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 ε2 d2f/dλ2 + …

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 ε2 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) = LimN → 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) = LimN → infinity [ (1 + k/N ) ]N

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

= ek.

We carry the notation over to describe the translation operator:

T(Δλ) = eΔλ 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) = ed/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 Vp in the tangent space of M at p; Vp 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 Φp: 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 TpM onto our curved manifold, M.

Φp(λ) = (x1p(λ), x2p(λ), ..., xnp(λ)) in a specific coordinate representation {xi}.

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 Φp are determined by demanding that the velocity of each curve Φ(λ) is equal to the vector field evaluated at that point, VΦ(λ). This demand can be represented in a coordinate-dependent manner:

dxi/dλ = Vi(x1(λ), x2(λ), ..., xn(λ)).

This is simply a set of first-order ordinary differential equations for xi(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 = Vi ∂/∂xi, 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 ↔ (x10), x20), ..., xn0)). Then, explicitly, we have:

xi0 + ε) = xi0) + ε dxi/dλ + ...

= [ 1 + ε d/dλ + ½ ε2 d2/dλ2 + ... ]|λo xi

xi0 + ε) = eε d/dλ xi

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:

xi0 + 1) = ed/dλ xi

This is the exponential map of d/dλ acting on xi. We could be more explicit by expressing d/dλ as Vk ∂/∂xk:

xi0 + 1) = exp { Vk ∂/∂xk } xi.

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

xi0 + 1) = [ xi0) + Vi|λo + ½ Vk ∂Vi/∂xk|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 ∂/∂xk, then the integral curves produced are simply the coordinate curves xi ≠ k(λ) = constant, xk = 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 S2 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 TpM to the more global structure of M itself.

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