Gradient of solution to heat equation under evolving metric

Question

The following simple question came to me when I was studying the heat equation on a Riemannian manifold: Suppose $M$ is a closed Riemannian manifold and $g_t$ is a smooth family of Riemannian metrics on $M$. It's a well-known question the study the evolving heat equation on $M$: \begin{equation} \dfrac{\partial}{\partial t}f=\Delta_{g(t)} f \end{equation} Suppose that $V\subset TM$ is a fixed sub-bundle of the tangent space of $M$. Given that at time $t=0$, the gradient of $f_0$ belongs to $V$. Is it true that the gradient of $f$ remains in $V$ the whole time? I have a feeling that this is true because of the equalizing nature of the heat equation, but I don't know how to formally prove or disprove it. Does anyone have any thoughts or helpful references? Please apologize if this is a trivial matter.