We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times.

(2) the product H × H of 2 hyperbolic planes. In short, among all the symmetric spaces of noncompact type with rank ≥ 2, there are only two manifolds that are minimal with respect to the partial order defined by totally geodesic embeddings. Our main theorem provides an analogue of this result for noncompact finitevolume spaces that are locally symmetric, rather than globally symmetric, but, in ...

It is still an open problem whether Riemannian manifolds all of whose local geodesic symmetries are volume–preserving (i.e., D’Atri spaces) or more generally, ball–homogeneous spaces, and C-spaces are locally homogeneous or not. We provide some partial positive answers by proving that five–dimensional locally φ–symmetric spaces can be characterized as Sasakian spaces which are ball–homogeneous ...

In this paper, we briefly introduce different types of buildings such as spherical buildings, Euclidean buildings, twin buildings, R-buildings and describe some of their applications to many subjects: (1) differential geometry such as Mostow strong rigidity, rank rigidity of nonpositively curved manifolds, Margulis superrigidity, quasi-isometry rigidity, classification of isoparametric manifold...

Consequences of the existence of conformal vector fields in (locally) symmetric and conformal symmetric spaces, have been obtained. An attempt has been made for a physical interpretation of the consequences in the framework of general relativity.

We consider the Kudla-Millson lift from elliptic modular forms of weight (p + q)/2 to closed q-forms on locally symmetric spaces corresponding to the orthogonal group O(p, q). We study the L-norm of the lift following the Rallis inner product formula. We compute the contribution at the Archimedian place. For locally symmetric spaces associated to even unimodular lattices, we obtain an explicit ...