How to bound from below the distance between the images of two points $x,y$ (within convexity radius) with a given distance $||x-y||$ under the exponential map in a Riemannian manifold?

Let $M$, $\dim M=2$, be a Riemannian manifold, $p\in M$, $T=T_pM$, $\exp_p:T\to M$ the exponential map, $E\subset T$ a disk around $0$ of radius not exceeding the convexity radius, i.e., for any $x,y\in E$, the shortest geodesic between $\exp_p(x)$ and $\exp_p(y)$ lies in $\exp_p(E)$.

Let $||x-y||=a$ (in my specific case $a=\sqrt3\,\text{radius}(E)$). How can I bound from below $d(\exp_p(x),\exp_p(y))$, where $d(\cdot,\cdot)$ is the distance in $M$?

I expect that the bound would involve the maximum sectional curvature $K$ in $E$ (supposing it is bounded) and the radius of $E$. Or maybe only the radius of $E$, since it is already bounded by the maximum sectional curvature.

My intuition is that the worst case is the sphere of the sectional curvature $K$, and of all points, the worst case is when both points are at the equator of the sphere, i.e., at the boundary of the convexity radius (supposing that the injectivity radius is large enough). If so, then a simple exercise in trigonometry gives the desired lower bound (the distance between two points at the equator of the sphere), i.e., for any manifold (with maximum sectional curvature $K$) other than the sphere and for any pair of points other than at the equator, the distance will be only greater.

However, I lack the expertise to formalize the above intuition about the "worst case" -- so my question is how to formalize it (not how to calculate the distances on the sphere, I can do it).

  • 3
    $\begingroup$ The thing to use is the Hinge Comparison theorem, however, you would have to use a smaller domain $D$ than your $exp(E)$. The comparison theorem requires an upper curvature bound, say, $K$, on the domain $D$. Then your distance is bounded from below by the same lower bound you get on the simply connected complete surface of constant curvature $K$. The latter you can compute using hyperbolic, Euclidean or spherical trigonometry. The standard reference is "Comparison Theorems in Riemannian Geometry" by Cheeger and Ebin. You can also read this material in Petersen's "Riemannian Geometry". $\endgroup$ – Moishe Kohan Jan 29 at 16:57

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