Step 2 O(3) Sym 0 (R 3 ), : a + := λ 1 λ 2 λ 3 a λ 1 λ 2 λ 3. a +. X a +, O(3).X. O(3).X = O(3)/O(3) X, O(3) X. 1.7 Step 3 O(3) Sym 0 (R 3 ),

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1 ,, O(2) R 2 O(2).p, {0} r > 0. O(3) R 3 O(3).p, {0} r > 0.,, O(n) ( SO(n), O(n) ): Sym 0 (R n ) := {X M(n, R) t X = X, tr(x) = 0}. 1.3 O(n) Sym 0 (R n ) : g.x := gxg 1 (g O(n), X Sym 0 (n, R)).. SO(n), O(n), n = Step 0 O(3) Sym 0 (R 3 ), : X, Y := tr( t XY ) = tr(xy ). O(3), S Step 1 O(3) Sym 0 (R 3 ), : λ 1 a := λ 2 λ 3 tr = 0.

2 Step 2 O(3) Sym 0 (R 3 ), : a + := λ 1 λ 2 λ 3 a λ 1 λ 2 λ 3. a +. X a +, O(3).X. O(3).X = O(3)/O(3) X, O(3) X. 1.7 Step 3 O(3) Sym 0 (R 3 ), X := diag(λ 1, λ 2, λ 3 ) a +, : (1) λ 1 = λ 2 = λ 3 (= 0), O(3) X = O(3). (2) λ 1 > λ 2 = λ 3, O(3) X = O(1) O(2). (3) λ 1 = λ 2 > λ 3, O(3) X = O(2) O(1). (4) λ 1 > λ 2 > λ 3, O(3) X = O(1) O(1) O(1)., {0}, RP 2, F 1,2 (R 3 ). S 4 RP 2 Veronese, S 4 F 1,2 (R 3 ) Cartan. 1.8 F 1,2 (R 3 ) = O(3)/(O(1) O(1) O(1)).,, (SL(n, R), SO(n), σ) with σ(g) := t g (G, K, σ) ( ), : G, K G, σ Aut(G), σ 2 = id, Fix(σ, G) 0 K Fix(σ, G) (SL(n, R), SO(n), σ), : σ(g) := t g g, [, ] : g g g. (g, [, ]), : (i) X, Y g, [X, Y ] = [Y, X]. (ii) X, Y, Z g, [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0.

3 1.4 3 g gl(n, R) := M(n, R) : [X, Y ] := XY Y X., gl(n, R) g. g g, : (i) g g. (ii) X, Y g, [X, Y ] g., gl(n, R) : (1) sl(n, R) := {X gl(n, R) tr(x) = 0}. (2) o(p, q) := {X gl(p + q, R) t XI p,q + I p,q X = 0}., ( ), sl(n, C) (g, k, θ) ( ), : g, k g, θ Aut(g), σ 2 = id, k = Fix(θ, g) (sl(n, R), o(n), θ), : θ(x) := t X., g, θ Cartan. (.), sl(n, R), θ Cartan g g g, : (i) g g. (ii) X g, Y g, [X, Y ] g.

4 4 1, sl(n, R) gl(n, R) g, dim g > 1, sl(n, R) n = Killing, g Killing, B : g g R Killing : B(X, Y ) := tr(ad X ad Y ) Killing B, : B([X, Y ], Z) + B(Y, [X, Z]) = 0 ( X, Y, Z g) Lie sl n (R) Killing B : B(X, Y ) = 2ntr(XY ) g, Killing B h 3 = Span{x, y, z}, : [x, y] = z, [y, z] = [z, x] = 0 ( 3 Heisenberg ). h 3 Killing,. 1.7 Cartan, g Lie, B g Killing. θ : g g, θ Lie, θ 2 = id : θ : g g Cartan, B θ B θ (X, Y ) := B(X, θ(y )) (for X, Y g).

5 1.8 5 Cartan θ, ± Cartan θ g = k p Cartan., k 1, p sl(n, R), θ(x) := t X Cartan., Cartan sl(n, R) = k p, : k = so(n), p = {X sl(n, R) X = t X} Cartan g = k p : [k, k] k, [k, p] p, [p, p] k g, Cartan., Cartan (G ) (G, K, σ), (g, k, θ), θ Cartan. : (1) G g. G < GL(n, R) ( ), g gl(n, R), g.x := gxg 1. (2) K g, Cartan g = k p. K p., sl(n, R),.

