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)


\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<t<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.