Gradient vector fields defined with respect to two different metrics and Morse theory

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 $f$.

• It looks to me like your first question can be answered by looking up the properties of gradient-like'' vector fields for $f$ on $M$. Intuitively, such a vector field should have 0s of the right type at the critical points, its flows should increase g, and so on. You may be interested in this question: mathoverflow.net/questions/123989/… Or is it more specific than that? – Elizabeth S. Q. Goodman Jun 7 '15 at 0:48
• If you are asking how to intuitively see what it's like to change the metric, say for a torus: imagine embedding the torus in $\mathbb R^3$, and let $f$ be the height function; the gradient vectors are orthogonal projections of vertical ones onto the torus. Then imagine tilting the torus, so that all the level sets are the same: this will change the projected vectors showing the gradient of the new inherited metric. One such way of tilting'' is a sheer linear transformation such as $T(x, y, z)=(x+y, y, z)$. – Elizabeth S. Q. Goodman Jun 7 '15 at 0:53