( ), Jürgen Berndt,.,. 1, CH n.,,. 1.1 ([6]). CH n (n 2), : (i) CH k (k = 0,..., n 1) tube. (ii) RH n tube. (iii). (iv) ruled minimal, equidistant. (v) normally homogeneous submanifold F k tube. (vi) normally homogeneous submanifold F k,ϕ tube., (i), (ii). (iii) (iv), Berndt ([1]). (v) (vi), Berndt Brück ([2])., Berndt ([6]), CH n,.,,.,., [6], [13]. 1.1, CH n. 1.2. (M, g) N M, N (, H Isom(M, g) s.t. N = H.p).,,. CP n, ([10])., 1.1, CP n CH n., CP n 2007 (, 2007/11/22 24) tamaru@math.sci.hiroshima-u.ac.jp 1
, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n,,. 1.3.., cohomogeneity ( ).,, ( )., cohomogeneity one.,, cohomogeneity one., cohomogeneity one,., cohomogeneity one.,, Isom(CH n, g) = SU(1, n),., su(1, n),,., CP n CH n., Isom(CH n ),., Isom(CP n ),., CH n. 2 : CH n, n, c < 0., CH n,. CH n., CH n = U(1, n)/u(1) U(n)., - ([9]) Goldman ([8]). 2
2.1 Ball model CH n, C n. 2.1. B n := {z C n z, z < 1} C n ball. z, w := z k w k., B n, CH n., [8], [9]. c,,. c = 1 ( )., [ 1, 1/4]. 2.2 Projective model CH n ball model,,. CH n, CP n., CP n C n+1 \ {0} (z C n+1 \ {0}, [z] := Cz CP n )., : F : C n+1 C n+1 C : (z, w) z 0 w 0 + n k=1 z kw k. 2.2. C n ball B n, M := {[z] CP n F (z, z) < 0}.., CP n (U, ϕ) : U = {[z 0 : : z n ] P (C n+1 ) z 0 0}, ϕ : U C n : [z 0 : : z n ] (z 1 /z 0,..., z n /z 0 ). M U. ϕ : M B n., CP n M CH n., : U(1, n) := {g GL(C n+1 ) F (gz, gw) = F (z, w) ( z, w C n+1 )}. 2.3. U(1, n), C n+1, CH n.. U(1, n) C n+1,, U(1, n) CP n (i.e., g.[z] := [g.z])., U(1, n) F, M = CH n.. 2.3 CH n, projective model, U(1, n). 2.4. CH n = U(1, n)/u(1) U(n).. U(1, n) CH n,. o := [1 : 0 : : 0] CH n. o U(1, n) o = U(1) U(n) 3
,., U(1, n) CH n., ( ) : o, CH n U(1, n).o = U(1, n)/u(1) U(n),. U(1, n).,., SU(1, n) := {g U(1, n) det(g) = 1},., CH n = SU(1, n)/s(u(1) U(n))., S(U(1) U(n)) := SU(1, n) (U(1) U(n)). 2.4 CH n = U(1, n)/u(1) U(n). 2.5. U(1, n) : z t ξ u(1, n) = ξ B z u(1), ξ Cn, B u(n).. F, I 1,n := diag( 1, 1,..., 1), : F (z, w) = t zi 1,n w., u(1, n) = {X M n+1 (R) t XI 1,n + I 1,n X = 0}.. o K := U(1) U(n). g K CH n, o T o CH n ( isotropy ).,. 2.6. u(1, n) = k + p : z k = B = u(1) u(n), p = ξ t ξ = C n., [k, k] k, [k, p] p, [p, p] k. k K., K p., isotropy., 2.7. (dπ) e p : p T o CH n, K-., π : U(1, n) U(1, n)/u(1) U(n) = CH n, (dπ) e : u(1, n) T o CH n. T o CH n = p, p K-., p = C n, p ( Killing form ). 4
3 : cohomogeneity one, 1 ( )..,. 3.1. H M cohomogeneity one, H p ν p (H.p)., H p H p M, H.p p H-, ν p (H.p) p H.p., ( ), cohomogeneity one. 4 1.1 (i), (ii)., CH n = G/K 3.1., G = U(1, n), K = U(1) U(n). g, k. 4.1., CH n. (i) CH k (k = 0,..., n 1) tube. (ii) RH n tube.. cohomogeneity one. (i), H := U(1, k) U(n k) CH n. H, h : z t ξ h := ξ B z u(1), ξ C k, B u(k), B u(n k) B. o H o, o H- H.o, H o = H K = U(1) U(k) U(n k), H.o = U(1, k)/u(1) U(k) = CH k CH n.. H cohomogeneity one, CH k tube (H ). o. ( 2.6) T o CH n = p, t ξ t ξ T o CH k = ξ ξ C k, ν och k = ξ C n k., H o ν o (CH k ), U(n k) C n k., 3.1., (ii), SO 0 (1, n). RH n, SO(n) R n. ξ 5
