I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2005, VOLUME 11, NUMBER 1, PAGES 141-158
Kähler geometry of hyperbolic type on the manifold of
V. V. Konnov
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A nondegenerate -pair in an -dimensional projective
consists of an -plane and an -plane in ,
which do not intersect.
of all nondegenerate -pairs is
is the homogeneous space
equipped with an internal Kähler structure of hyperbolic type.
Therefore, the manifold is a hyperbolic analogue
of the complex Grassmanian .
In particular, the manifold of -pairs is
a hyperbolic analogue of the complex projective space .
Similarly to , the
is a Kähler manifold of constant
nonzero holomorphic sectional curvature (relative to a hyperbolic
In this sense,
is a hyperbolic spatial form.
It was proved that the manifold of -pairs
is globally symplectomorphic to the total space
of the cotangent bundle over the projective space .
A generalization of this result is as follows: the manifold of
nondegenerate -pairs is
globally symplectomorphic to the total space
of the cotangent bundle over the Grassman manifold of
subspaces of the space .
In this paper, we study the canonical Kähler structure
We describe two types of submanifolds in , which
are natural hyperbolic spatial forms holomorphically isometric to
manifolds of -pairs in and
We prove that for any point of the manifold , there
exist a -parameter family
hyperbolic spatial forms of first type and a -parameter family of
hyperbolic spatial forms of second type passing through this point.
We also prove that natural hyperbolic spatial forms of first type
are in bijective correspondence with points of
and natural hyperbolic spatial forms
of second type on
are in bijective correspondence with
points of the manifolds .
Last modified: April 27, 2005