We prove some rigidity results for compact manifolds with boundary. In particular for a compact Riemannian manifold with nonnegative Ricci curvature and simply connected mean convex boundary, it is shown that if the sectional curvature vanishes on the boundary, then the metric must be ‡at. In [Schroeder and Strake 1989, Theorem 1], Schroeder and Strake proved the following rigidity theorem. Let...

Complete noncompact Riemannian manifolds with nonnegative sectional curvature arise naturally in the Ricci flow when one takes the limits of dilations about a singularity of a solution of the Ricci flow on a compact 3-manifold [ H-95a]. To analyze the singularities in the Ricci flow one needs to understand these manifolds in depth. There are three invariants, asymptotic scalar curvature ratio, ...

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W+ is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S2 × S2 with both (i) K > 0 and (ii) 1 6 s−W+ ≥ 0. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of ...

We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal (most negative) eigenvalue of the curvature operator at any point in the Teichmüller space Teich(Sg) of a closed surface Sg of genus g is uniformly bounded away...

1. Introduction. Perhaps the most significant aspect of differential geometry is that which deals with the relationship between the curvature properties of a Riemannian manifold M and its topological structure. One of the beautiful results in this connection is the (generalized) Gauss-Bonnet theorem which relates the curvature of compact and oriented even-dimensional manifolds with an important...

In this article, we study the smoothness of Riemannian submersions for open manifolds with non-negative sectional curvature. Suppose thatM is a C-smooth, complete and non-compact Riemannian manifold with nonnegative sectional curvature. Cheeger-Gromoll [ChG] established a fundamental theory for such a manifold. Among other things, they showed that M admits a totally convex exhaustion {Ωu}u≥0 of...

In pointwise differential geometry, i.e., linear algebra, we prove two theorems about the curvature operator of isometrically immersed submanifolds. We restrict our attention to Euclidean immersions because here the results are most easily stated and the curvature operator can be simply expressed as the sum of wedges of second fundamental form matrices. First, we reprove and extend a 1970 resul...

We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define analogues of curvature in non-smooth or discrete spaces, beginning with Alexandrov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvat...

We classify the left-invariant metrics with nonnegative sectional curvature on SO(3) and U(2).