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

2 2 1 (1),, ( ). (2),,. (3),.,, : Hashinaga, T., Tamaru, H.: Three-dimensional solvsolitons and the minimality of the corresponding submanifolds, preprint. ArXiv: Hashinaga, T., Tamaru, H., Terada, K.: Milnor-type theorems for left-invariant Riemannian metrics on Lie groups, J. Math. Soc. Japan 68 (2016), Kodama, H., Takahara, A., Tamaru, H.: The space of left-invariant metrics on a Lie group up to isometry and scaling, Manuscripta Math. 135 (2011), ,., G, M. 1.1 Φ : G M M, g.p := Φ(g, p). Φ G M, : (i) (ii) g, h G p M (gh).p = g.(h.p) p M e.p = p e G. G M G M 1.2 RH 2 := {z C Im(z) > 0}, SL(2, R) RH 2 : ( a b c d 1.3 : ).z := az + b cz + d. (1) SL(2, R) RH 2 (i). (2) GL(2, R) RH 2..

3 3 2,., M, M = G/K.,,. 2.1,. 2.1 M(n, R) n n, : (1) GL(n, R) := {g M(n, R) det(g) 0}. (2) SL(n, R) := {g GL(n, R) det(g) = 1} p, q Z 0., I n, : I p,q := ( Ip I q ). 2.2, : (1) O(p, q) := {g GL(p + q, R) t gi p,q g = I p,q } ( ) (2) SO(p, q) := SL(p + q, R) O(p, q) ( ) O(n) := O(n, 0), SO(n) := SO(n, 0).,, R p+q ( ) x, y p,q := t xi p,q y. 2.3 g M(p + q, R) (i) (ii) g O(p, q) g, p,q v, w R p+q gv, gw p,q = v, w p,q

4 4 2 (iii) g = (v 1,..., v p+q ) {v 1,..., v p+q } R p+q (p, q) G, M. 2.4 Φ : G M M, g.p := Φ(g, p). Φ G M, : (i) g, h G p M (gh).p = g.(h.p) (ii) p M e.p = p G M G M,.,. 2.5 Φ : G M M G M g G : Φ(g 1, ) Φ(g, ). 2.6 Φ : G M M G M (1) G G Φ G M M (2) M M G G.M M Φ G M : G M M G M., :=, n, (1) GL(n, R) R n R n : (g, v) gv (2) O(n) R n (3) O(n) S n 1 := {x R n x, x = 1} 2.8, O(p + 1, q) M(p, q) := {x R p+q+1 x, x p+1,q = 1} M(n, 0), M(0, n). M(p, 1), M(1, q). 2.9 G k (R n ) := {V R n V k } GL(n, R) G k (R n ) G k (R n ) : (g, V ) g.v := {gv v V },,.,.

5 , G M p, q M g G g.p = q 2.11 o M G M p M g G g.o = p 2.12 n 1 O(n + 1) SO(n + 1) S n, SO(n + 1) S n., GL(n, R) SL n (R) O(n) SO(n) G k (R n ) O(p + 1, q) M(p, q) SL(2, R) S RH 2 S := {( a b 0 a 1 ) } a > 0, b R.,.,.

6 , G M G G M M G M = G/K, G/K 2.17 G K G K g h : g 1 h K. g G [g] [g] = gk := {gk k K} G K G/K G K, M G, M = G/K.,. G/K G/K K G., G/K G., G G/K : g.[h] := [gh]., well-defined,, ( ) M G, p M., : f : G/G p M : [g] g.p., G p := {g G g.p = p}. G p.

7 M G/K., G, p M n 1 S n {[ ] } S n 1 = O(n + 1)/ α O(n) α {[ ] } 1 = SO(n + 1)/ α SO(n). α,., : G G, G M = G/G p, G p = G G p ( ) (k, n k) G k (R n ) {[ G k (R n ) = GL(n, R)/ 0 {[ = SL(n, R)/ 0 {[ ] α 0 = O(n)/ = SO(n)/ 0 β {[ α 0 0 β ] ] } det 0 } det = 1 } α O(k) β O(n k) ] } α O(k) β O(n k) det(α) det(β) = RH 2 = SL(2, R)/SO(2) = S/{e}. M(p, q) = O(p + 1, q)/o(p, q) M := {, : R n }., (1) GL(n, R) M : g., := g 1 ( ), g 1 ( ). (2) M = GL(n, R)/O(n) ; : (1) {J GL(2n, R) J 2 = I 2n }. (2) {Ω : R n R n R : (i.e.,,, )}.

8 ,. ( ), H M,., p M, p : H.p := {h.p h H}.,.,, SO(2) R 2,. RH 2 = SL(2, R)/SO(2), : A := {( ) } {( a 0 1 b 0 a 1 a > 0, N := 0 1 ) } b R., O(3) M(3, R) : g.x := gxg 1. : (1) O(3) Sym 0 (3, R) := {X M(3, R) t X = X, tr(x) = 0}. (2) X := diag(1, 1, 2), O(3).X = O(3)/(O(1) O(2)) ( = RP 2 ). (3) M(3, R) X, Y := tr( t XY ), O(3). (4) O(3) S 4 ( Sym 0 (3, R) = R 5 ). S 4,. (5), O(3), RP 2 S 4., ( )., RP 2 ( S 4 ) Veronese.

