Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications

For a compact Riemannian m -manifold ( M m , g ) , m > 1 , endowed with a nontrivial conformal vector field ζ with a conformal factor σ , there is an associated skew-symmetric tensor φ called the associated tensor, and also, ζ admits the Hodge decomposition ζ = ζ ¯ + ∇ ρ , where ζ ¯ satisfies div ζ ¯ = 0 , which is called the Hodge vector, and ρ is the Hodge potential of ζ . The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on M m . The first result of this article states that a compact Riemannian m -manifold M m is an m -sphere S m ( c ) if and only if (1) for a nonzero constant c , the function − σ / c is a solution of the Poisson equation Δ ρ = m σ , and (2) the Ricci curvature satisfies R i c ζ ¯ , ζ ¯ ≥ φ 2 . The second result states that if M m has constant scalar curvature τ = m ( m − 1 ) c > 0 , then it is an S m ( c ) if and only if the Ricci curvature satisfies R i c ζ ¯ , ζ ¯ ≥ φ 2 and the Hodge potential ρ satisfies a certain static perfect fluid equation. The third result provides another new characterization of S m ( c ) using the affinity tensor of the Hodge vector ζ ¯ of a conformal vector field ζ on a compact Riemannian manifold M m with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold M m , m > 2 , is a Euclidean m -space if and only if it admits a nontrivial conformal vector field ζ whose affinity tensor vanishes identically and ζ annihilates its associated tensor φ .