This question is related to this one (https://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) I've asked at math.stackexchange. Let $(M,g)$ be a compact Riemannian manifold with no conjugate points and $(\tilde{M},\tilde{g})$ its universal covering. Let $\hat{g}$ the Sasaki metric on $TM-\{0\}$ and $d_{\hat{g}}$ its associated distance function.

Fix $\tilde{p}\in\tilde{M}$ and $R=1$. Let $\tilde{H}:=\tilde{M}-\overline{B_1(p)}$. For $x\in\tilde{H}$, consider the geodesic sphere centered at $p$ and of radius $r_x=d_g(x,p)$. I will denote this as $S_{r_x}(p)$. Let $N(x)$ be the (inner) normal unit vector at $x\in S_{r_x}(p)$ pointing towards $p$. As we fixed $p$, We have defined a function $x\longmapsto N(x)$ whose domain is $\tilde{H}$. My question is the following:

Is there some constant $k$ such that \begin{equation}d_{\hat{g}}(N(x),N(y))\leqslant k\,d_{\tilde{g}}(x,y) \end{equation} for all $x,y\in\tilde{H}$?

Roughly asking, is the normal vector field of "all large spheres" uniformly Lipschitz? I see that if we fix the radius $r_x$, this is true by the smoothness of geodesic spheres, but I'm trying to get some estimatives letting $r_x\in(1,+\infty)$.

Other way of viewing this question is the following: is there some upper bound for the normal curvature of large geodesic spheres of universal coverings of compact manifolds without conjugate points?

In some sense, I see that the abscense of conjugate points could play some role: if we take the sphere $\mathbb{S}^n$ with the usual metric and fix $p$, the normal vector field of each geodesic sphere centered in $p$ is Lipschitz, but as we take the radius going to $\pi$ (and the geodesic spheres get "closer" to $-p$) we see the normal curvatures growing arbitrarily. On the other hand, I see that in the hyperbolic space $\mathbb{H}^n$, the round spheres have constant mean curvature (a result by Alexandrov - Uniqueness theorems for surfaces in the large, V. Vestnik Leningrad Univ., 13, No. 19, A.M.S. (Series 2), 21 (1958), 412–416.).

As I pointed out in https://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres, I'm aware that the normal curvature satisfies a Riccati equation and then, maybe, one could be able to use some results describing bounds to solutions of this equations in abscense of conjugate points, but I could not see so far if it can help.

So I would like to ask the community for some result/reference on the subject. Maybe this is a really naive questions, but I've got no intuition so far.

Thanks for all members of the community.



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