23 5.3...................................... 24 5.4...................................... 25

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.

2 1 1.1,.. 1.3 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. 0 0 1 1.6 3 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.,,., :

1.2 3 1.9 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.,. 1.10 1.9., 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. 1.12 1.11.,.,, Aut(M). Aut(M) M. 1.13 G V ϕ : G GL(V ), representation.,. 1.14 G M, G.p := {g.p M g G} G p M orbit. M.,,.

4 1 1.3,. 1.15 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.,. 1.19 M, M G. G-. 1.4. 1.20 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.

1.5 5 1.21 M G-. p M, G p := {g G g.p = p}, : G/G p M : [g] g.p. G p isotropy subgroup. 1.22 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. 1.23 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 ). 1.24 G G/K, G G/K : G G/K G/K : (g, [h]) g.[h] := [gh]. 1.5.. 1.25 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 }.. 1.26 : RP n G 1 (R n+1 ) : [v] Rv.

6 1,,... 1.27 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,.,,. 1.28 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.

7 2 M G/K.., M, G/K, M = G/K,. G/K, G.,. :, +.,.,.,.,.,,., [KO, 5 ], [O2], [W, Chapter 3]. 2.1,,.,. C -. 2.1 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 -.

8 2 2.2 : (1) R n. (2) GL n (R). (3) 3 Heisenberg. R n. GL n (R), M n (R) = R n2. 2.3 3 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).

2.2 9 2.8 : SL n (R), dim SL n (R) = n 2 1. 2.9 : 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 = 1. 2.2,. 2.10 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. 2.12 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),. 2.14 [X, Y ] := XY Y X : gl n (C) := M n (C), u(n) := {X gl n (C) X + t X = 0} unitary Lie algebra,

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 0 0 0 x, y, z R.,.,. 2.16 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,,. 2.17 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. 2.18 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

2.4 11, R n : (dl a ) g ( x i ) g = ( x i ) a+g. 2.20 G, g, G. X, Y, [X, Y ],. 2.21 G = R n, g = span R { x 1,..., x n }, n. 2.22 : α : g T e G : X X e., g T e G.,. 2.4,.,,,. 2.23 g, g. ϕ : g g, : X, Y g, ϕ([x, Y ]) = [ϕ(x), ϕ(y )].,. 2.24 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. 2.25 G, G H, : (1) G H. (2) G H. (3) (1), (2) H.

12 2., i : H G,, i. 2.26 i : H G, di : h g., h di(h)., h g., GL n (R), gl n (R),. 2.27 O(n) o(n). 2.26, O(n). O(n) 2.7,. 2.28 2.27 : (1) SL n (R) sl n (R). (2) GL n (C) gl n (C). (3) U(n) u(n). 2.5,.,., G, g. 2.29 X e T e G, 1 c X : R G : s.t. ċ X (0) = X e. c X 1. 2.30 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.,.,,.

2.5 13 2.32 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. 2.34 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 >0. 2.35 X o(2), e X SO(2).,. 2.36 G GL n (R), g = {X gl n (R) t R, e tx G}., 2.27. 2.37 O(n) o(n). 2.38 2.37, : (1) SL n (R) sl n (R). (2) GL n (C) gl n (C). (3) U(n) u(n). 2.39 3 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. 0 0 1 0 0 0 :.

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]. 3.1. 3.1 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.

3.2 15.,. 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.

16 3 3.7 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.,. 3.10 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). 3.11 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).. 3.12 G 1,2,...,n 1 (R n ). 3.3 M = G/G p, G p T p M.,. 3.13 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.

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). 3.17 G/H reductive, : p g : (Ad H )- s.t. g = h p., reductive.,,. 3.18 G/H reductive g = h p., Ad H : H GL(p). : 3.19 O(n + 1) S n, O(n) R n. 3.20 SL 2 (R) H 2, : SO(2) R 2. S 2 = SO(3)/SO(2), SO(2) R 2.,., S 2 H 2. 3.21 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

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.

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-. 4.5 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,, 3. 4.6 G 1, G 2, G 3., G N., G G, G.,.,.

20 4 4.2 S n, O(n + 1) 1.. 4.7 G S n S n 1 (r) (r 0): {( ) } 1 G := α O(n) = O(n). α, r > 0 1. r = 0, 0. 4.8 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)).., 2. 4.3 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.

4.3 21 4.10 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.. 4.12 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,.

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.

5.2 23 5.3 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 ).

24 5 5.8 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. 5.10 σ g 2, {X, Y } σ. K σ := R(X, Y )X, Y σ. σ. 5.3 RH 2.. 5.11 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 ( 1 0 0 1 } 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 := 1. 5.12 g bracket : [A, X] c := cx., f : (g, [, ],, 1/c ) (g, [, ] c,, ) f(a) = ca, f(x) = X, f.,, bracket.,.

5.4 25 5.13 (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,. 5.14 (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,. 5.15 (g, [, ] c,, ), : (1) Ric(X, Y ) = c 2 X, Y. (2) σ := g, K σ = c 2. (1), Ricci Einstein. (2), 2. 5.4.. 5.16 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, [, ],, ).,,. 5.17 (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.. 5.18 (g, [, ],, ), : (1) R(A, X i )A = c 2 X i.

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). 5.21 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,.

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