KAEHLER GEOMETRY OF HYPERBOLIC TYPE ON THE MANIFOLD OF NONDEGENERATE m-PAIRS

摘要

A nondegenerate m-pair (A, E) in an n-dimensional projective space RP_n consists of an m-plane A and an (n-m-l)-plane E in RP_n, which do not intersect. The set n_m~n of all nondegenerate m-pairs RP_n is a 2(n - m)(n - m -1)-dimensional, real-complex manifold. The manifold n_m~n is the homogeneous space n_m~n = GL(n +1l,R)/GL(m + 1,R) × GL(n - m,R) equipped with an internal Kahler structure of hyperbolic type. Therefore, the manifold n_m~n is a hyperbolic analogue of the complex Grassmanian CG_(m,n) = U(n+l)/U(m+1)×U(n-m). In particular, the manifold of 0-pairs n_0~n= GL(n+1,R)/GL(1,R)× GL(n,R) is a hyperbolic analogue of the complex projective space CP_n = U(n+1)/U(1) ×U(n). Similarly to CP_n, the manifold n_0~n is a Kaehler manifold of constant nonzero holomorphic sectional curvature (relative to a hyperbolic metrics). In this sense, n_0~n is a hyperbolic spatial form. It was proved in [6] that the manifold of 0-pairs n_0~n is globally symplectomorphic to the total space T~*RP_n of the cotangent bundle over the projective space RP_n. A generalization of this result (see [7]) is as follows: the manifold of nondegenerate m-pairs n_m~n is globally symplectomorphic to the total space T~*RG_(m,n) of the cotangent bundle over the Grassman manifold RG_(m,n) of m-dimensional subspaces of the space RP_n. In this paper, we study the canonical Kahler structure on n_m~n. We describe two types of submanifolds in n_m~n, which are natural hyperbolic spatial forms holomorphically isometric to manifolds of 0-pairs in RP_(m+1) and in RP_(n-m), respectively. We prove that for any point of the manifold n_m~n, there exist a 2(n - m)-parameter family of 2(m + 1)-dimensional hyperbolic spatial forms of first type and a 2(m + l)-parameter family of 2(n - m)-dimensional hyperbolic spatial forms of second type passing through this point. We also prove that natural hyperbolic spatial forms of first type on n_m~n are in bijective correspondence with points of the manifold n_(m+1)~n and natural hyperbolic spatial forms of second type on n_m~n are in bijective correspondence with points of the manifolds n_(m-1)~n.