Consider $(M,g)$ is a Riemannian Manifold (compact).
For a smooth function $f(x,t): M \times \mathbb{R} \to \mathbb{R}$,
It is known that we can induce a one parameter group of diffeomorphism such that $$\frac{\partial}{\partial t} \phi_t(x) = \nabla f (t, \phi_t (x)) $$ And $$\phi_0(x)=x$$
I want to show the trace free part of hessian of $f$ ,i.e. $$\nabla \nabla f -\frac{1}{2}\Delta f g$$ has length that is invariant under the diffeomorphism. Which means take $\phi_t^*g=\overline{g}$, $\phi_t^*\nabla=\overline{\nabla}$ , we have $$|\nabla \nabla f -\frac{1}{2}\Delta f g|^2=|\overline{\nabla}\overline{ \nabla} f -\frac{1}{2}\overline{\Delta} f \overline{g}|^2 $$
Any help will be appreciated.