9 M := {, : R n }., GL(n, R) M : g., := g 1 ( ), g 1 ( )., GL(n, R) R n. : (1) GL(n, R) Map(R n, R) := {F : R n R : } : (g.f )(x) := F (g 1 x). (2) (R n ) := {F : R n R : }., (g.f )(x) := F (g.x) (n 2 )., GL(n, R) (R n ) ( (R n ) R n ). 2, GL(n, R) M : (1) GL(n, R) Map(R n R n, R) := {F : R n R n R : } : (g.f )(x, y) := F (g 1 x, g 1 y). (2) 2 (R n ) := {F : R n R n R : }. (3), S 2 (R n ) := {F : R n R n R :, }. (4), M := {F : R n R n R :,, }. (2), (3),.,.

10 10 3, (g,, ),. ( ), Lie algebra. 3.1 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. g. [, ].,. 3.2 gl(n, R) := M(n, R) : [X, Y ] := XY Y X. 3.3 g, g, f : g g, : (i) f, (ii) f,, X, Y g, f([x, Y ]) = [f(x), f(y )]. 3.4 g, g, f : g g,., f : g g, g g.

11 ,., gl(n, R). 3.5 g. g g, : (i) g g. (ii) X, Y g, [X, Y ] g.,. 3.6 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}, n = p + q.., ( ) gl(3, R) ( 3 ): h 3 := ,. {x 1,..., x n },, [x i, x j ] (i < j),. 3.8 {x 1,..., x n } g., i < j [x i, x j ] bracket relation., 0., bracket relation,. 3.9 g, h 3 : g = span{e 1, e 2, e 3 }, [e 1, e 2 ] = e 3.

12 (g,, ), (g,, ), X, Y, Z g. (1) : g g g Levi-Civita : 2 X Y, Z = [X, Y ], Z + [Z, X], Y + X, [Z, Y ]. (2) R(X, Y )Z := X Y Z Y X Z [X,Y ] Z. (3) Ric(X) := R(X, e i )e i. {e i } g. (4) σ g 2, K σ := R(E 1, E 2 )E 2, E 1 σ. {E 1, E 2 } σ. 3.11, Levi-Civita.,, Koszul, (g,, ) X, Y, Z g, : (1) Ric. (2) X Y, Y = 0. X,. (3) R(X, X) = 0. (4) R(X, Y )Z, Z = 0. (5) K σ g,, R 0.,., U : g g g : 2 U(X, Y ), Z = [Z, X], Y + X, [Z, Y ] ( X, Y, Z g). U, X Y = (1/2)[X, Y ] + U(X, Y ).

13 3.4 : :, (g,, ), (1), K σ σ. (2) Einstein, Ric = c id (c R) g RH n := span{a, X 1,..., X n 1 } : [A, X i ] = X i (i {1,..., n 1}). 3.17, RH n., g RH n,,. (g RH n,, ) : (1) U(A, A) = 0, U(A, X i ) = (1/2)X i, U(X i, X j ) = δ ij A. (2) A = 0, Xi A = X i, Xi X j = δ ij A (g RH n,, ), : (1) K σ 1,. (2) Ric = (n 1)id, Einstein , 3.19 ( ).

14 Ricci soliton : Ricci soliton, h Der(g) g : Der(g) := {D : g g : D([, ]) = [D( ), ] + [, D( )]} h 3 = span{e 1, e 2, e 3 }, Der(h 3 ) = a 11 a 12 0 a 21 a 22 0 a 11 + a 22 = a 33. a 31 a 32 a (g,, ) Ricci soliton, : c R, D Der(g) : Ric = c id + D (g,, ) (G, g)., (g,, ) Ricci soliton, (G, g) Ricci soliton. (.) 3.25 h 3 = span{e 1, e 2, e 3 }, {e 1, e 2, e 3 },., U(e 1, e 1 ) = U(e 1, e 2 ) = U(e 2, e 2 ) = U(e 3, e 3 ) = 0, U(e 1, e 3 ) = (1/2)e 2, U(e 2, e 3 ) = (1/2)e 1. e1 e 1 = 0, e1 e 2 = (1/2)e 3, e1 e 3 = (1/2)e 2, e2 e 1 = (1/2)e 3, e2 e 2 = 0, e2 e 3 = (1/2)e 1, e3 e 1 = (1/2)e 2, e3 e 2 = (1/2)e 1, e3 e 3 = (h 3,, ) Ricci soliton. Ric = 1/ / / (g,, ), Einstein Ricci soliton, ( )., g,,,.

