Some rigidity results related to the Obata type equation

Let $({\mathrm{\Omega }}^{n+1},g)$ be an $(n+1)$-dimensional smooth complete connected Riemannian manifold with compact boundary $\mathrm{\partial }\mathrm{\Omega }=\mathrm{\Sigma }$ and $f$ a smooth function on $\mathrm{\Omega }$ which satisfies the Obata type equation ${\mathrm{\nabla }}^{2}f-fg=0$ 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 $\mathrm{\Omega }$ determined by the equation ${\mathrm{\nabla }}^{2}f-fg=0$ and appropriate curvature assumptions. We also apply a similar method to the Obata type equation ${\mathrm{\nabla }}^{2}f+fg=0$ and get a rigidity result on the standard sphere ${\mathbb{S}}^{n+1}$.