Geodesics and the exponential map

Question

I've been studying some hyperbolic deep learning and got stuck when learning about geodesics in riemannian manifolds.There are two questions that are in my mind:

• First, I would like to know why it is impossible to do a reparametrization of a geodesic with something other than an affine function. I mean, the math behind it is fine but intuitively this doesn't make sense to me. As I see it, geodesics are determined by their image, because a geodesic is a curve that minimizes distance, and the length of a curve is independent of the parametrization, right?
• The second thing is about the exponential map. Given $p \in M$ and $v \in T_pM$, the exponential map is defined as $Exp_p(v) = \gamma_v(1)$, where $\gamma_v$ is the unique maximal geodesic starting at $p$ with velocity $v$. The problem is: $\gamma_v$ may not be defined in $1$, but we can do a reparametrization of geodesics by any linear function, so what I'm asking is: do reparametrizations of the form $t \mapsto kt$ preserve the maximality of a geodesic?

Answer 1

Your questions are relevant for any curve, not just geodesics. A curve has many parameterizations. Given any parameterized curve $$ c: [a,b] \rightarrow M$$ and smooth monotone function $$ f: [\alpha,\beta] \rightarrow [a,b] $$, the curve $$ $\tilde{c}: [\alpha,\beta] \rightarrow M$$ given by $$\tilde{c} = c\circ f$$ is the same curve with a different parameterization. Note that if you change the parameterization, the curve remains the same but the domain changes.

However, some parameterizations are easier to work with than others. You can see this if you recall that the speed of a curve is the norm of velocity, $|c'|$. If a parameterized curve has wildly different varying speed, then doing numerical calculations with it can be very unstable because a small change in the parameter causes a large movement along the curve.

You might recall that a good parameterization is the arclength parameterization, where $|c'|=1$. This is often easier to work with than an arbitrary parameterization. Note that such a curve can also be called a unit speed curve.

However, it is often more convenient to fix the domain of the parameterization. This is especially true if you want to work with a family of curves $c_\delta$ that, say, always have the same start and end points. It turns out almost all of the advantages of working with a unit speed curve also hold for a constant speed curve, i.e., a curve such that $|c'|$ is constant. For example, there is always a constant speed parameterization so that the parameter domain is $[0,1]$. You'll see this parameterization used a lot in differential geometry textbooks, especially when discussing Jacobi fields.

Using the notation above, it's not hard to show that if $c$ has constant speed, then $\tilde{c}$ has constant speed if and only if $f(t) = pt + q$ for some constants $p$ and $q$.

For geodesics, it also has the advantage that the critical points of energy functional are the constant speed geodesics (with the speed uniquely determined if the end points of the geodesic are fixed). The energy functional is much easier to work with than the length functional, because there's no square root.

As for your second question, the maximality of a geodesic does not depend on its parameterization.

Answer 2

Of course you can reparametrize a geodesic. Depending on how you define the term geodesic the resulting curve may not be called a geodesic anymore, though (*). If you choose an affine parametrization it's parameter values will be proportional to it's length, which is not true for an arbitrarily chosen parametrization.

If $\gamma_v$ is not defined for $t=1$, then the same is true for the exponential map at the value $v$. If the manifold is "complete" though, then every geodesic (parametrized by arclength) can be extented to all of $\mathbb{R}$. Completeness may be defined by this requirement, but may also be defined by the requirement that the distance $d$ on $M$ is (Cauchy) complete, where

$$ d(p,q) = \inf\{\ell(c): c(0)= p, c(1) = q\}$$ and $\ell$ is the length of a curve $c:[0,1]\rightarrow M$. A famous theorem in Differential Geometry, the Hopf-Rinow theorem, states that these two definitions are equivalent.

The maximality of a geodesic does not depend on the parametrization, it's a geoemetric property of the geodesic. The concept itself, does, though, because it's about extenting solutions to a certain differential equation.

(*) you can define a geodesic as a locally length minimizing curve, in which case the term does not imply a specific parametrization. It's common, though, to define it as a critical point for the the length functional $\ell(c) = \int |c^\prime(t)| \,dt$, or for the energery functional $E(c)=\frac{1}{2}\int |c^\prime(t)|^2 \,dt$ . That last definition is quite common, since it the one which is easiest to deal with from the point of view of the Calculus of Variations. A critical point for the energy functional will be parametrized proportionally to arc length.