Conformal Vector Fields and Their Applications to Einstein-Type Manifolds

Abstract

In this paper, we investigated the properties of conformal vector fields defined on a Riemannian manifold. Given a conformal vector field X, we can define a skew-symmetric (1, 1)-tensor $\Phi$ associated to X which can be used to test whether the 1-form dual to X is closed. First, we show that complete divergence of the associated skew-symmetric (1, 1)-tensor $\Phi$ for a conformal vector field X is always vanishing, i.e., $\textrm{div}^2 \Phi = 0$, and $\textrm{div}\,\Phi = 0$ if and only if $\Phi = 0$ when M is compact. Second, we consider Riemannian manifolds admitting a conformal vector field whose conformal factor satisfies the critical point equation, and vacuum static spaces admitting a closed conformal vector field whose conformal factor satisfies the vacuum static equation. In both cases, we prove that the given Riemannian manifold is Einstein and is isometric to a standard sphere. These results generalize results in Deshmukh and Alsolamy (Balkan J Geom Appl 17(1):9–16, 2012) and da Silva Filho (Math Nach 293:2299–2305, 2020) in some sense.