Homotopy type of manifolds with partially positive curvature

Abstract: It is well-known that Riemannian metrics on a manifold will put some restriction on its topology. For instance, thanks to Cheeger-Gromoll-Meyer, a compact Riemannian manifold with positive sectional curvature and with convex boundary must be a topological disk. Later H. Wu, J. Sha, and Lawson-Michelsohn studied the homotopy type of manifolds with weaker conditions on sectional curvature and boundary convexity. In this talk I will present our recent work along this line. If we view the previous results as the Euclidean ones, our results may be viewed as the hyperbolic ones.