6 , g Lie, θ Cartan, g = k p Cartan. K p p a, : (1) a., X, Y a, [X, Y ] = 0. (2) a a p, a, a = a a = sl(n, R), : { } a := ak E kk tr = ( n = 3 ) p K-. SL(n, R), O(n) Sym 0 (R n ) a ( ). 1.10, g Lie, B Killing, θ Cartan, g = k p Cartan, a p X, Y, Z g, : (1) ad [X,Y ] = ad X ad Y ad Y ad X, (2) B([X, Y ], Z) + B(Y, [X, Z]) = 0, (3) X p, B θ (ad X (Y ), Z) = B θ (Y, ad X (Z))., X p, ad X B θ. ad X,. X a.,. a a α a, α : g α := {X g [H, X] = α(h)x ( H a)} α a (a ), : α 0, g α 0.,, = (g, a).

7 = (g, a), : (1) g = g 0 ( α g α) B θ, (2) [g α, g β ] g α+β ( α, β {0}), (3) θ(g α ) = g α ( α {0}). (4) g 0 = k 0 a, k 0 := g 0 k. (1) sl n (R), Cartan, : (1) a := { a k E kk tr = 0} p, (2) g 0 = a, (3) g εi ε j = span{e ij }, ε i ( a k E kk ) := a i, (4) = {ε i ε j i j} sl(2, C) ( ), Cartan sl(2, C) = su(2) p. p,, O(n) Sym 0 (R n ), a, a +., Λ := {α 1,..., α r }, : (i) Λ a, (ii) α, c 1,..., c r Z 0 c 1,..., c r Z 0 : α = c 1 α c r α r. 1.44, Λ = {α 1,..., α r }, α = c 1 α c r α r. (1) α, c 1,..., c r Z 0. (2) α, c 1,..., c r Z 0. (3) α, : c 1α c rα r, c 1 c 1,..., c r c r.

8 sl n (R), α i := ε i ε i+1 (i = 1,..., n 1), {α 1,..., α n 1 }., α = ε 1 ε n = α α n 1. = (g, a),., = (g, a),., {α 1,..., α r }, H a, : a + := {X a α i (X) 0 ( i)} g = sl(3, R), a,, Λ, λ 1 a + = λ 2 λ 3 a λ 1 λ 2 λ 3.

9 RH 2 1, : {( ) } {( ) } a 0 1 b A := 0 a 1 a > 0, N := b R g. (1) [g, g] := { n i=1 [X i, Y i ] X i, Y i g} derived ideal. (2) g, g 1 := [g, g],..., g k := [g, g k ] {0}. (3) g, [g, g]. 2.3 : s := {( a b 0 a ) } a, b R. : sl(2, R) = o(2) s. 2.4 gl(n, R) ( n = 3 Heisenberg ),.

10 , g, g = k p Cartan, a p, = (g, a), Λ = {α 1,..., α r }. 2.5 α = c 1 α c r α r, level(α) := c c r α. 2.6 g k := level(α)=k g α, : [g k, g l ] g k+l. 2.7 : (1) n := α>0 g α, (2) s := a n ( [a, n] n), (3) k s = g = k a n, ( ).,,., G = KAN. 2.9 S, s G. M := G/K = S/{e} = S. RH 2 = SL(2, R)/SO(2), : (1) α > 0, g α g α = (1 + θ)g α (1 θ)g α, (2) p = a α>0 (1 θ)g α, (3) dim n = dim p (= dim G/K). (4) M S.[e] = S/S [e] = S/(S K).

11 2.4 ( ) ( ) 2.8, 2.10., α > 0, : (1) (1 + θ)g α = (1 + θ)g α (=: k α ), (2) (1 θ)g α = (1 θ)g α (=: p α )., g g = g 0 α g α (1 ± θ), ( ) : (1) k = k 0 α>0 k α, (2) p = a α>0 p α., α > 0 g α k α g α., 2.8 ( g k a n) M = G/K ( (g, k, θ), g θ Cartan ). 1 : (K), (A),, (N),., (A)-type, (N)-type, S, S 1. S S 1,. S, 1, 1. (B θ ).

12 H a. s H := s span{h} = (a span{h}) n 1. S H M = G/K 1, α Λ ( ), 0 X g α. s X := s span{x} = a (n span{x}) 1. S X M = G/K 1, Berndt-T. (2003) S H (N)-type, S X (A)-type., (A)-type (N)-type 1,., G, G M, : f : M M : (or diffeo, ) : p M, f(g.p) = G.f(p).

1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, webpage,.,,.

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