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1 ( ),.,,., 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, (2) s 2 p = id ( id ), (3) s p ( )., p ( s p (p) = p),, Fix(s p, M) := {x M s p (x) = x} {p}., p s p, R 2 ( ),. p s p : s p (x) = 2p x. tamaru math.sci.hiroshima-u.ac.jp 1

2 (1),. 0 R 2, x- s(x, y) := (x, y)., Fix(s, M) x-, {0} S n := {x R n+1 x = 1},. p s p : s p (x) = x + 2 x, p p. S n s p, 0 p G. p s p : s p (x) = px 1 p.,.., e s e (x) = x 1. p, L g : G G : x gx, p e, : s p (x) = L p s e L p 1(x) = px 1 p., (, ). 2, (M, g), (G, K).,,. 2.1,.., M, G., Aut(M) := {f : M M : }.., Aut(M) ϕ : G Aut(M) : g ϕ g G M (action), G M., M, G, Aut(M) Isom(M),. 2

3 2.2. O(n) S n : ϕ g (x) := gx., SL 2 (R) RH 2 := {z C Im(z) > 0} : [ ] a b g =, ϕ c d g (z) := az + b cz + d.., RH G M ϕ (transitive), : p, q M, g G : ϕ g (p) = q. (homogeneous space)., (, ) , G, K. G : g h : g 1 h K. G/ G K, G/K.,.,, G, M., (1) G ϕ M., p M, G p := {g G ϕ g (p) = p}, M = G/G p ( ). (2) M G (, M = G/K)., G M : ϕ g ([h]) := [gh]. G p, p (isotropy subgroup).,., : p, q M, g G : g 1 G p g = G q (, G p G q ) : S n = O(n + 1)/O(n).., S n = SO(n + 1)/SO(n) ( ) : RH 2 = SL 2 (R)/SO(2).,. 3

4 2.10. : RP n = O(n + 1)/O(1) O(n).,, RP n := {l R n+1 l 0 }., R n+1 \ {0} : v w : c 0 : v = cw., RP n = (R n+1 \ {0})/ = S n /. 2.2,., M = G/K, (M, g), G := Isom(M, g) 0 M., Isom(M, g) 0 Isom(M, g). M = R n, 2 p, q,, p q. M = S 1, 2 p, q, pq, p q., : 2 p, q. 2, p q m (, )., m s m p q., Isom(M, g) M. M, G., (M, g), M = G/K., K o M (, K = G o ). G K, G, K G. (G, K) (Riemannian symmetric pair), : (1) σ : G G : Fix(σ, G) 0 K Fix(σ, G). (2) K. σ (involution), σ 2 = id. (Cartan involution)., K,., (M, g), G := Isom(M, g) 0, K, (G, K)., K o, σ : G G : σ(g) := s o gs o.,. 4

5 2.14. (SL n (R), SO(n)). σ(g) = t g 1. n = 2, RH 2 = SL 2 (R)/SO(2) (SO(n + 1), SO(n)) (SO(n + 1), S(O(1) O(n))). σ(g) = I 1,n gi 1,n, I 1,n := diag( 1, 1,..., 1) ( ). : S n = SO(n + 1)/SO(n), RP n = SO(n + 1)/S(O(1) O(n)) (SU(n + 1), S(U(1) U(n))). σ(g) = I 1,n gi 1,n. : CP n := {l C n+1 l 1 }., C n+1 \ {0} : v w : c C : v = cw., CP n = (C n+1 \ {0})/ = SU(n + 1)/S(U(1) U(n)). 2.3, (G, K). M := G/K, G-,..,, σ : G G., o := [e] : s o : M M : [g] [σ(g)]., s o (o) = o s 2 o = id. p M, ( 1.5). 2.7, G M = G/K. ϕ, p = [g] M : s p := ϕ g s o ϕ g S 1 = SO(2)/SO(1) = SO(2)/{e}, 2.15 o := (1, 0), s o (x, y) = (x, y)., G, (G G, diag(g))., diag(g) := {(g, g) g G}, σ(g, h) := (h, g). G G G : ϕ (g,h) (x) := gxh 1., G = G G/diag(G)., ( 1.5).,,

