Ok, we know what connections and covariant derivatives are. Now we can use them to get a map from the tangent space ${T_p(M)}$ at one point to the manifold ${M}$ which is a local isomorphism. This is interesting because it gives a way of saying, “start at point ${p}$ and go five units in the direction of the tangent vector ${v}$,” in a rigorous sense, and will be useful in proofs of things like the tubular neighborhood theorem—which I’ll get to shortly.

Anyway, first I need to talk about geodesics. A geodesic is a curve ${c}$ such that the vector field along ${c=(c_1, \dots, c_n)}$ created by the derivative ${c'}$ is parallel. In local coordinates ${x_1, \dots, x_n}$, here’s what this means. Let the Christoffel symbols be ${\Gamma^k_{ij}}$. Then using the local formula for covariant differentiation along a curve, we get $\displaystyle D(c')(t) = \sum_j \left( c_j''(t) + \sum_{i,k} c_i'(t) c_k'(t) \Gamma^j_{ij}(c(t)) \right) \partial_j,$

so ${c}$ being a geodesic is equivalent to the system of differential equations $\displaystyle c_j''(t) + \sum_{i,k} c_i'(t) c_k'(t) \Gamma^j_{ij}(c(t)) = 0, \ 1 \leq j \leq n.$

The fundamental theorem of ODEs does not, unfortunately, apply to this though. However, there is an elementary trick we can use. Let ${y_1, \dots, y_n, z_1, \dots, z_n}$ be functions of ${t}$ and consider the ${2n}$ equations $\displaystyle y_j'(t) = z_j(t)$

and $\displaystyle z_j'(t) = - \sum_{i,k} z_i(t) z_k(t) \Gamma^j_{ij}(y_1(t), \dots, y_n(t)) .$ Now this is a system of ODEs, and given a solution to this, we can take ${c(t) = (y_1, \dots, y_n)}$ as a geodesic. In particular, given initial conditions ${c(0)}$ and ${c'(0)}$ (respectively the ${y_i(0)}$ and ${z_i(0)}$) we can construct a geodesic ${c}$ locally. More precisely, using the local smoothness theorem, we find:

Proposition 1 Fix ${p \in M}$. Then there is a neighborhood ${V \subset T_p(M)}$ of ${0}$ and a smooth map ${G: (-2,2) \times V \rightarrow M}$ such that if ${v \in V}$, then ${G(t,v)}$ for ${-2 is a geodesic starting at ${p}$ with initial tangent vector ${v}$.

There’s a slight problem here I haven’t yet addressed: strictly speaking, the local smoothness theorem only gives the proposition with ${(-2,2)}$ replaced by ${(\delta,\delta)}$ for some ${\delta>0}$ small. But geodesics can be rescaled—if ${\gamma_v(t)}$ is a geodesic for the tangent vector ${v}$, then ${\gamma_v(\epsilon t)}$, which is defined on an interval ${\frac{1}{\epsilon}}$ times as large, is one for the tangent vector ${\epsilon v}$. So just shrink ${V}$ by a small constant factor to allow the geodesics to be defined on a larger interval.Another reason that geodesics are important is that, as I will eventually prove, with respect to a Riemannian metric and the associated Levi-Civita connection, geodesics have the chracteristic property of being locally distance-minimizing.

Now fix the map ${G}$ as before and let ${\exp_p(v) := G(1, v)}$. Then ${\exp_p}$ is a smooth map ${V \rightarrow M}$ where ${V}$ is as above, called the exponential map. I claim that the differential of ${\exp_p}$ at ${0}$ is the identity, if we identify ${T_0(T_p(M)) \simeq T_p(M)}$ in the usual way. In particular,

Proposition 2 ${\exp_p}$ is a local isomorphism.

To prove the claim, take a tangent vector ${v \in T_p(M)}$ and consider the corresponding path ${tv, -1 \leq t \leq 1}$, in ${T_p(M)}$. Then ${\exp_p(M)}$ sends this path to $\displaystyle G(1, tv) = G(t,v),$ by the above rescaling remarks, and this traces out a geodesic in ${M}$ whose tangent vector at ${0}$ is ${v}$. Thus the assertion on the differential is proved.

Where next? I want to spend a post on the tubular neighborhood theorem just because it’s fun, and provides an illustration of these concepts. Next I think it might be interesting to discuss what the differential of the exponential map is other than the origin. There is a formula of Helgason that describes that for real-analytic manifolds. Then I should get to some Riemannian geometry.