I'm currently reading about a problem regarding second variations of some functional defined on a Riemannian Manifold $M$ equipped with the Levi-Civita connection, and am confused about how to express a certain quantity.
For some curve $x$ and vector fields $V, W$ along $x$, consider the two-parameter family of curves defined by $\alpha(t; r, s) = \exp_{x(t)}(rV(t) + sW(t)),$ where $\exp$ is the Riemannian exponential map.
Given a smooth scalar field $f$ on $M$, how would I go about calculating $\frac{\partial^2}{\partial r \partial s}\Big\vert_{(r, s)=(0,0)}f(\alpha(t; r, s))$?
My thought is first to write:
\begin{align*} \frac{\partial^2}{\partial r \partial s}\Big\vert_{(r, s)=(0,0)}f(\alpha(t; r, s)) &= \frac{\partial}{\partial r}\Big\vert_{r=0}\left(\frac{\partial}{\partial s}\Big\vert_{ s=0}f(\alpha(t; r, s))\right) \\ &= \frac{\partial}{\partial r}\Big\vert_{r=0} \left<\text{grad} f(\alpha(t; r, 0)), \frac{\partial \alpha}{\partial s}(t; r, 0) \right> \\ &= \left<\frac{D}{\partial r}\Big\vert_{r=0}\text{grad} f(\alpha(t; r, 0)), W(t) \right> + \left<\text{grad} f(x(t)), \frac{\partial}{\partial r}\Big\vert_{r=0}\left(\frac{\partial \alpha}{\partial s}(t; r, 0)\right) \right> \end{align*} Where $\text{grad}f$ is the gradient vector field of $f$ and $\left<\cdot, \cdot\right>$ is the Riemannian metric. From here, I'm unsure. How do I calculate the quantity $\frac{D}{\partial r}\Big\vert_{r=0}\text{grad} f(\alpha(t; r, 0))$?
I worked out the quantity a bit further a believe I found an answer.
Suppose $\gamma_t(r) = \alpha(t; r, 0)$. Then since $\text{grad}f$ is globally defined (and hence extendible) and $\frac{d}{dr}\Big\vert_{r=0}\gamma_t(r) = \dot{\gamma}_t(0) = V(t)$, we may write
\begin{align*} \frac{D}{\partial r}\Big\vert_{r=0} \text{grad}f(\exp(rV(t))) &= \frac{D}{\partial r}\Big\vert_{r=0} \text{grad}f(\gamma_t(r)) \\ &= \nabla_{\dot{\gamma_t}(r)} \text{grad}f\Big\vert_{r=0} \\ &= \nabla_{V(t)} \text{grad}f \end{align*}
I'm not entirely convinced that the last equality is justified, but it makes sense to me. Hence the quantity I wanted would become:
$$\left<\nabla_{V(t)} \text{grad}f, W(t) \right> + \left<\text{grad} f(x(t)), \frac{\partial^2}{\partial r \partial s}\alpha(t; r, s)\Big\vert_{(r,s)=(0,0)}\right>$$
It's still not clear if the final term can be written down in some "nice" way using the particular form of the variation I chose, but I suspect not and that term does not factor into the analysis in my particular problem.