An open manifold M with nonnegative sectional curvature contains a compact totally geodesic submanifold S called the soul. In his solution of the Cheeger-Gromoll conjecture, G. Perelman showed that the metric projection π : M → S was a C Riemannian submersion which coincided with a map previously constructed by V. Sharafutdinov. In this paper we improve Perelman’s result to show that π is in fa...

In this paper, we present an elementary proof of the following result. Theorem A. Let Mn denote a closed Riemannian manifold with nonpositive sectional curvature and let M̃n be the universal cover of Mn with the lifted metric. Suppose that the universal cover M̃n contains no totally geodesic embedded Euclidean plane R2 (i.e., Mn is a visibility manifold ). Then Gromov’s simplicial volume ∥Mn∥ is ...

We generalize wave maps to biwave maps. We prove that the composition of a biwave map and a totally geodesic map is a biwave map. We give examples of biwave nonwave maps. We show that if f is a biwave map into a Riemannian manifold under certain circumstance, then f is a wave map. We verify that if f is a stable biwave map into a Riemannian manifold with positive constant sectional curvature sa...

In this survey paper, we shall derive the following result. Theorem A. Let M denote a closed Riemannian manifold with nonpositive sectional curvature and let M̃ be the universal cover of M with the lifted metric. Suppose that the universal cover M̃ contains no totally geodesic embedded Euclidean plane R (i.e., M is a visibility manifold ). Then Gromov’s simplicial volume ‖M‖ is non-zero. Conseque...

A complete noncompact manifold M with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact submanifold S called the soul of M . When S is a round sphere we show that the clutching map of this bundle is restricted; this is used to deduce that there are at most finitely many isomorphism types of such bundles with sectional curvature lying in a fixed interval [0, κ]. ...

Let M = M1 ×M2 be a product of complex manifolds. We prove that M cannot admit a complete Kähler metric with sectional curvature K < c < 0 and Ricci curvature Ric > d, where c and d are arbitrary constants. In particular, a product domain in Cn cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe that there are complete Kähler metrics with nega...