6 2.20. (SO(n), SO(k) SO(n k)) (SO(n), S(O(k) O(n k))). σ(g) = I k,n k gi k,n k, I k,n k := diag( 1 k, 1 n k ). (SO(n), S(O(k) O(n k))), : G k (R n ) := {V R n V k }. G 1 (R n ) = RP n 1., (SO(n), SO(k) SO(n k)),. 3,.,. 3.1., M, f : M R. f, M (manifold), {(U α, ϕ α )} : (1) M = α U α, (2) ϕ α : U α ϕ α (U α ) R n, (3) U α U β, ϕ β ϕ 1 α : ϕ α (U α U β ) ϕ β (U α U β ) C. {(U α, ϕ α )}. (2) n. (1), (2), p M, U α M, f : M R. f p M, : U α (p U α ), f ϕ 1 : ϕ α (U α ) R ϕ α (p). f : M N,. 3.1 (3), U α,.,, U α S 1 1., x > 0, x < 0, y > 0, y < 0 4., S 1 4 y = ± 1 x 2, x = ± 1 y 2,.. 6

7 3.4. U R n C - f : U R m, M := {(x, y) U R m y = f(x)} n., : ϕ : M U : (x, y) x. M f U R n C - F : U R m, M := {p U F (x) = 0}. p M, rank(jf ) p = k, M n k. rank(jf ) x.,., (F (p) = 0), (y = f(x)),, (, F (x, y) = ax + by + c, ). 3.2, G (Lie group) : (1) G, (2) G, (3) G G G : (g, h) gh G G : g g 1 C GL n (R) n 2. GL n (R), M n (R) = R n2,, C. n G (3 Heisenberg ): 1 x z G := 0 1 y x, y, z R R 3.,,., C - F : GL n (R) R m, G := {g GL n (R) F (g) = 0}. g G, dim ker(df ) g = k, G k. F : (df ) g : R n2 R m : X lim (F (g + tx) F (g)). t t 0 1 7

8 n = m = 1. X., (df ) g,, Jacobi (JF ) g., rank(jf ) g = dim Image(dF ) g = n 2 dim ker(df ) g., 3.5, G O(n) n(n 1)/2., O(n) F (g) = t gg I n. : ker(df ) e = {X M n (R) X + t X = 0}. 3.3., g [, ] : g g g (Lie algebra), : (1) [, ], (2) (, X, Y g, [X, Y ] = [Y, X]), (3) Jacobi (, X, Y, Z g, [[X, Y ], Z]+[[Y, Z], X]+[[Z, X], Y ] = 0) g, [X, Y ] := 0 ( ) gl n (R) := M n (R) [X, Y ] := XY Y X ( )., Jacobi., GL n (R)., : g h, : X, Y h, [X, Y ] h., ( ) gl n (R) : (1) sl n (R) := {X gl n (R) tr(x) = 0} ( ), (2) o(n) := {X gl n (R) X + t X = 0} ( ). 8

9 , SL n (R) O(n). : G. G T e G. [, ]. : M, p M., (1) c : ( ε, ε) M : C c(0) = p, ċ(0) p. (2) M p : T p M := {ċ(0) c : ( ε, ε) M : C, c(0) = p}., T p M, dim T p M = dim M.,.,,., : exp(x) := n=0 X n n! = I n + X + X2 2! + X3 3! T e O(n) = o(n)., X o(n), c(t) := exp(tx). c O(n), ċ(0) = X., det(exp(x)) = e tr(x), : T e SL n (R) = sl n (R)., F. : U R n C - F : U R m, M = {x U F (x) = 0}. : T p M = ker(df ) p., F (g) = t gg I n, (df ) e (X) = t X + X ( 3.10 ) U(n) := {g GL n (C) t ḡg = I n }, : u(n) := {X gl n (C) t X + X = 0}.,,, Heisenberg : 0 x z 0 0 y x, y, z R

10 3.4, (g, k), : (1) θ : g g : θ 2 = id., k = Fix(θ, g). (2) k. θ, bracket (i.e., θ([x, Y ]) = [θ(x), θ(y )]). θ. k, K (sl n (R), o(n)). θ(x) = t X., (SL n (R), SO(n)) (o(n + 1), o(n)). θ(x) = I 1,n XI 1,n, I 1,n := diag( 1, 1,..., 1) ( )., (SO(n + 1), SO(n)) (SO(n + 1), S(O(1) O(n))).,., ( ) (G, K)., g, k, (g, k). (G, K) σ, θ = (dσ) e (g, k). (dσ) e, σ e., C - f : M N, f x M, : (df) x : T x M T f(x) N : ċ(0) d (f c)(0). dt , 3.25 θ, σ. 3.5 (g, k), g, g, g = k p, : (1) [k, p] p, [p, p] k. (2) k. 10

