Distance between two geodesics originating from separate but nearby points

Question

Let $(M,g)$ be a complete Riemannian manifold with sectional curvatures constrained within $[\kappa_{\min},\kappa_{\max}]$. Suppose $x,y\in M$ are two points in $M$ and $v_x\in T_{x}M$ is a tangent vector. Define $\Gamma_{x\to y}:T_{x}M\to T_{y}M$ as the parallel transport map along the shortest geodesic connecting $x$ and $y$. Let $v_{y}=\Gamma_{x\to y}v_x\in T_{y}M$. Our question is whether the following inequality holds: $$ d({\rm Exp}(x,t v_x),{\rm Exp}(y,tv_y))\leq c_1e^{c_2t}\cdot d(x,y),\quad \forall t\geq 0, \forall x,y\in M, $$ where $c_1,c_2$ are positive constants independent of $t,x,y$ and $d(\cdot,\cdot)$ is the geodesic distance.

The objective of this question is to show that when $x$ tends to $y$, the distance between two geodesics tends to zero in a certain uniform sense. Hence, the constants $c_1,c_2$ should be independent of $t,x,$ and $y$.

Answer 1

Let $\sigma(s)$ denote a unit speed minimizing geodesic from $x$ to $y$. Let $l=d(x,y)$ be the length of $\sigma$.

Define a vector field $v(s)$ along $\sigma$ by parallel transport. Then set $$ \Gamma(s,t) = \exp_{\sigma(s)}(tv(s)). $$ Note that for each $s$ fixed $\Gamma_s:t\mapsto\Gamma(s,t)$ is a geodesic. In particular $S:=\partial_s\Gamma$ is a Jacobi field along each $\Gamma_s$.

Note also that $S(s,0)=\sigma'(s)$ and $$ D_tS(s,0) = D_sT(s,0) = D_sv(s) = 0. $$ Here we wrote $T=\partial_t\Gamma$ as usual, and used the symmetry lemma $D_tS=D_sT$. (Note that $T(s,0)=v(s)$ is parallel along $\sigma$ by construction.)

By the curvature bounds we can obtain $|S(t,s)|\leq e^{ct}$ (see here for example).

Finally, we write $$ d(\Gamma(l,t),\Gamma(0,t)) \leq L(s \mapsto \Gamma(s,t)) = \int_0^l |S(s,t)| ds\leq e^{ct} l. $$ This is the desired bound.