On submanifolds in locally symmetric spaces of noncompact type

Abstract

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 $\left[Y^m\right]\ne 0\in H_m\left(X^n;ℝ\right)$; in low dimensions, our condition is also necessary. We provide conditions under which there exist a tangential map of pairs from a finite cover $\left(\bar{X},\bar{Y}\right)$ to the nonnegatively curved duals $\left(X_u,Y_u\right)$.