Example of a function in Riemannian manifold

Question

Is there any example of a function $f:M\rightarrow\mathbb{R}$ such that the following condition holds $$\nabla^2f_p(X,Y)\geq Ric_p(X,Y)f(p)\ \forall p\in M \text{ and } X,Y\in T_pM$$ where $\nabla^2$ is the Hessian operator and $Ric_p$ is the Ricci tensor at $p$. I know that every convex function in Euclidean space satisfies the condition but I can not find any other example in other Riemannian manifold. Please help.

Answer 1

Let $M$ be a complete manifold with sectional curvature $L\leq 0$. The function $f$ defined on the warped product manifold $R\times_{e^t} M$ by $$f(t,x)=e^t$$ is positive and has a positive definite Hessian. I think this function satisfies the condition. For more details, you may see the interesting article Here