Conformal Vector Fields and Their Applications to Einstein-Type Manifolds

초록

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 Φ 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 Φ for a conformal vector field X is always vanishing, i.e., div 2Φ = 0 , and divΦ=0 if and only if Φ = 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. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

키워드