Given a connected, compact, totally geodesic submanifold of noncompact type inside a compact locally symmetric space of noncompact type , we provide a sufficient condition that ensures that ; in low dimensions, our condition is also necessary. We provide conditions under which there exist a tangential map of pairs from a finite cover to the nonnegatively curved duals .
"On submanifolds in locally symmetric spaces of noncompact type." Algebr. Geom. Topol. 6 (5) 2455 - 2472, 2006. https://doi.org/10.2140/agt.2006.6.2455