The exponential map in

mathematics provides a concrete relationship between the

tangent space of a manifold, and the

manifold itself. There are many ways to conceptually approach the exponential map, and therefore a few different "definitions" are provided.

**From the Infinitesimal to the Macroscopic**

Often in mathematics (and very often in physics), one deals with the effect on a function f(λ) when we change λ by a small parameter, ε. Formally, we can expand f(λ + ε) in a taylor series about λ:

f(λ + ε) = f(λ) + ε df/dλ + ½ ε^{2} d^{2}f/dλ^{2} + …

Now, if ε is infinitesimally small, by which we mean it is as small as any number you can imagine, we can ignore terms of order greater than ε^{2} and the result is simply:

f(λ + ε) = (1 + ε d/dλ) f(λ)

We can think of this as an operator, T(ε) acting on f(λ), whose operation is to translate λ by an infinitesimal amount, ε.

f(λ + ε) = T(ε) f(λ), where T(&epsilon) = 1 + ε d/dλ

What if we wanted a more general operator, T(Δλ), whose operation translated λ by a finite amount, Δλ? We could read off this operator from the fully-expanded taylor series above, but it is more instructive to think of this finite-translation operator as a product of a large amount of successive infinitesimal-translation operators:

T(Δλ) = [ T(ε) ]^{N}, where Δλ = N × ε.

We then take the limit as N → infinity (and ε → 0). In other words,

T(Δλ) = Lim_{N → infinity} [ (1 + (Δλ d/dλ) /N ) ]^{N}

Does this formula look familiar yet? Let us pretend for the time being that the operator Δλ d/dλ is just a number, k. Then the formula looks like:

T(Δλ) = Lim_{N → infinity} [ (1 + k/N ) ]^{N}

This you should recognize as one definition of the exponential function,

= e^{k}.

We carry the notation over to describe the translation operator:

T(Δλ) = e^{Δλ d/dλ}

This is typically evaluated by expanding the exponential in its usual power series. It is straightforward to check that its action on a function just gives the full taylor series expansion of that function.

Now, we make a small conceptual transition. Instead of thinking of this as an operator-valued function with respect to the interval, Δλ, it is more natural to think of it as an operator-valued function on derivatives, Δλ d/dλ = d/dt. In other words, we can vary the translation distance by varying our reparameterization t = f(λ). So, we symbolically rewrite this as

T(d/dt) = e^{d/dt}

This is a more natural form of our translation operator, the exponential map. You can think of this as generating a translation T, given a derivative d/dt.

**From Vector Fields to Integral Curves**

On an arbitrary differential manifold M, imagine a smoothly varying family of curves Φ(p), covering the manifold (or at least filling some open set in the manifold) without intersecting. Such a family of curves is known as a congruence of curves. At each point p in the manifold, there is exactly one curve passing through p. Such a curve is associated with a particular vector **V**_{p} in the tangent space of M at p; **V**_{p} is the velocity of the curve. Since we can do this at every point p ∈ M, this determines a smoothly varying vector field **V**(p).

We can go the other direction, too. On a differential manifold M, a smooth vector field **V**(p) determines a smoothly varying family of curves Φ_{p}: R → M, called the integral curves of **V**. You can think of this set of curves as the effect of trying to smush our flat tangent space T_{p}M onto our curved manifold, M.

Φ_{p}(λ) = (x^{1}_{p}(λ), x^{2}_{p}(λ), ..., x^{n}_{p}(λ)) in a specific coordinate representation {x^{i}}.

This family of curves provides a map from the tangent space to the manifold, which we call the exponential map, exp: TM → M. The curves Φ_{p} are determined by demanding that the velocity of each curve Φ(λ) is equal to the vector field evaluated at that point, **V**_{Φ(λ)}. This demand can be represented in a coordinate-dependent manner:

dx^{i}/dλ = V^{i}(x^{1}(λ), x^{2}(λ), ..., x^{n}(λ)).

This is simply a set of first-order ordinary differential equations for x^{i}(λ). There always exists a unique solution about a sufficiently small neighborhood of p. Note that this requirement implies that the directional derivative d/dλ = V^{i} ∂/∂x^{i}, i.e. that the curve parameter λ appears in the directional derivative associated with the vector field **V**.

For notational use, we make the association p ↔ (x^{1}(λ_{0}), x^{2}(λ_{0}), ..., x^{n}(λ_{0})). Then, explicitly, we have:

x^{i}(λ_{0} + ε) = x^{i}(λ_{0}) + ε dx^{i}/dλ + ...

= [ 1 + ε d/dλ + ½ ε^{2} d^{2}/dλ^{2} + ... ]|_{λo} x^{i}

x^{i}(λ_{0} + ε) = e^{ε d/dλ} x^{i}

As before, we notice that ε d/dλ = **ε V** is a vector by itself. That is, instead of thinking of this as a map which inputs a vector **V** and gives us a curve, and inputs a distance ε and moves us this distance along the curve to produce a point in M, we can think of this as a map which inputs vectors **ε V** and outputs the point on our manifold found by moving a unit distance along its integral curve. We can cut through all the unnecessary notation by simply evaluating our expression at ε = 1:

x^{i}(λ_{0} + 1) = e^{d/dλ} x^{i}

This is the exponential map of d/dλ acting on x^{i}. We could be more explicit by expressing d/dλ as V^{k} ∂/∂x^{k}:

x^{i}(λ_{0} + 1) = exp { V^{k} ∂/∂x^{k} } x^{i}.

This expression may seem strange-looking, as we are taking partial derivatives with respect to x^{k} of x^{i}, which we expect to just give us a kronecker delta, δ^{i}_{k}, but don't forget that V^{k} is also dependent on the {x^{i}}. Thus, the expansion of this formula should look like:

x^{i}(λ_{0} + 1) = [ x^{i}(λ_{0}) + V^{i}|_{λo} + ½ V^{k} ∂V^{i}/∂x^{k}|_{λo} + ... ]

Now, this formula was only guaranteed to work in a small neighborhood of p (meaning we cannot justify setting ε = 1 the way we did), but we can get around this by restricting the domain, i.e. requiring that our vector fields be small enough to keep within some neighborhood of p in M. Moreover, we can often find solutions which cover a large portion of the manifold M. For example, if we just choose a coordinate vector field ∂/∂x^{k}, then the integral curves produced are simply the coordinate curves x^{i ≠ k}(λ) = constant, x^{k} = constant + λ. This exponential map will be well-defined as far as the coordinate chart reaches, which may nearly be the entire manifold (For example, the sphere S^{2} can be covered minus one point, by stereographic projection). For this reason, the exponential map is often thought of as a map from the local structure of T_{p}M to the more global structure of M itself.