Some rigidity results related to the Obata type equation

Let $(\Omega^{n+1},g)$ be an $(n + 1)$-dimensional smooth complete connected Riemannian manifold with compact boundary $\partial\Omega=\Sigma$ and $f$ a smooth function on $\Omega$ which satisfies the Obata type equation $\nabla^2 f -fg =0$ with Robin boundary condition $f_{\nu} = cf$, where $c=\coth{\theta}>1$. In this paper, we provide some rigidity results based on the warped product structure of $\Omega$ determined by the equation $\nabla^2 f -fg =0$ and appropriate curvature assumptions. We also apply a similar method to the Obata type equation $\nabla^2 f +fg =0$ and get a rigidity result on the standard sphere $\mathbb{S}^{n+1}$.