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

View as HTML
View as gif image

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

Location: http://mech.math.msu.su/~fpm/eng/k05/k051/k05105h.htm

Last modified: April 27, 2005