# Harmonic vector fields on a weighted Riemannian manifold arising from a Finsler structure

Harmonic vector fields on a weighted Riemannian manifold arising from a Finsler structure AbstractIn the present work, the harmonic vector field is defined on closed Finsler measure spaces through different approaches. At first, the weighted harmonic vector field is obtained as the solution space of a PDE system. Then a suitable Dirichlet energy functional is introduced. A σ-harmonic vector field is considered as the critical point of related action. It is proved that a σ-harmonic vector field on a closed Finsler space with an extra unit norm condition is an eigenvector of the defined Laplacian operator on vector fields. Moreover, we prove that a unit weighted harmonic vector field on a closed generalized Einstein manifold is a σ-harmonic vector field. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Pure and Applied Mathematics de Gruyter

# Harmonic vector fields on a weighted Riemannian manifold arising from a Finsler structure

, Volume 9 (2): 11 – Apr 1, 2018
11 pages

/lp/de-gruyter/harmonic-vector-fields-on-a-weighted-riemannian-manifold-arising-from-KbtpVQSTbC
Publisher
de Gruyter
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6090
eISSN
1869-6090
DOI
10.1515/apam-2016-0099
Publisher site
See Article on Publisher Site

### Abstract

AbstractIn the present work, the harmonic vector field is defined on closed Finsler measure spaces through different approaches. At first, the weighted harmonic vector field is obtained as the solution space of a PDE system. Then a suitable Dirichlet energy functional is introduced. A σ-harmonic vector field is considered as the critical point of related action. It is proved that a σ-harmonic vector field on a closed Finsler space with an extra unit norm condition is an eigenvector of the defined Laplacian operator on vector fields. Moreover, we prove that a unit weighted harmonic vector field on a closed generalized Einstein manifold is a σ-harmonic vector field.

### Journal

Advances in Pure and Applied Mathematicsde Gruyter

Published: Apr 1, 2018