A note on closed vector fields

Abstract

Special vector fields, such as conformal vector fields and Killing vector fields, are commonly used in studying the geometry of a Riemannian manifold. Though there are Riemannian manifolds, which do not admit certain conformal vector fields or certain Killing vector fields, respectively. Closed vector fields exist in abundance on each Riemannian manifold. In this paper, we used closed vector fields to study the geometry of the Riemannian manifold. In the first result, we showed that a compact Riemannian manifold $ (M^{n}, g) $ admits a closed vector field $\boldsymbol{\omega }$ with $ \mathrm{div} \boldsymbol{\omega }$ non-constant and an eigenvector of the rough Laplace operator, the integral of the Ricci curvature $ Ric(\boldsymbol{\omega }, \boldsymbol{\omega }) $ has a suitable lower bound that is necessarily isometric to $ S^{n}(c) $ and that the converse holds. In the other result, we found a characterization of an Euclidean space using a closed vector field $\boldsymbol{\omega }$ with non-constant length that annihilates the rough Laplace operator and squared length of its covariant derivative that has a suitable upper bound. Finally, we used the closed vector field provided by the gradient of the non-trivial solution of the Fischer-Marsden equation on a complete and simply connected Riemannian manifold $ (M, g) $ and showed that it is necessary and sufficient for $ (M, g) $ to be isometric to a sphere and that the squared length of the covariant derivative of this closed vector field has a suitable upper bound.

Keywords

• closed vector field
• $ n $-sphere
• euclidean space
• fischer-marsden equation