Related papers: The curvature operator at the soul

We look at smooth manifolds equipped with a possibly singular Riemannian metric. We give sufficient conditions for the existence of scalar curvature measures and Dirac operators.

We show that parameterized versions of splitting theorems in Morse theory can be effectively used to generalize some famous bifurcation theorems for potential operators. In particular, such generalizations based on the author's recent…

In this work we study operator splitting methods for a certain class of coupled abstract Cauchy problems, where the coupling is such that one of the problems prescribes a "boundary type" extra condition for the other one. The theory of…

We introduce a metric notion of Ricci curvature for $PL$ manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers Theorem, for surfaces as well as for a large class of higher dimensional manifolds.

On an oriented 4-manifold, we examine the geometry that arises when the curvature operator of a Riemannian or Lorentzian metric $g$ commutes, not with its own Hodge star operator, but rather with that of another semi-Riemannian metric $h$…

We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form…

In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of…

Using a new type of Jacobi field estimate we will prove a duality theorem for singular Riemannian foliations in complete manifolds of nonnegative sectional curvature.

In this note we obtain a formula for the sectional curvature on an arbitrary two-dimensional smooth manifold $M$ equipped with a Lorentzian metric $g$.

In this article, we investigate the interplay between the curvature operator, Weyl curvature, and the Hopf conjecture on compact Riemannian manifolds of even dimension. By decomposing the curvature operator into Hermitian components, we…

We investigate the curvature of invariant metrics on G-manifolds with finitely many non-principal orbits. We prove existence results for metrics of positive Ricci curvature and non-negative sectional curvature, and discuss some families of…

In this paper we continue to study the property of separability of functional space C(X) with the open-point and bi-point-open topologies.

We study the mapping properties of a large class of elliptic operators $P_T$ in gluing problems where two non-compact manifolds with asymptotically cylindrical geometry are glued along a neck of length $2T$. In the limit where $T…

For an asymptotically hyperbolic metric on the interior of a compact manifold with boundary, we prove that the resolvent and scattering operators are continuous functions of the metric in the appropriate topologies.

We classify the complete metrics of nonnegative sectional curvature on M^2xR^2, where M^2 is any compact 2-manifold.

We prove a warped product splitting theorem for manifolds with Ricci curvature bounded from below in the spirit of [Croke-Kleiner, \emph{Duke Math.\;J}.\;(1992)], but instead of asking that one boundary component is compact and mean-convex,…

Let (M,g) be a complete noncompact riemannian manifold with bounded geometry and parallel Ricci curvature. We show that some operators, "affine" relatively to the Ricci curvature, are locally invertible, in some classical Sobolev spaces,…

We define an analytic index and prove a topological index theorem for a non-compact manifold $M\_0$ with poly-cylindrical ends. We prove that an elliptic operator $P$ on $M\_0$ has an invertible perturbation $P+R$ by a lower order operator…

We establish curvature obstruction theorems for manifolds with boundary. Our main theorems show that, for dimensions up to 7, a topologically nontrivial compact manifold with boundary cannot have a metric of positive $m$-intermediate…

A celebrated 1922 theorem of Kuratowski states that there are at most 14 distinct sets arising from applying the operations of complementation and closure, any number of times, in any order, to a subset of a topological space. In this paper…