Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold

Amira Ishan,Gabriel-Eduard Vîlcu,Sharief Deshmukh

doi:10.3390/math9080863

Amira Ishan, Gabriel-Eduard Vîlcu + Show 1 more

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https://doi.org/10.3390/math9080863

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Abstract

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere Sm(c) of constant curvature c. The first characterization uses the well known de-Rham Laplace operator, while the second uses a nontrivial solution of the famous Fischer–Marsden differential equation.

Highlights

Conformal vector fields and conformal mappings play important roles in the geometry ofRiemannian manifolds as well as in the general relativity.The characterization of important spaces, such as Euclidean spaces, Euclidean spheres and hyperbolic spaces, represents one of the most fascinating problems in Riemannian geometry
The aim of the present work was to study whether the existence of a nontrivial conformal vector field on an n-dimensional compact Riemannian manifold satisfying some very natural conditions influences the geometry of this space
Investigating this question, we arrived at two characterizations of the standard n-spheres with the help of nontrivial conformal vector fields, using the de-Rham Laplace operator and the Fischer–Marsden differential equation

Summary

Introduction

Conformal vector fields and conformal mappings play important roles in the geometry of (pseudo-)Riemannian manifolds as well as in the general relativity (see, e.g., [1,2,3,4,5]). The sphere Sm (c) admits a nontrivial conformal vector field that is an eigenvector of the de-Rham Laplace operator with eigenvalue (m − 2)c (see Equation (5)). (i) Is a compact Riemannian manifold ( M, g) that admits a nontrivial conformal vector field u, which is eigenvector of de-Rham Laplace operator corresponding to a positive eigenvalue, necessarily isometric to a sphere?. (ii) Is a compact Riemannian manifold ( M, g) that admits a nontrivial conformal vector field u with potential function a nontrivial solution of the Fischer–Marsden differential equation, necessarily isometric to a sphere?. We answer the above two problems, showing that the first question has an affirmative answer (cf. Theorem 1), while an affirmative answer for the second question requires an additional condition on the Ricci curvature (cf. Theorem 2)

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Conclusions