MDPI - Publisher of Open Access Journals

15 pages, 266 KiB

Open AccessArticle

Lifting Dual Connections with the Riemann Extension

by Stéphane Puechmorel

Mathematics 2020, 8(11), 2079; https://doi.org/10.3390/math8112079 - 21 Nov 2020

Cited by 4 | Viewed by 2296

Let

(M, g)

be a Riemannian manifold equipped with a pair of dual connections

(\nabla, \nabla^{*})

. Such a structure is known as a statistical manifold since it was defined in the context of information geometry. [...] Read more.

Let

(M, g)

be a Riemannian manifold equipped with a pair of dual connections

(\nabla, \nabla^{*})

. Such a structure is known as a statistical manifold since it was defined in the context of information geometry. This paper aims at defining the complete lift of such a structure to the cotangent bundle

T^{*} M

using the Riemannian extension of the Levi-Civita connection of M. In the first section, common tensors are associated with pairs of dual connections, emphasizing the cyclic symmetry property of the so-called skewness tensor. In a second section, the complete lift of this tensor is obtained, allowing the definition of dual connections on

T T^{*} M

with respect to the Riemannian extension. This work was motivated by the general problem of finding the projective limit of a sequence of a finite-dimensional statistical manifold. Full article

(This article belongs to the Special Issue Geometry and Topology in Statistics)

33 pages, 410 KiB

Open AccessFeature PaperArticle

Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

by Oğul Esen, Miroslav Grmela, Hasan Gümral and Michal Pavelka

Entropy 2019, 21(9), 907; https://doi.org/10.3390/e21090907 - 18 Sep 2019

Cited by 20 | Viewed by 3142

Abstract

Geometrical and algebraic aspects of the Hamiltonian realizations of the Euler’s fluid and the Vlasov’s plasma are investigated. A purely geometric pathway (involving complete lifts and vertical representatives) is proposed, which establishes a link from particle motion to evolution of the field variables. [...] Read more.

T^{*} T^{*} Q

) of (both compressible and incompressible) Euler’s fluid and Vlasov’s plasma are derived. Poisson mappings relating the momentum realizations with the usual field equations are constructed as duals of injective Lie algebra homomorphisms. The geometric pathway is then used to construct the evolution equations for 10-moments kinetic theory. This way the entire Grad hierarchy (including entropic fields) can be constructed in a purely geometric way. This geometric way is an alternative to the usual Hamiltonian approach to mechanics based on Poisson brackets. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

Search Results (2)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (2)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI