On submanifolds in locally symmetric spaces of noncompact type
  Given a connected, compact, totally geodesic submanifold Y^m of noncompact type inside a compact locally symmetric space of noncompact type X^n, we provide a sufficient condition that ensures that [Y^m] is nonzero in H_m(X^n; 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 (X-bar,Y-bar) to the nonnegatively curved duals (X_u,Y_u). 
Jean-Francois Lafont, Benjamin Schmidt