Kähler geometry of hyperbolic type on the manifold of nondegenerate $m$-pairs
V. V. Konnov Finance Academy under the Government of the Russian Federation
Abstract:
A nondegenerate
$m$-pair $(A,\Xi)$ in an
$n$-dimensional projective space
$\mathbb RP_n$ consists of an
$m$-plane
$A$ and an
$(n-m-1)$-plane
$\Xi$ in
$\mathbb RP_n$, which do not intersect. The set
$\mathfrak N_m^n$ of all nondegenerate
$m$-pairs
$\mathbb RP_n$ is a
$2(n-m)(n-m-1)$-dimensional, real-complex manifold. The manifold
$\mathfrak N_m^n$ is the homogeneous space $\mathfrak N_m^n=\matrm{GL}(n+1,\mathbb R)/\matrm{GL}(m+1,\mathbb R)\times\matrm{GL}(n-m,\mathbb R)$ equipped with an internal Kähler structure of hyperbolic type. Therefore, the manifold
$\mathfrak N_m^n$ is a hyperbolic analogue of the complex Grassmanian $\mathbb CG_{m,n}=\mathrm U(n+1)/\mathrm U(m+1)\times\mathrm U(n-m)$. In particular, the manifold of 0-pairs $\mathfrak N_0^n=\matrm{GL}(n+1,\mathbb R)/\matrm{GL}(1,\mathbb R)\times\matrm{GL}(n,\mathbb R)$ is a hyperbolic analogue of the complex projective space $\mathbb CP_n=\mathrm U(n+1)/\mathrm U(1)\times\mathrm U(n)$. Similarly to
$\mathbb CP_n$, the manifold
$\mathfrak N_0^n$ is a Kähler manifold of constant nonzero holomorphic sectional curvature (relative to a hyperbolic metrics). In this sense,
$\mathfrak N_0^n$ is a hyperbolic spatial form. It was proved that the manifold of 0-pairs
$\mathfrak N_0^n$ is globally symplectomorphic to the total space
$T^*\mathbb RP_n$ of the cotangent bundle over the projective space
$\mathbb RP_n$. A generalization of this result is as follows: the manifold of nondegenerate
$m$-pairs
$\mathfrak N_m^n$ is globally symplectomorphic to the total space
$T^*\mathbb RG_{m,n}$ of the cotangent bundle over the Grassman manifold
$\mathbb RG_{m,n}$ of
$m$-dimensional subspaces of the space
$\mathbb RP_n$. In this paper, we study the canonical Kähler structure on
$\mathfrak N_m^n$. We describe two types of submanifolds in
$\mathfrak N_m^n$, which are natural hyperbolic spatial forms holomorphically isometric to manifolds of 0-pairs in
$\mathbb RP_{m+1}$ and in
$\mathbb RP_{n-m}$, respectively. We prove that for any point of the manifold
$\mathfrak N_m^n$, there exist a
$2(n-m)$-parameter family of
$2(m+1)$-dimensional hyperbolic spatial forms of first type and a
$2(m+1)$-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
$\mathfrak N_m^n$ are in bijective correspondence with points of the manifold
$\mathfrak N_{m+1}^n$ and natural hyperbolic spatial forms of second type on
$\mathfrak N_m^n$ are in bijective correspondence with points of the manifolds
$\mathfrak N_{m-1}^n$.
UDC:
514.76
Fulltext:
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