Ray Singer torsion is a numerical invariant associated with a compact manifold equipped with a flat bundle, a Riemannian metric on the manifold and a Hermitian structure on the bundle. In this note we show how one can remove the dependence on the Riemannian metric and the Hermitian structure with the help of a base point and of an Euler structure, and obtain a topological invariant. A numerical...

We study umbilic (space-like) submanifolds of pseudo-Riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo Euclidean space and relate this notion to umbilicity. Finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.A pseudo-Riemannian submanifold M in (a...

We study compact complex 3-manifolds admitting holomorphic Riemannian metrics. We prove a uniformization result: up to a finite unramified cover, such a manifold admits a holomorphic Riemannian metric of constant sectionnal curvature.

Wepresent a general procedure for handling equality constraints in optimization problems that is of particular use in direct search methods. First we will provide the necessary background in differential geometry. In particular, we will see what a Riemannian manifold is, what a tangent space is, how to move over a manifold and how to pullback functions from a manifold to the tangent spaces. The...

Recall that the radius of a compact metric space (X, dist) is given by rad X = minx∈X maxy∈X dist(x, y). In this paper we generalize Berger’s 1 4 -pinched rigidity theorem and show that a closed, simply connected, Riemannian manifold with sectional curvature ≥ 1 and radius ≥ π2 is either homeomorphic to the sphere or isometric to a compact rank one symmetric space. The classical sphere theorem ...

We propose a new framework based on optimization on manifolds to approximate the solution of a Lyapunov matrix equation by a low-rank matrix. The method minimizes the error on the Riemannian manifold of symmetric positive semi-definite matrices of fixed rank. We detail how objects from differential geometry, like the Riemannian gradient and Hessian, can be efficiently computed for this manifold...

In this paper we study geometrical structures on the manifold of FIR lters and their application to multichannel blind deconvolution First we introduce the Lie group and Riemannian metric to the manifold of FIR lters Then we derive the natu ral gradient on the manifold using the isometry of the Riemannian metric Using the natural gradient we present a novel learning algorithm for blind deconvol...

We study the relation between the exponential stability of an invariant manifold and the existence of a Riemannian metric for which the flow is “transversally” contracting. More precisely, we investigate how the following properties are related to each other: i). A manifold is “transversally” exponentially stable; ii). The “transverse” linearization along any solution in the manifold is exponen...

We consider an inverse problem for a second order hyperbolic initial boundary value problem on a compact Riemannian manifold M with boundary. Assume that we know ∂M and the Cauchy data on ∂M × [0, T ] of the solutions with vanishing initial data. In the paper we consider two problems. Firstly, when T is sufficiently large and the Riemannian manifold satisfies an additional geometrical condition...

1 We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes will permit us to show that a regular Poisson manifold whose symplectic foliat...