Closures of totally geodesic immersions into locally symmetric spaces of noncompact type

On submanifolds in locally symmetric spaces of noncompact type

Given a connected, compact, totally geodesic submanifold Ym of noncompact type inside a compact locally symmetric space of noncompact type Xn , we provide a sufficient condition that ensures that [Ym] 6= 0 ∈ Hm(X; R); in low dimensions, our condition is also necessary. We provide conditions under which there exist a tangential map of pairs from a finite cover (X̄, Ȳ) to the nonnegatively curved ...

Totally geodesic submanifolds in Riemannian symmetric spaces

In the first part of this expository article, the most important constructions and classification results concerning totally geodesic submanifolds in Riemannian symmetric spaces are summarized. In the second part, I describe the results of my classification of the totally geodesic submanifolds in the Riemannian symmetric spaces of rank 2. To appear in the Proceedings volume for the conference V...

Minimal Immersions of Symmetric Spaces into Spheres

Closures of Totally Geodesic Immersions in Manifolds of Constant Negative Curvature

Let 800(1, n) denote the connected component of the group of linear transformations of IRn+l preserving the bilinear form Abstract Using techniques of Lie groups and ergodic theory, it can be shown that in a compact manifold of constant negative curvature, the closure of a totally geodesic, complete (immersed) submanifold of dimension atleast 2 is a totally geodesicimmersed submanifold. The mai...

On the Isoperimetric Constant of Symmetric Spaces of Noncompact Type

From this result one can easily deduce I (H) = n 1. For a detailed discussion of Cheeger’s constant and related results, one can consult [Cha, Chapter 6]. In general, it is very di¢ cult to know if the isoperimetric constant is positive or not and it is almost impossible to compute it explicitly if it is known to be positive. In this short note, we prove that the isoperimetric constant is posit...