11 3.29. (g, k), θ ( 1)- p, g = k p., g = k p, (g, k).., θ(x) = X k X p. X k X p, X k- p (sl n (R), o(n)) θ(x) = t X, sl n (R) = o(n) {X sl n (R) t X = X}., (o(n + m), o(n) o(m)) θ(x) = I n,m XI n,m, o(n + m) = (o(n) o(m)) {( 0 t C C 0 ) } C M n,m (R)., (o(n + m + l), o(n) o(m) o(l)) ( ) (su(n), o(n)) θ(x) = X ( ), su(n) = o(n) {X su(n) X = X}. su(n) = {X u(n) tr(x) = 0} ( )., X u(n), X = X t X = X. 4,.,,,. 4.1 g, g C., g C = g 1g, bracket, g g = k p, g := k 1p. g g C,., bracket ( [p, p] k)., [p, p] k, g. (g, k), (g, k).,,. 11

12 4.2. (sl n (R), o(n)) (su(n), o(n)).,,.,., SL n (R)/SO(n), SU(n)/SO(n) (, ). 4.2, K p ( )., (G, K), g = k p., ϕ : K GL(p) : g ϕ g := (di g ) e p., I g : G G : h ghg 1.. I g (e) = e T e G = g, I g (di g ) e : g g. p ϕ g., (di g ) e (p) p (g K ). ϕ K V, ϕ : K GL(V ),. v V, K.v := {ϕ g (v) V g K} v K., K. V,., K SO(n) GL n (R) ( ). v R n, v.,, ϕ i : K GL(V i ) ( i = 1, 2), : F : V 1 V 2 : s.t. g K, F (ϕ 1 ) g = (ϕ 2 ) g F., S n = SO(n + 1)/SO(n), SO(n) R n. S n 1 R n g = k p, g = k 1p. 12

13 4.9. SL n (R)/SO(n) ϕ : SO(n) GL(p) : ϕ g (X) = gxg 1., p = {X sl n (R) t X = X}. n = 3, (1) X = diag(2, 1, 1) SO(3).X = SO(3)/S(O(1) O(2)) = RP 2. (2) X = diag(1, 0, 1) SO(3).X = SO(3)/S(O(1) O(1) O(1)). (1), RP 2 R 5., SO(n) p, 1., RP 2 S 4. Veronese. 4.3 g. g = k p a p (maximal abelian), : (1) [a, a] = 0, (2) a a p [a, a ] = 0 a = a.,. p V 1, V 2, ϕ,, g K : ϕ g (V 1 ) = V 2. a, p p, (rank) (sl n (R), o(n)), a := {diag(a 1,..., a n ) a a n = 0} p. n (o(n + 1), o(n)) ,, a., ϕ a p, : X p, H a : K.X = K.H ( )., (sl n (R), o(n)),. 4.4, g,.,. g = k a, a p. 13

14 4.16. α : a R, g α := {X g H a, [H, X] = α(h)x}., α a, α 0 g α {0}. α g α. g α, Ad H : g g : X [H, X] ( H a)., H a, Ad H. = (g, a) a, (1) [g α, g β ] g α+β, (2) g = g 0 α g α ( ) (sl 3 (R), o(3)), 4.13 a, ε i : a R : diag(a 1, a 2, a 3 ) a i (i = 1, 2, 3). a = {±(ε 1 ε 2 ), ±(ε 2 ε 3 ), ±(ε 1 ε 3 )}. g = k p, k p θ, k α := {X + θx X g α } p α := {X θx X g α }., (1) θ(g α ) = g α. k α = k α, p α = p α. (2) k α = k (g α g α ), p α = p (g α g α ). (3) k = k 0 α>0 k α, p = a α>0 p α. (4) dim k α = dim p α., k H, H a., : k H = k 0 α(h)=0 k α., k H = {X k [X, H] = 0} sl 3 (R) = o(3) p, (1) H = diag(2, 1, 1), k H = k 0 k ε2 ε 3 k ε2 +ε 3. (2) H = diag(1, 0, 1), k H = k H a 3 : (1) α, α(h) 0. (2) K.H. (3) dim K.H = dim p dim a., 2, R n = p 2., ( ), S n 1 1.,,., 1,. 14

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