Let \(M\) be a compact manifold and \(G\) a connected simple Lie group with finite center and \(\mathbb{R}\)-rank\((G)\geq 2\). The authors prove that if there is a real analytic action of \(G\) on \(M\) preserving a real analytic connection and a finite measure, then the fundamental group \(\pi_ 1(M)\) is not isomorphic to the fundamental group of any complete Riemannian manifold \(N\) with negative curvature bounded away from 0 and \(-\infty\). Under the additional assumption that \(N\) is locally symmetric, the authors’ result follows from the results established in a previous paper of the second author [J. Am. Math. Soc. 2, 201-215 (1989; Zbl 0676.57017)]. In the special case that \(M=G/\Gamma\), for \(\Gamma\subset G\) a cocompact lattice, and the symmetric space associated to \(G\) is Hermitian, the result has been obtained by J. Jost and S.-T. Yau [Pitman Monogr. Surv. Pure Appl. Math. 52, 241-259 (1991; Zbl 0729.58024)]. However, the authors’ techniques, based on algebraic ergodic theory, are completely different from those of Jost and Yau. Under some varied hypotheses, other generalizations of the results of Jost and Yau concerning the image of a homomorphism of \(\pi_ 1(M)\) into the isometry group of \(N\) are given.