H = U(1, k) U(n k), CH k (G ) normalizer., CH k. U(1, k), CH k cohomogeneity one., CH n, tube (, ), (i), (ii)., CH n, normalizer ([2]).,, tube ([5]). 5 1.1 (iii), (iv)., cohomogeneity one., CH n. 5.1 CH n, G = U(1, n) (, )., 2.6 g = u(1, n) = k + p., p {P j, Q j j = 1,..., n} : P j := E j+1,1 + E 1,j+1, Q j := 1E j+1,1 1E 1,j+1. P 1, a := span R {P 1 }. a, p. 5.1. ad P1 0, ±1, ±2., (1) 0 g 0, z g 0 = z z u(1), B u(n 1) a. B (2) g = g 2 + g 1 + g 0 + g 1 + g 2 gradation., [g j, g k ] g j+k ( j, k).. (1),.,., [P 1, [P 1, P j ]] = P j, [P 1, [P 1, Q 1 ]] = 4Q 1, [P 1, [P 1, Q j ]] = Q j. k g k,, g ±1 = span R {P j ± [P 1, P j ], Q j ± [P 1, Q j ] j = 2,..., n}, g ±2 = span R {2Q 1 ± [P 1, Q 1 ]}., g = g 2 + g 1 + g 0 + g 1 + g 2, ad P1 0, ±1, ±2. (2) gradation, Jacobi. gradation, u(1, n) a (, CH n 1 ). 6
5.2. s := a + g 1 + g 2, : (1) n := g 1 + g 2 Heisenberg., {X j, Y j, Z} : [X j, Y j ] = Z, bracket 0. (2) s := a + n Damek-Ricci., A a : [A, X j ] = (1/2)X j, [A, Y j ] = (1/2)Y j, [A, Z] = Z.. (1), n : X j := 2P j+1 + 2[P 1, P j+1 ], Y j := 2Q j+1 + 2[P 1, Q j+1 ], Z := 2Q 1 + [P 1, Q 1 ]., A := 2P 1, (2). g = k + a + n ( G = KAN). 5.3. s := a + g 1 + g 2, CH n.. s := a + g 1 + g 2 S (S U(1, n), CH n )., o S o = S K = {1} ( s k = {0}, 5.1 g 1, g 2 )., CH n S.o = S/S o = S. dim S = 2n = dim CH n, S = S.o = CH n,., CH n = S ( ). CH n, s = a + n. 5.4. CH n, s J,,., JA = Z, JX j = Y j.,,, {A, X i, Y i, Z}. c = 1, s ( Damek-Ricci ([7])). 5.2 CH n, cohomogeneity one. s = span R {A, X j, Y j, Z j = 1,..., n 1}. 5.5., CH n. (iii). (iv) ruled minimal, equidistant.. CH n S., s 1 : n = span R {X j, Y j, Z}, s := s RX 1., S, cohomogeneity one ( 1 )., n. ruled minimal, s o. 7
, CH n cohomogeneity one, ( )., cohomogeneity one ([4])., cohomogeneity one, 1 (CH n ) 2,. 5.3, ( cohomogeneity one, )., (s,, ). s s ξ := s Rξ 1 ( ξ )., (s,, ) Levi-Civita, A ξ : s ξ s ξ., Koszul s ξ, : 2 A ξ X, Y = 2 X Y, ξ = [ξ, X], Y + X, [ξ, Y ]. (5.1) 5.6. 2, ruled minimal 3.. A ξ., ξ = A. ad ξ, (5.1), A ξ : n n : X [ξ, X]. A ξ g 1 g 2, 2. ruled minimal, ξ = X 1. A ξ : s s, (5.1) A ξ (A) = (1/2)X 1, A ξ (Y 1 ) = (1/2)Z, A ξ (Z) = (1/2)Y 1, A ξ (X j ) = 0, A ξ (Y j ) = 0. A ξ 0, ±1/2 3. S.p,, ( [1], [4] ). ( ),. 6 1.1 (v), (vi)., cohomogeneity one., CH n. 6.1 CH n : CH n = U(1, n)/u(1) U(n) = SU(1, n)/s(u(1) U(n)) = S. 8
S, U(1, n), CH n., S H U(1, n) H, H CH n.,. 6.1. 5.1 gradation u(1, n) = g 2 + g 1 + g 0 + g 1 + g 2. q := su(1, n) (g 0 + g 1 + g 2 ), su(1, n)., q SU(1, n),., u(1, n) su(1, n), ( u(1, n) ).,, SU(1, n) ( CH n 1 ). 6.2. SU(1, n) Q, CH n., (1) CH n = Q/U(n 1), z u(n 1) = z B B u(n 1), 2z = tr(b). (2) q = u(n 1) + a + n,, [u(n 1), a + n] a + n. (3) (dπ) e a+n : a + n T o CH n, U(n 1)-., π : Q Q/U(n) = CH n, (dπ) e : q T o CH n.. (1) 5.1. (2). (3). cohomogeneity one,.,,. 6.2 CH n = Q/U(n), CH n cohomogeneity one. q = su(1, n) (g 0 +g 1 +g 2 ). k 0 := g 0 s(u(1) + u(n)) = u(n 1), K 0. 6.3. V g 1 ( dim V 2) N K 0(V ) V. N K 0(V ) K 0 V normalizer., (1) s V := s V N k 0(V ) + s V, s = a + n. (2) N k 0(V ) + s V cohomogeneity one, o.. (1), V g 1, q,. (2), N k 0(V ) + s V o. 6.2, s = T o CH n U(n 1)-. o F, T o F = s V, ν o F = V. N k 0(V ) V., 3.1, cohomogeneity one. F, T o F = s V dim V 2. 9
