Search results for "Riemannian manifold"

Scalable implementation of measuring distances in a Riemannian manifold based on the Fisher Information metric

2019

This paper focuses on the scalability of the Fisher Information manifold by applying techniques of distributed computing. The main objective is to investigate methodologies to improve two bottlenecks associated with the measurement of distances in a Riemannian manifold formed by the Fisher Information metric. The first bottleneck is the quadratic increase in the number of pairwise distances. The second is the computation of global distances, approximated through a fully connected network of the observed pairwise distances, where the challenge is the computation of the all sources shortest path (ASSP). The scalable implementation for the pairwise distances is performed in Spark. The scalable…

The role of virtual work in Levi-Civita's parallel transport

2015

The current literature on history of science reports that Levi-Civita’s parallel transport was motivated by his attempt to provide the covariant derivative of the absolute differential calculus with a geometrical interpretation (For instance, see Scholz in The intersection of history and mathematics, Birkhauser, Basel, pp 203–230, 1994, Sect. 4). Levi-Civita’s memoir on the subject was explicitly aimed at simplifying the geometrical computation of the curvature of a Riemannian manifold. In the present paper, we wish to point out the possible role implicitly played by the principle of virtual work in Levi-Civita’s conceptual reasoning to formulate parallel transport.

Quantum Groups, Star Products and Cyclic Cohomology

1993

After some historical remarks, we start with a rapid overview of the star-product theory (deformation of algebras of functions on phase space) and its applications to deformation-quantization. We then concentrate on Poisson-Lie groups and their “quantization”, give a star-product realization of quantum groups and discuss uniqueness and the rigidity as bialgebra of a universal model for the quantum SL(2) groups. In the last part we develop the notion of closed star-product (for which a trace can be defined on the algebra), show that it is classified by cyclic cohomology, permits to define a character and that there always exists one; finally we show that the pseudodifferential calculus on a …

Regular 1-harmonic flow

2017

We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to Lipschitz initial data. We prove uniqueness and, in the case of a convex domain, local existence of solutions to the flow equations. If the target manifold has non-positive sectional curvature or in the case that the datum is small, solutions are shown to exist globally and to become constant in finite time. We also consider the case where the domain is a compact Riemannian manifold without boundary, solving the homotopy problem for 1-harmonic maps under some …

Comparing the relative volume with a revolution manifold as a model

1993

Given a pair (P, M), whereM is ann-dimensional connected compact Riemannian manifold andP is a connected compact hypersurface ofM, the relative volume of (P, M) is the quotient volume(P)/volume(M). In this paper we give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature ofM and the mean curvature ofP, with respect to that of a model pair\(\left( {\mathcal{P},\mathcal{M}} \right)\) where ℳ is a revolution manifold and\(\mathcal{P}\) a “parallel” of ℳ.

Nonexistence of solutions to higher order evolution inequalities with nonlocal source term on Riemannian manifolds

2022

We establish sufficient conditions for the nonexistence of nontrivial solutions to higher order evolution inequalities, with respect to the time variable. We consider a nonlocal source term, and work on complete noncompact Riemannian manifolds. The obtained conditions depend on the parameters of the problem and the geometry of the manifold. Our main result recovers some nonexistence theorems from the literature, established in the whole Euclidean space.

A topological obstruction to the geodesibility of a foliation of odd dimension

1981

Let M be a compact Riemannian manifold of dimension n, and let ℱ be a smooth foliation on M. A topological obstruction is obtained, similar to results of R. Bott and J. Pasternack, to the existence of a metric on M for which ℱ is totally geodesic. In this case, necessarily that portion of the Pontryagin algebra of the subbundle ℱ must vanish in degree n if ℱ is odd-dimensional. Using the same methods simple proofs of the theorems of Bott and Pasternack are given.

Linear invariants of Riemannian almost product manifolds

1982

Using the decomposition of a certain vector space under the action of the structure group of Riemannian almost product manifolds, A. M. Naveira (9) has found thirty-six distinguished classes of these manifolds. In this article, we prove that this decomposition is irreducible by computing a basis of the space of invariant quadratic forms on such a space.

Unit Vector Fields that are Critical Points of the Volume and of the Energy: Characterization and Examples

2007

In the last few years, many works have appeared containing examples and general results on harmonicity and minimality of vector fields in different geometrical situations. This survey will be devoted to describe many of the known examples, as well as the general results from where they are obtained.

The isoperimetric inequality and the geodesic spheres. Some geometric consequences

1986