Given a differentiable manifold $M$, we can equip $M$ with a Riemannian metric $g$ or $g'$ to generate a pair of Riemannian manifolds $(M,g)$ and $(M,g')$, respectively. The gradient vector fields $X_f,X'_f \in TM$ of a function $f: M \to \mathbb{R}$ satisfy, for all $Y \in TM$,
\begin{equation*}
g(X_f,Y) = df(Y) \ ,
\end{equation*}

and
\begin{equation*}
g'(X'_f,Y) = df(Y) \ .
\end{equation*}
My first question is: What intuition can be used to describe the flow of $X'_f$ on $(M,g)$?

Now assume that $f$ is Morse "with respect to all of the appropriate combinations". Second question: What would it mean to attempt Morse theory with the flow of $X'_f$ using the Hessian $H_f$ constructed using the Levi-Civita connection $\nabla$ of $g$, i.e. defined for all $Y,Z \in TM$ by

\begin{equation*} H_f(Y,Z) =g(\nabla_Z X'_f,Y) \ . \end{equation*} The idea being that this Hessian rather than the usual one would be used to index critical points of $f$.