dg.differential geometry - Exponential map and covariant derivative

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