Naturally reductive Riemannian $S$-manifolds

A Geometry Preserving Kernel over Riemannian Manifolds

Abstract- Kernel trick and projection to tangent spaces are two choices for linearizing the data points lying on Riemannian manifolds. These approaches are used to provide the prerequisites for applying standard machine learning methods on Riemannian manifolds. Classical kernels implicitly project data to high dimensional feature space without considering the intrinsic geometry of data points. ...

Connections on Naturally Reductive

Heterogeneous Riemannian Manifolds

We define a Riemannian metric g on a manifold M to be heterogeneous if no two distinct points of M have isometric neighborhoods. Intuitively, a heterogeneous metric is as far as possible from being homogeneous. Heterogeneity can be reformulated in terms of a multijet transversality condition so that by an application of the standard transversality theorems, the genericity of heterogeneous metri...

Flowers on Riemannian manifolds

In this paper we will present two upper bounds for the length of a smallest “flower-shaped” geodesic net in terms of the volume and the diameter of a manifold. Minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a Riemannian manifold. Let Mn be a closed Riemannian manifold of dimension n. We prove that there exists a minimal geodesic net that ...

Degenerations of Riemannian Manifolds

This is an expositiry article on collapsing theory written for the Modern Encyclopedia of Mathematical Physics (MEMPhys). We focus on describing the geometric and topological structure of collapsed/non-collapsed regions in Riemannian manifold under various curvature assumptions. Numerous applications of collapsing theory to Riemannian geometry are not discussed in this survey, due to page limit...