Hessian tensor equation? I am studying some curvature conditions on a warped product manifold $M\times_fN$ where $f$ is a positive real-valued function defined on $M$. I got the following equation
$$\frac 1f H^f(\nabla_XY,Z)+\frac 1f H^f(Y,\nabla_XZ)=X\left(\frac 1f H^f(Y,Z)\right)$$
where $X,Y,Z$ are vector fields over $M$, $H^f$ is the Hessian tensor (defined on $M$) of $f$  and $\nabla$ is the Levi-Civita connection on $M$. .
My questions:  


*

*What does this equation imply?

*Under what conditions could this equation have a non-trivial solution $f$? What are the (non-constant if any) solutions $f$ of this equation?

 A: (1) If $T:= \frac{1}{f} H^f$, then $T$ is a parallel.
If $M$ is a closed manifold, then $$g+sT,\ |s|<\epsilon \ \ast$$ is
positive definite so that it is a family of metrics on $M$ whose
connections are same.
If $M$ is not closed manifold, then in $\ast$ we can not find
$\epsilon$
If $M$ is an Einstein manifold with nonzero $C$, then $${\rm Ric}\
(g)=C(g+sT)$$ It is a contradiction. That is existence of $T$, which
is not a multiple of $g$, implies that $M$ is far away from
Einstein.
[Add] $g_s:= g+sT$ so that \begin{align*} g_s(\nabla^s_XY,Z) &=
\frac{1}{2} \{ X g_s( Y,Z) + Yg_s(Z,X) -Zg_s(X,Y) \\&+ g_s([X,Y],Z)
- g_s([Y,Z],X) -g_s(
 [X,Z],Y) \}\\&=\frac{1}{2} \{ X g(
Y,Z) + Yg(Z,X) -Zg(X,Y) \\&+ g([X,Y],Z) - g([Y,Z],X) -g(
 [X,Z],Y) \} + T(\nabla^s_XY,Z)\end{align*}
since $T$ is parallel. Hence $\nabla^s=\nabla$
(2) In flat $\mathbb{R}^2$, $\nabla T=0$ iff $f_kf_{ij}=ff_{ijk}$ where index means a partial derivatives. Here if $f$ is a
liner function, then $T=0$, but I do not know nontrivial $f$.
