Short survey on the existence of slices for the space of Riemannian metrics

We review the well-known slice theorem of Ebin for the action of the diffeomorphism group on the space of Riemannian metrics of a closed manifold. We present advances in the study of the spaces of Riemannian metrics, and produce a more concise proof for the existence of slices.Comment: 21 pages; added references [DR19] and [Kan05]; corrected typos; added propositions 2.2 and 4.6, remarks 2.6, 2.18 and 2.22; to appear in the Proceedings of the IV Meeting of Mexican Mathematicians Abroad 201

The Interaction of Curvature and Topology

In this snapshot we will outline the mathematical notion of curvature by means of comparison geometry. We will then try to address questions as the ways in which curvature might influence the topology of a space, and vice versa

Spaces of positive intermediate curvature metrics

In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional Spin-manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.Comment: 34 page

Spaces of Riemannian metrics

Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the notions we use to characterize the "shape" of a manifold. The space of Riemannian metrics is a mathematical object that encodes the many possible ways in which we can geometrically deform the shape of a manifold

Bundles with even-dimensional spherical space form as fibers and fiberwise quarter pinched Riemannian metrics

Let $E$ be a smooth bundle with fiber an $n$-dimensional real projective space $\mathbb{R}P^n$. We show that, if every fiber carries a positively curved pointwise strongly $1/4$-pinched Riemannian metric that varies continuously with respect to its base point, then the structure group of the bundle reduces to the isometry group of the standard round metric on $\mathbb{R}P^n$.Comment: 11 pages, to appear in Proceedings of the American Mathematical Societ