Let M be a complete locally compact CAT(0)-space, and X an ultralimit of M . For γ ⊂M a k-dimensional flat, let γω be the k-dimensional flat in X obtained as an ultralimit of γ. In this paper, we identify various conditions on γω that are sufficient to ensure that γ bounds a (k + 1)-dimensional half-flat. As applications we obtain (1) constraints on the behavior of quasi-isometries between loca...

The paper treats properties of pseudo-Riemannian spaces admitting non-trivial geodesic mappings. A symmetric pair is a with coinciding values covariant derivatives for their Riemann tensors. It proved that the spaces, which are not constant curvatures, defined unequivocally by lines. research carried out locally, using tensors, no restrictions to sign metric tensor and signature space.

In several areas of mathematics, homogeneous spaces are fundamental objects as they often serve as models for more general objects: Various examples from differential geometry (Riemannian symmetric spaces, principal bundles) and topology (geometric 3-manifolds), to algebraic and complex geometry (uniformization theorems, flag manifolds) etc. underline the importance of spaces, furnished with a ...

Heckman-Opdam system of differential equations is holonomic , with regular singularities and has locally |W |-dimensional space of solutions ( cf. corollary 3.9 of [12]), where |W | is the cardinality of the Weyl group W . The system is a generalization of radial parts of Laplace-Casimir operators on symmetric Riemannian spaces of nonpositive curvature and is isomorphic to Calogero-Sutherland m...

A pair of points in a riemannian manifold makes a secure configuration if the totality of geodesics connecting them can be blocked by a finite set. The manifold is secure if every configuration is secure. We investigate the security of compact, locally symmetric spaces.

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3 manifolds with polarisation of degree 2d and split type is of general type if d ≥ 12. MSC 2000: 14J15, 14J35, 32J27, 11E25, 11F55

We survey contributions of Robert MacPherson to the theory of arithmetic groups. There are two main areas we discuss: (i) explicit reduction theory for Siegel modular threefolds, and (ii) constructions of compactifications of locally symmetric spaces. The former is joint work with Mark McConnell, the latter with Lizhen Ji.

We derive upper Gaussian bounds for the heat kernel on complete, non-compact locally symmetric spaces M = Γ\X with non-positive curvature. Our bounds contain the Poincaré series of the discrete group Γ and therefore we also provide upper bounds for this series.

For a compact Riemannian manifold, Weyl’s law describes the asymptotic behavior of the counting function of the eigenvalues of the associated Laplace operator. In this paper we discuss Weyl’s law in the context of automorphic forms. The underlying manifolds are locally symmetric spaces of finite volume. In the non-compact case Weyl’s law is closely related to the problem of existence of cusp fo...

We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.