Exponential map and covariant derivative What is the meaning of the exponential map of a covariant derivative on manifolds?
 A: There are many ways to talk about covariant derivatives and the exponential maps on manifolds. I will discuss the exponential map from the coordinate point of view.
The covariant derivative can be used to give an equation for geodesics, curves with minimal length (on a Riemannian manifold) or curves with maximum length (on a Lorentzian manifold). Given a particular point $p\in M$, where $M$ is the manifold the exponential map takes a vector $v$ to the point which is a unit distance along the geodesic whose derivative at $p$ is $v$.
In other words $\exp_p:U\subset T_pM\to M$ a homeomorphism, where $U$ is some subset, $\exp_p(v)=\gamma_v(1)$ where $\gamma_v$ is the curve given as the solution to the differential equation $\nabla_{\gamma'_v}\gamma'_v=0$ with initial conditions $\gamma'_v(0)=v$ and $\gamma_v(0)=p$ and $\nabla$ is the covariant derivative. Thus, for example, $\exp_p(tv)=\gamma_{tv}(1)=\gamma_v(t)$.
We need to use a subset $U$ of $T_pM$ since $\exp$ may fail to be a homeomorphism on all of $T_pM$.
Basically the exponential map of a covariant derivative is a uniquely defined homeomorphism from a subset of
the tangent space at $p$ to the manifold $M$. I like to think of it as giving a representation of the covariant derivative on the manifold, but this isn't exactly correct.
As I mentioned there are many ways to understand the exponential map, particular in the context of fibre bundles and Lie groups, may I suggest Kobayashi and Nomizu's excellent book, "Foundations of Differential Geometry" particularly volume 1 for the Riemannian case, where as O'Neill's "Semi-Riemannian Geometry With Applications to Relativity" does adequate justice to the Lorentzian and semi-Riemannian cases. 
A: I think what the questioner is getting at here is the relation between a connection (or covariant derivative) on the tangent bundle of a manifold and what is called a "geodesic spray" (which is a more convenient way of representing the "exponential map"). This is the subject of a very old paper by Ambrose, Singer, and myself (in 1960). Here is Kuranishi's Math Reviews article for that paper.
MR0126234 (23 #A3530) 53.45 (53.55)
Ambrose,W.; Palais, R. S.; Singer, I. M.
Sprays.
An. Acad. Brasil. Ci. 32 1960 163–178
Let M be a $C^1$ manifold. When an affine connection of M is given, we can associate, for each
tangent vector x at a point m in M, the geodesic $\alpha_x$ with tangent x at m. Conversely, we define a spray on M by saying that it is an assignment which gives, for each tangent vector x of M,
a $C^1$ curve $\alpha_x$ in M satisfying certain conditions which are satisfied by the geodesics. A spray obtained by an affine connection is called a geodesic spray. The first theorem says that any spray is a geodesic spray and, moreover, we can prescribe the torsion form of the affine connection which gives the spray.
Let $M_m^k$ be the space of kth order tangent vectors of M at m. $M_m^k$ contains $M_m^1$. By an dissectionof M, we mean an assignment which gives, for each point m in M, a linear complementary
space $M_m^c$ of $M_m^1$ in $M_m^2$ such that the assignment is of class $C^\infty$. Elements in $M_m^c$ are called pure. Clearly a dissection gives rise to a spray. Namely, we demand that the second-order tangent vectors of $\alpha_x$ are pure. The second theorem says that this correspondence of dissections and sprays is injective as well as surjective.
Reviewed by M. Kuranishi
Copyright American Mathematical Society 1962, 2010
