## Abstract

A nondegenerate *m-pair* (*A*, Ξ) in an *n*-dimensional projective space ℝ*P*
_{
n
} consists of an *m*-plane *A* and an (*n − m* − 1)-plane Ξ in ℝ*P*
_{
n
}, which do not intersect. The set \(\mathfrak{N}_m^n \) of all nondegenerate *m*-pairs ℝ*P*
_{
n
} is a 2(*n − m*)(*n − m* − 1)-dimensional, real-complex manifold. The manifold \(\mathfrak{N}_m^n \) is the homogeneous space \(\mathfrak{N}_m^n = {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(m + 1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(m + 1,\mathbb{R})}} \times GL(n - m,\mathbb{R})\) equipped with an internal Kähler structure of hyperbolic type. Therefore, the manifold \(\mathfrak{N}_m^n \) is a hyperbolic analogue of the complex Grassmanian ℂ*G*
_{
m,n
} = U(*n*+1)/U(*m*+1) × U(*n−m*). In particular, the manifold of 0-pairs \(\mathfrak{N}_m^n {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(1,\mathbb{R})}} \times GL(n,\mathbb{R})\) is a hyperbolic analogue of the complex projective space ℂ*P*
_{
n
} = U(*n*+1)/U(1) × U(*n*). Similarly to ℂ*P*
_{
n
}, the manifold \(\mathfrak{N}_m^n \) 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 in [6] that the manifold of 0-pairs \(\mathfrak{N}_0^n \) is globally symplectomorphic to the total space *T**ℝ*P*
_{
n
} of the cotangent bundle over the projective space ℝ*P*
_{
n
}. A generalization of this result (see [7]) is as follows: the manifold of nondegenerate *m*-pairs \(\mathfrak{N}_m^n \) is globally symplectomorphic to the total space *T**ℝ*G*
_{
m,n
} of the cotangent bundle over the Grassman manifold ℝ*G*
_{
m,n
} of *m*-dimensional subspaces of the space ℝ*P*
_{
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 ℝ*P*
_{
m
}+1 and in ℝ*P*
_{
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 \(\mathfrak{N}_{m + 1}^n \).

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Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.

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### Cite this article

Konnov, V.V. Kähler geometry of hyperbolic type on the manifold of nondegenerate *m*-pairs.
*J Math Sci* **141, **1004–1015 (2007). https://doi.org/10.1007/s10958-007-0027-3

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### Keywords

- Manifold
- Projective Space
- Plane Generator
- Cotangent Bundle
- Ahler Manifold