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 $\zeta$ with a conformal factor $\sigma$, there is an associated skew-symmetric tensor $\phi$ called the associated tensor, and also, $\zeta$ admits the Hodge decomposition $\zeta =\overline{\zeta }+\nabla \rho$, where $\overline{\zeta }$ satisfies $\mathrm{div}\overline{\zeta }=0$, which is called the Hodge vector, and $\rho$ is the Hodge potential of $\zeta$. 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 $S^m\left(c\right)$ if and only if (1) for a nonzero constant c, the function $-\sigma /c$ is a solution of the Poisson equation $\Delta \rho =m\sigma$, and (2) the Ricci curvature satisfies $Ric\left(\overline{\zeta },\overline{\zeta }\right)\ge {∥\phi ∥}^2$. The second result states that if $M^m$ has constant scalar curvature $\tau =m(m-1)c>0$, then it is an $S^m\left(c\right)$ if and only if the Ricci curvature satisfies $Ric\left(\overline{\zeta },\overline{\zeta }\right)\ge {∥\phi ∥}^2$ and the Hodge potential $\rho$ satisfies a certain static perfect fluid equation. The third result provides another new characterization of $S^m\left(c\right)$ using the affinity tensor of the Hodge vector $\overline{\zeta }$ of a conformal vector field $\zeta$ 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 $\zeta$ whose affinity tensor vanishes identically and $\zeta$ annihilates its associated tensor $\phi$.