FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 1, PAGES 141-158

Kähler geometry of hyperbolic type on the manifold of nondegenerate $m$-pairs

V. V. Konnov

Abstract

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A nondegenerate $m$-pair $(A,$X) in an $n$-dimensional projective space $$RP_n consists of an $m$-plane $A$ and an $(n$- m - 1)-plane $$X in $$RP_n, which do not intersect. The set $\$\backslash mathfrak\; N$_mⁿ$ of all nondegenerate $m$-pairs $$RP_n is a $2(n$- m)(n - m - 1)-dimensional, real-complex manifold. The manifold $\$\backslash mathfrak\; N$_mⁿ$ is the homogeneous space $\$\backslash mathfrak\; N$_mⁿ = GL(n + 1, R)/GL(m + 1, R) × GL (n - m, R)$ equipped with an internal Kähler structure of hyperbolic type. Therefore, the manifold $\$\backslash mathfrak\; N$_mⁿ$ is a hyperbolic analogue of the complex Grassmanian $$CG_m,n = U(n + 1)/U(m + 1) × U(n - m). In particular, the manifold of $0$-pairs $\$\backslash mathfrak\; N$₀ⁿ = GL(n + 1, R)/GL(1, R) × GL(n, R)$ is a hyperbolic analogue of the complex projective space $$CP_n = U(n + 1)/U(1) × U(n). Similarly to $$CP_n, the manifold $\$\backslash mathfrak\; N$₀ⁿ$ is a Kähler manifold of constant nonzero holomorphic sectional curvature (relative to a hyperbolic metrics). In this sense, $\$\backslash mathfrak\; N$₀ⁿ$ is a hyperbolic spatial form. It was proved that the manifold of $0$-pairs $\$\backslash mathfrak\; N$₀ⁿ$ is globally symplectomorphic to the total space $T$*RP_n of the cotangent bundle over the projective space $$RP_n. A generalization of this result is as follows: the manifold of nondegenerate $m$-pairs $\$\backslash mathfrak\; N$_mⁿ$ 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 Kähler structure on $\$\backslash mathfrak\; N$_mⁿ$. We describe two types of submanifolds in $\$\backslash mathfrak\; N$_mⁿ$, 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 $\$\backslash mathfrak\; N$_mⁿ$, there exist a $2(n$- m)-parameter family of $2(m\; +\; 1)$-dimensional hyperbolic spatial forms of first type and a $2(m\; +\; 1)$-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 $\$\backslash mathfrak\; N$_mⁿ$ are in bijective correspondence with points of the manifold $\$\backslash mathfrak\; N$_m+1ⁿ$ and natural hyperbolic spatial forms of second type on $\$\backslash mathfrak\; N$_mⁿ$ are in bijective correspondence with points of the manifolds $\$\backslash mathfrak\; N$_m-1ⁿ$.

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Last modified: April 27, 2005