I would like to ask the community for a reference on the following property of geodesic spheres. Let $(M,g)$ be a compact Riemannian manifold without conjugate points and $\tilde{M}$ its universal covering with the natural metric $\tilde{g}$. Given a point $\tilde{p}\in\tilde{M}$ and $R>1$, Is there some result like
The inner normal field of the geodesic sphere centered at $\tilde{p}$ and with radius $R$ is a Lipschitz vector field?
I am aware that the normal curvature satisfies a Riccati equation along a geodesi radius starting at the center of the sphere, namely $\tilde{p}$. Perhaps I could get some control of the normal vector field of a geodesic sphere using this equation, but I don't know if there is this control.
If someone could give me any reference on the subject, it would be really appreciated.