Geodesics with respect to time-dependent Riemannian Metric I'm not sure where to look to solve a problem of this variety.  Does it potentially have to do with Ricci flow?
Suppose we consider Euclidean space $\mathbb{R}^n$ and append to it a time-dependent metric of $A(t)$ where $A: \mathbb{R} \to SPD(n)$ is a smooth curve in the manifold of symmetric positive definite matrices $SPD(n)$.  Define the metric on $\mathbb{R}^n$ as $\langle u, v\rangle_t = u^TA(t)v$, in other words a time-dependent metric.  I'm trying to figure out how to find geodesics in $\mathbb{R}^n$ with respect to the same parameter $t$ that the metric is parameterized by.  I know the geodesic equations for an ordinary Riemmanian manfiold are given by 
$$
\frac{d^2\gamma^k}{dt^2} + \Gamma_{ij}^k \frac{d\gamma^i}{dt} \frac{d\gamma^j}{dt} \;\; =\;\; 0
$$
with 
$$
\Gamma_{ij}^k \;\; =\;\; \frac{1}{2} g^{km} \left ( \frac{\partial g_{im}}{\partial x_j} + \frac{\partial g_{jm}}{\partial x_i} - \frac{\partial g_{ij}}{\partial x_m} \right ).
$$
It's not clear to me how to approach this problem or if it is even well-posed.  Another way I suppose I can phrase this is: how does a curve $\gamma$ naturally "flow" with respect to this metric given the initial conditions $\gamma(0)$ and $\gamma'(0)$?  Can anyone offer any insights or references?  I would appreciate it if I knew a general approach to this problem, or if the problem needs to be posed differently.
 A: The answer really depends on what you mean by geodesics; you are using a word as if there is a meaning for geodesics of "time dependent metrics" as you described them. There isn't. Below I give three possible interpretations of the notion of geodesics, and associated to them are four possible solution curves, they are all different in general. 
Geodesics are solutions to a certain time-dependent ODE
One way to understand geodesics on a Riemannian manifold is via the geodesic spray, which is in fact defined for any connection, not necessarily one coming from a metric. The idea is that the geodesic equation on a fixed Riemannian manifold can be written as a vector field defined on its tangent bundle. Geodesics are then projections of the integral curves of this vector field to the underlying manifold. 
In other words, for every fixed time $t$ there exists a vector field $V$ on $\mathbb{R}^n \times \mathbb{R}^n$ associated to the metric $A(t)$, and the position/velocity $(p,v)$ of your "time dependent" curve can be taken to mean the naive notion that they solve
$$ \frac{d}{dt} (p,v) = V(t; p,v)$$
As it turns out, however, since the metric $A(t)$ is the constant metric for any fixed $t$, the associated geodesic spray $V(t)$ is always the same: $V(t;p,v) = (v,0)$. This means that the "geodesics" are always straight lines. 
Geodesics are to be interpreted in an overarching manifold
An alternative way to interpret this "time-dependent" geodesic notion is that instead of considering geodesics on the manifold $\mathbb{R}^n$, you consider geodesics of the manifold 
$$ \mathbb{R}\times\mathbb{R}^n $$
equipped with the Riemannian metric $dt^2 + A_{ij}(t) dx^i~dx^j$. You can compute the associated geodesic evolution using the usual way for Riemannian manifolds, by computing the connection coefficients (for example) for the metric mentioned above. In particular you find that the velocities satisfy
$$ \dot{v}^0 - \frac12 \dot{A}_{ij}(t) {v}^i {v}^j = 0 $$
and
$$ \dot{v}^i + \frac12 (A^{-1})^{ij} \dot{A}_{jk} {v}^0 {v}^k = 0 $$
In general solutions to this, after projected to $\mathbb{R}^n$, are not straight lines. 
Geodesics as energy minimizers
An alternative formulation is that we let geodesics be critical points of the energy of length integrals. The energy integral with a "varying metric" is 
$$ \int \langle \dot{\gamma}(t),\dot{\gamma}(t)\rangle_t ~\mathrm{d}t$$
while the length integral is 
$$ \int \sqrt{\langle \dot{\gamma}(t),\dot{\gamma}(t)\rangle_t} ~\mathrm{d}t $$
The corresponding Euler-Lagrange equation in the energy case is the linear ODE
$$ \ddot{\gamma}(t) + A^{-1}(t) \dot{A}(t) \dot{\gamma}(t) = 0 $$
and in the length case is something slightly more complicated. In general solutions to these equations are also not straight lines, and they are different from the curves defined in the second case above. 
