What is the meaning of the exponential map of a covariant derivative on manifolds?

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 secondorder 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 


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 "SemiRiemannian Geometry With Applications to Relativity" does adequate justice to the Lorentzian and semiRiemannian cases. 

