This paper advocates Riemannian multi-manifold modeling in the context of network-wide non-stationary timeseries analysis. Time-series data, collected sequentially over time and across a network, yield features which are viewed as points in or close to a union of multiple submanifolds of a Riemannian manifold, and distinguishing disparate time series amounts to clustering multiple Riemannian su...

In 1912 Bieberbach proved that every compact flat Riemannian manifold M is finitely covered by a flat torus. More precisely, M has the form (F\G)/H where G is a group of translations of Euclidean space, F c G is a discrete subgroup, and H is a finite group of isometries of the space of right cosets F\G. For a proof see e.g. Wolf [18]. The condition that M has a flat Riemannian metric can be sep...

The main purpose of the following article is to introduce a Lie theoretical approach to the problem of classifying pseudo quaternionic-Kähler (QK) reductions of the pseudo QK symmetric spaces, otherwise called generalized Wolf spaces. The history of QK geometry starts with the celebrated Berger’s Theorem [Ber55] which classifies all the irreducible holonomy groups for not locally symmetric pseu...

Let F be a singular Riemannian foliation on a compact Riemannian manifold M . By successive blow-ups along the strata of F we construct a regular Riemannian foliation F̂ on a compact Riemannian manifold M̂ and a desingularization map ρ̂ : M̂ → M that projects leaves of F̂ into leaves of F . This result generalizes a previous result due to Molino for the particular case of a singular Riemannian folia...

In this paper, we prove a generalized integral inequality for submanifolds with parallel mean curvature vector in an arbitrary Riemannian manifold, and from which we obtain a pinching theorem for compact oriented submanifolds with parallel mean curvature vector in a complete simply connected pinched Riemannian manifold, which generalizes the results obtained by Alencar-do Carmo and Hong-Wei Xu.

Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data. Among them, persistent homology is a well-known tool to extract robust topological features, and outputs as persistence diagrams. Unfortunately, persistence diagrams are point multi-sets which can not be used in machine learning algorithms for vector data. To de...

Abstract. A real Bott manifold is the total space of iterated RP 1 bundles starting with a point, where each RP 1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a real toric manifold and admits a flat riemannian metric invariant u...

We prove that given a C∞ Riemannian manifold with boundary, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result holds extends to C manifolds, and that in dimensions 2, 3 and 4 it also holds for PL manifolds. We employ the main result to prove that given any orientable C∞ Riemannian manifold with boundary admits quasimeromorphic mappings ont...

We quantify conditions that ensure that a signed measure on a Riemannian manifold has a well defined centre of mass. We then use this result to quantify the extent of a neighbourhood on which the Riemannian barycentric coordinates of a set of n+1 points on an n-manifold provide a true coordinate chart, i.e., the barycentric coordinates provide a di↵eomorphism between a neighbourhood of a Euclid...

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distance...