15 15 4 Milnor, (g,, ), Einstein Ricci soliton.,, Einstein Ricci soliton. 4.1 Milnor 4.1 g unimodular, : X g, tr(ad X ) = Milnor, 1976 g 3 unimodular, :, : g, {x 1, x 2, x 3 } :,, λ 1, λ 2, λ 3 R : [x 1, x 2 ] = λ 3 x 3, [x 2, x 3 ] = λ 1 x 1, [x 3, x 1 ] = λ 2 x , 3 unimodular. (λ 1, λ 2, λ 3 ), (+ + +) so(3), (+ + ) sl(2, R), (+00) Heisenberg. 4.4, 3 unimodular., Einstein Ricci soliton. 4.5 dim = 3., [, ] : 2 g g, 3 2 g = g, L : g g. L. 4.6, g 3, M := GL(3, R)/O(3). dim M = 6. 3., [, ] ( Aut(g) ).

16 16 4 Milnor 4.2 g Aut(g),. 4.7 g : Aut(g) := {φ : g g : φ([, ]) = [φ( ), φ( )]} 4.8 : g := span{e 1, e 2, e 3 }, [e 1, e 2 ] = e 2., Aut(g) = a 21 a 22 0 GL(3, R). a 31 0 a 33, g RH 2 R h 3 = span{e 1, e 2, e 3 }, Aut(h 3 ) = a 11 a 12 0 a 21 a 22 0 GL(3, R) a 11 a 22 a 12 a 21 = a 33. a 31 a 32 a H M p M, p : H.p := {h.p h H} U M., U H-, : U H-., U H-, : p M, U H.p SO(2) R 2,. M := G/K. H G, H M H M = G/K, o := [e]., U G., U, : {h.o h U} H-.

17 RH 2 = SL(2, R)/SO(2), : A := {( ) } {( a 0 1 b 0 a 1 a > 0, N := 0 1 ) } b R. ; (1) A RH 2, N. (2) N RH 2, A.,, H M = G/K, U G., U : g G, U HgK. HgK := {hgk h H, k K}.., [[g]] M := GL(3, R)/O(3) : H := GL(3, R). U : U := λ 0. 0 λ M := GL(3, R)/O(3) : H := GL(3, R)., U := {I 3 } ( ).

18 18 4 Milnor 4.4 {e 1,..., e n } g, g = R n.,, : GL(n, R) M(g) := {, : g }, g., := g 1 ( ), g 1 ( ). {e 1,..., e n } g, 0., GL(n, R), 0, ( ) O(n). : M(g) = GL(n, R)/O(n)., M(g), : R Aut(g) := {cφ GL(n, R) c R \ {0}, φ Aut(g)} R Aut(g) M(g), U. :, : g, k > 0, φ Aut(g), g U : {φge 1,..., φge n } k,., Milnor., : g := span{e 1, e 2, e 3 }, [e 1, e 2 ] = e 2. :,, λ 0, k > 0, {x 1, x 2, x 3 } : k, : [x 1, x 2 ] = x 2 + λx 3. Milnor. U,. 4.20, Einstein ( Ricci soliton )., k = 1, {x 1, x 2, x 3 } : (1 + (λ 2 /2)) Ric = 0. λ 2 /2

19 (,, )., (, ).,.., Milnor ( 4.18), 3 unimodular, Milnor.,,,, ( ) g n, p + q = n. g (p, q) GL(p + q, R)/O(p, q)., R Aut(g), Milnor.,., Aut(g), g := h 3 R. R Aut(g) GL(p + q, R)/O(p, q), ( (p, q) = (1, 3), (2, 2)).,,., ( 2.27 ) g 2n. g ( ).,, ( ) Milnor g 2n. g 2.,, Milnor. Milnor, Einstein.,,.

20 20 [1] Hashinaga, T.: On the minimality of the corresponding submanifolds to fourdimensional solvsolitons, Hiroshima Math. J. 44 (2014), [2] Hashinaga, T., Tamaru, H.: Three-dimensional solvsolitons and the minimality of the corresponding submanifolds, preprint. ArXiv: [3] Hashinaga, T., Tamaru, H., Terada, K.: Milnor-type theorems for left-invariant Riemannian metrics on Lie groups, J. Math. Soc. Japan 68 (2016), [4] Kodama, H., Takahara, A., Tamaru, H.: The space of left-invariant metrics on a Lie group up to isometry and scaling, Manuscripta Math. 135 (2011), [5] Kubo, A., Onda, K., Taketomi, Y., Tamaru, H.: On the moduli spaces of leftinvariant pseudo-riemannian metrics on Lie groups, Hiroshima Math. J. 46 (2016), to appear. [6] Taketomi, Y., Tamaru, H.: On the nonexistence of left-invariant Ricci solitons a conjecture and examples, Transf. Groups, to appear. [7] Tamaru, H.: The space of left-invariant metrics on a generalization of Milnor frames. In: Proceedings of The Seventeenth International Workshop on Diff. Geom. 17 (2013), [8] Tamaru, H.: Group actions on symmetric spaces related to left-invariant geometric structures. In: Development of group actions and submanifold theory, RIMS Kokyuroku 1929 (2014), [9] Tamaru, H.: The space of left-invariant Riemannian metrics. In: Geometry and Topology of Manifolds, Springer Proc. Math. Stat. 154 (2016), [10], : ( ).

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 ),

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 ), 1 1 1.1,,. 1.1 1.2 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