TITLE: Higher derivatives than Jacobi fields
QUESTION [7 upvotes]: The first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the exponential map (again both of the full $\exp:TM\to M$ map and of the map $\exp_p:T_pM\to M$) may be computed with the aid of Jacobi fields, i.e, solutions to Jacobi's equation.
I have a scenario where I need second derivatives of the `full' exponential map $\exp:TM\to M$. That is, denoting the pushforward of a differentiable map by a  '$ _*$', I would like to know a thing or two about $\nabla_X\exp_*\mathcal{V}$ (where $\mathcal{V}$ is a section of $TTM$ and $X$ is an appropriate vector field). In particular, I think I will require some comparison techniques analagous to those for Jacobi fields (e.g., Rauch's comparison theorems).
Can anyone point me in the right direction?

REPLY [3 votes]: Recently in the context of optimization on manifolds, the covariant derivative of the differential of the exponential map $\nabla \text{d } \text{exp}$ was studied in two papers:
1 ''An accelerated first-order method for non-convex optimization on manifolds'' and
2 ''Curvature-Dependant Global Convergence Rates for Optimization on Manifolds of Bounded Geometry''
As @Deane Yang describes, both papers show that this object satisfies a second-order inhomogenous linear ODE, see Proposition B.1 in 1, and Proposition 4.1 in 2.  The following is from 2:

Let $(M, g)$ be a Riemannian manifold, and let $\gamma$ be a geodesic
with initial unit vector $v \in T_p M$.  For two vectors $w_1, w_2$
perpendicular to $v$, the vector field along $\gamma$ $$K(t) = (\nabla
 \text{d } \text{exp}_p)_{tv}(t w_1, t w_2)$$ satisfies the following
ODE along $\gamma$ $$\ddot{K} + R(K, \dot{\gamma})\dot{\gamma} + Y =
 0, \quad K(0) = 0, \dot{K}(0) = 0$$ where $Y$ is the vector field
along $\gamma$ given by $$Y = 2 R(J_1, \dot{\gamma}) \dot{J}_2 +
 2R(J_2, \dot{\gamma})\dot{J}_1 + (\nabla R_{\dot{\gamma}})(J_2,
 \dot{\gamma})J_1 + (\nabla R_{J_2})(J_1, \dot{\gamma})\dot{\gamma}$$
and $J_1, J_2$ are the Jacobi fields along $\gamma$ with initial
conditions $J_i(0) = 0, \dot{J}_i(0) = w_i$.

Using this ODE, both papers derive bounds on $||\nabla \text{d } \text{exp}||$ based on bounds for the sectional curvatures and $\nabla R$.  Paper 1 uses a bootstrapping technique, while paper 2 uses ODE comparison techniques which yield comparatively tighter bounds albeit requiring a little bit more analysis.  For these results, see Theorem B.2 in 1, and Theorems 4.10 and 4.11 in 2.  The following is from 2:

Let $$\text{sn}_{\kappa}(t) = \left\{ \begin{array}{lII}
 \frac{\sin(\sqrt{\kappa}t)}{\sqrt{\kappa}} & \quad \kappa > 0 \\ t & \quad
 \kappa = 0 \\ \frac{\sinh(\sqrt{-\kappa} t)}{\sqrt{-\kappa}} & \quad
 \kappa < 0 \end{array} \right\}$$ and $$\pi_{\kappa} = \left\{
 \begin{array}{lI} \frac{\pi}{\sqrt{\kappa}} & \quad \kappa > 0 \\
 \infty & \quad \kappa \leq 0 \end{array} \right\}.$$
Let $(M, g)$ be a Riemannian manifold with sectional curvatures
bounded below by $\delta$ and above by $\Delta$.  Also assume
$$||(\nabla_u R)(w, u)w + (\nabla_w R)(w, u)u|| \leq 2 \Lambda ||u||^2
 ||w||^2 \quad \forall p \in M \quad u, w \in T_p M.$$
For a geodesic $\gamma : [0, r] \rightarrow M$ with initial unit
vector $v$, $r < \pi_{(\Delta + \delta)/2}$, and any two vectors $w_1,
 w_2 \in T_p M$, we have that $$||(\nabla \text{d } \text{exp}_p)_{r
 v}(w_1, w_2)|| \leq \frac{8}{3r^2} \text{sn}_{\delta}
 \left(\frac{r}{2} \right)^2\left(\Lambda \cdot
 \text{sn}_{\delta}\left(\frac{r}{2}\right)^2 + 2 \max\{|\Delta|,
 |\delta|\}\text{sn}_{\delta}(r) \right) ||w_1|| \cdot ||w_2||.$$

Finally, these bounds are shown to be tight, in a certain sense -- see the discussion after Theorem B.2 in 1, and subsection 4.6.1 in 2.