Let $S^{d-1}$ be the sphere in $\mathbb{R}^d$.
Given a $C^\infty$ function $f \colon S^{d-1} \to \mathbb{R}$, define $g \colon S^{d-1} \to S^{d-1}$ as $g(x) = \exp_x(\nabla f(x))$, where $\nabla f(x)$ is the Riemannian gradient of $f$ and $\exp$ is the exponential map on the sphere.
The map $g$ looks similar to the transport plan map appearing in Optimal Transport when pushing one absolutely continuous measure to another one. I'm interested in something different, where $g$ pushes some set of positive measure into something of zero measure. More precisely:
Can we build some $f$ such that there exists a set $S$ with $\mu(S) > 0$ and $\mu(g(S)) = 0$ and such that $ I + \nabla^2 f(x) \succ 0$ ?
Some motivation: in the Euclidean case, it's quite easy to see that this is impossible. If we take $f$ to be $C^2$ a c-convex function with respect the quadratic cost function $c(x, y) = \text{dist}(x, y)^2 / 2$ and such that $I + \nabla^2 f(x) \succ 0$ (the c-convexity only ensures that $I + \nabla^2 f(x) \succeq 0$, if I am not mistaken), then $g$ cannot collapse a positive measure set into a null set, since $Dg(x) = I + \nabla^2 f(x) \succ 0$.
[This is a cross-post of this, where I did not receive any answers.]
