A note on the characteristic classes of negatively curved manifolds

We give conceptual proofs of some well known results concerning compact non-positively curved locally symmetric spaces. We discuss vanishing and non-vanishing of Pontrjagin numbers and Euler characteristics for these locally symmetric spaces. We also establish vanishing results for Stiefel-Whitney numbers of (finite covers of) the Gromov-Thurston examples of negatively curved manifolds. We mention some geometric corollaries: the MinVol question, a lower bound for degrees of covers having tangential maps to the non-negatively curved duals, and estimates for the complexity of some representations of certain uniform lattices.