Some characterizations of Riemannian manifolds endowed with a conformal vector fields

2024/12/03 by A. Barros, Barros, A., Israel Evangelista +3
Mathematics · Physics and Astronomy · Computer Science · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2412.02910

Abstract

The aim of this article is to investigate the presence of a conformal vector ξ with conformal factor ρ on a compact Riemannian manifold M with or without boundary ∂ M. We firstly prove that a compact Riemannian manifold (Mⁿ, g) ,n ≥ 3, with constant scalar curvature, with boundary ∂ M totally geodesic, in such way that the traceless Ricci curvature is zero in the direction of ∇ ρ, is isometric to a standard hemisphere. In the 4-dimensional case, under the condition ∫_M|\mathringRic|²⟨ ξ,∇ ρ⟩ dM≤0, we show that, either M is isometric to a standard sphere, or M is isometric to a standard hemisphere. Finally, we give a partial answer for the cosmic no-hair conjecture.

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