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

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 $\mathfrak N_{m-1}^n$.