, dim V = 1, cohomogeneity one., ( 5 ). 6.4., CH n. (v) normally homogeneous submanifold F k tube. F k, k = 2,..., n 1,. (vi) normally homogeneous submanifold F k,ϕ tube. F k,ϕ, 2k (k = 1,..., n 1), Kähler ϕ (0, π/2).., U(n 1), g 1 = span R {X i, Y i } = C n 1., 6.3. (v), V := span R {X 1,..., X k }., N U(n 1) (V ) SO(k) V = R k, 6.3, N k 0(V ) + s V cohomogeneity one. o F k, k, ν o F k = V (, g 1 k, V U(n 1)- ). (vi), V := span R {X 2i 1, cos(ϕ)y 2i 1 + sin(ϕ)y 2i i = 1,..., k}., N U(n 1) (V ) = SO(2k) V = R 2k, 6.3, N k 0(V ) + s V cohomogeneity one. o F k,ϕ, 2k, ν o F k,ϕ = V Kähler ϕ (, g 1 2k, Kähler ϕ, V U(n 1)- ). k = n, F k RH n, ϕ = 0, F k,0 CH k., F k, F k,ϕ. 6.3 (v), (vi),, Berndt Díaz-Ramos ([3]). (v), 4, (r = log(2 + 3)) 3. (vi), 5, k = 1 4.,. 6.5 ([3])., 2, 3, 4, 5. 6.4 CH n cohomogeneity one, g 1 = C n 1, U(n 1)-. k,, U(n 1) G k (R 2n 2 ) ( G k (R 2n 2 ) k Grassmann )., 10
Hermann,. 1 cohomogeneity one, CH n, : RH n : SO(n 1) G k (R n 1 ). ( ) HH n : Sp(n 1) G k (R 4n 4 ). ( ) OH 2 : Spin(7) G k (R 8 ). ( ) HH n, Hermann,. ( ), HH n.,,,, CH n. [1] Berndt, J.: Homogeneous hypersurfaces in hyperbolic spaces. Math. Z. 229 (1998), 589 600. [2] Berndt, J., Brück, M.: Cohomogeneity one actions on hyperbolic spaces. J. Reine Angew. Math. 541 (2001), 209 235. [3] Berndt, J., Díaz-Ramos, J.C.: Homogeneous hypersurfaces in complex hyperbolic spaces. Preprint. arxiv:math/0612157. [4] Berndt, J., Tamaru, H.: Homogeneous codimension one foliations on noncompact symmetric spaces. J. Differential Geom. 63 (2003), 1 40. [5] Berndt, J., Tamaru, H.: Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit. Tôhoku Math. J. 56 (2004), 163 177. [6] Berndt, J., Tamaru, H.: Cohomogeneity one actions on noncompact symmetric spaces of rank one. Trans. Amer. Math. Soc. 359 (2007), no. 7, 3425 3438. [7] Berndt, J., Tricerri, F., Vanhecke, L.: Generalized Heisenberg groups and Damek-Ricci harmonic spaces. Lecture Notes in Mathematics 1598, Springer-Verlag, Berlin, 1995. [8] Goldman, W.M.: Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1999. [9] Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. [10] Takagi, R.: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10 (1973), 495 506. [11] Tamaru, H.: Cohomogeneity one actions on symmetric spaces with a totally geodesic singular orbit (in Japanese). 1292 (2002), 106 114. [12] Tamaru, H.: Cohomogeneity one actions on symmetric spaces. 45 (2003), 105 120. [13] Tamaru, H.: Cohomogeneity one actions on noncompact symmetric spaces of rank one (in Japanese). 2003, 43 48. 11