Coordinate-Free Classic Geometries

Abstract

This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian structure. Their basic geometrical objects have linear nature and provide natural compactifications of classic spaces. The usual riemannian concepts are easily derivable from the strong structure and thus gain their coordinate-free form. Many examples illustrate fruitful features of the approach. The framework introduced here has already been shown to be adequate for solving problems concerning particular classic spaces.2000 Mathematics Subject Classification. 53A20 (53A35, 51M10).

Bounds for Laplace eigenvalues of Kähler metrics

“…The analysis of geodesics in CP m shows that it is a hyperplane formed by all 1-dimensional subspaces [W ] orthogonal to [Z]. We refer to [1,5], where these and related facts are discussed in detail.…”

Grassmannians and Conformal Structure on Absolutes

“…For example, V /p will denote V /pP . So, our notation and convention are similar to those used in [AGr1].…”

“…Then the equality (3.1) t 1 , t 2 := ± tr(t * 1 t 2 ) defines the (hermitian) metric on T p Gr K (k, V ). Many examples with k = 1 were dealt with in [AGr1] and some of them appear in Examples 3.3-5 below. One can introduce a more subtle structure by considering the coefficients char(t * 1 t 2 ) ∈ K k−d of the characteristic polynomial of t * 1 t 2 .…”

“…The material in this subsection is well-known and closely related to that in [Cal]. Our intention is simply to give it a complete description in the spirit of [AGr1] and of the present paper.…”

“…Many projective (k = 1) geometries are well known: real/complex hyperbolic, spherical, Fubini-Study, de Sitter, anti de Sitter, etc. The basic geometrical objects in these geometries, such as geodesics, totally geodesic subspaces, equidistant loci, etc., have linear nature (see [AGr1] and the references therein). In other words, grassmannians constitute a natural habitat for important geometrical objects in classical spaces.…”