Tag Archives: metric

Geodesics

Straight Lines

When we think of a straight line, we usually think of a line in the Euclidean sense; that is, c(t)=p+tX, where p is a point contained in the line, t is a real number, and X is a vector that points parallel to the line. If we consider Euclidean space as a manifold, we would say that X is in the tangent space T_{c(t)}(\mathbb E^n), because c'(t)=X. One important observation to make is that all along c(t), X never changes; i.e., we never accelerate. That is, if we move along the curve, we never speed up or slow down, and we never turn.

In the language of my post on covariant derivatives, this is easy to express:

\nabla_{c'} c' \equiv 0

The geometric interpretation is simple here: in the direction of the velocity vector, the velocity vector doesn’t change. You can probably see the punchline coming by now. If we generalize to a curve c(t) on a manifold M, c(t) is a geodesic if \nabla_{c'} c' \equiv 0.

Now, you may notice that we can trace out the same curve if we tweak the parameter t so that we could accelerate on the curve (we wouldn’t turn, but we could speed up or slow down). That is, we could have an alternate parametrization. But in order to have a geodesic, we need \nabla_{c'} c' \equiv 0, so \nabla_{c'}\left<c',c' \right> = 2\left<\nabla_{c'} c',c' \right> = 0, and therefore \|c'\| is a constant along the curve. This gives us a unique parametrization of the curve, up to a constant scaling factor on the parameter. In fact, if we consider such a scaling factor, we get that \nabla_{c'(st)} c'(st) = \nabla_{s c'(t)} s c'(t) = s^2\nabla_{c'(t)} c'(t) = 0, so a geodesic with a constant scaling factor on its parameter is still a geodesic (and obviously has the same image). This motivates the following definition: if \|c'\| = 1 then the geodesic is called a normal geodesic.

The Exponential Map

Say that some curve \gamma(t) is a geodesic. Then \nabla_{\gamma'}\gamma' = 0 is a second-order differential equation in t. If we assume that \gamma(0) = p, and \gamma'(0) = v, then we have the required conditions for existence and uniqueness of a solution to the differential equation. That is, given a point p\in M and tangent vector v\in T_p(M), there is a unique geodesic \gamma_v that passes through p with velocity v.

The exponential map \text{exp}_p:T_p(M)\to M is defined as \text{exp}_p(v) = \gamma_v(1), assuming that 1 is in the domain of \gamma_v. The exponential map is fairly important when talking about Riemannian manifolds, and it turns out that it is smooth and a local diffeomorphism. The latter means that there is a neighborhood around p where its unique inverse exists. This inverse is the logarithmic map, or \text{log}_p:M\to T_p(M).

The exponential map is so important, in fact, that it appears in many of the important theorems in Riemannian geometry, like the Hopf-Rinow Theorem and the Cartan-Hadamard Theorem. It’s also essential to understanding the effects of curvature on a Riemannian manifold.

Arc Length

At this point we can ask about the relationship between arc length and geodesics. Assume that we have some smooth function \alpha : [a,b]\times(-\epsilon,\epsilon)\to M. We can compute the change in arc length L[c_s] over the family of curves c_s = \alpha | [a,b]\times\{s\}:

\frac d{ds}L[c_s] = \frac d{ds}\int_a^b\left<c_s'(t),c_s'(t)\right>^{1/2}dt = \int_a^b\nabla_S\left<T,T\right>^{1/2}dt
= \frac 1 2\int_a^b\left<T,T\right>^{-1/2}\nabla_S\left<T,T\right>dt = \int_a^b\left<T,T\right>^{-1/2}\left<\nabla_S T,T\right>dt

The variables S,T that we substitute here are fields of tangent vectors corresponding to the differential of \alpha with respect to the variables s,t. The rest is just calculus. Since s,t are independent of each other, we know that their derivatives commute and so we can say that [T,V] = 0. This means that we can make the switch \nabla_S T = \nabla_T S:

\frac d{ds}L[c_s] = \int_a^b\left<T,T\right>^{-1/2}\left<\nabla_T S,T\right>dt
= \int_a^b\left<T,T\right>^{-1/2}\left(T\left<S,T\right>-\left<S,\nabla_T T\right>\right)dt

If we consider the curve c_0, and consider that we can always reparametrize a curve without loss of generality so that l = \left<T,T\right>^{1/2} is a constant,

\frac d{ds}L[c_s]\mid_{s = 0} = l^{-1} \left(\left<S,T\right>\mid_a^b-\int_a^b\left<S,\nabla_T T\right>dt\right)

This is called the first variation formula. The function \alpha is called a variation. If we assume that all the c_s are curves that join two points in M, then we know that S vanishes at the endpoints. If we further assume that c_0 is a geodesic, then the integral vanishes (because \nabla_T T = 0). What this means is that geodesics are critical points of the arc length function L for curves that join two points.

We can’t claim that a geodesic segment minimizes the distance between two points (though there is a unique minimizing geodesic segment; for that we need the second variation formula, which I won’t get into in this post). To see this, consider the case when M is a sphere, with the usual angular metric. If we consider any two distinct points, there is a great circle path that joins them that is of length the angular distance between them, \delta. However, there is also a path of length 2\pi - \delta that goes around “the long way” that joins the points as well. This path happens to be the longest one that you can take, and it’s also a geodesic segment. Obviously this would be a maximum of the first variation formula.

It’s easy to see that the first variation formula gives us a lot of power in talking about the geometry of a Riemannian manifold. The source that I use actually motivates the definition of a geodesic from an effort to minimize the first variation formula. I prefer to motivate it from the “straight line” perspective.

Sources

Much of this material comes from Comparison Theorems in Riemannian Geometry by Jeff Cheeger and David G. Ebin.

Convexity Using Metric Balls

I figure that I owe my readers a technical post, so while I’m riding home on the bus, I’ll write it up. This occurred to me when I was trying to figure out what to do on the ride. I have a nice gadget with a WordPress app, so why not?

The project that I’m working on now involves defining a notion of convexity for a non-Euclidean space. There are any number of difficulties that you can run into when you attempt to define convexity on an arbitrary space, but I do have a few guarantees:

  • I’m on a manifold, so shapes make “sense,” albeit in a squishy way
  • I don’t have any limitations of convexity; that is, I can make a convex set as large as I like
  • Metric balls are convex

So now I want to define a convex hull of a set of points in this space. I can do this in one of two ways. I can say that the convex hull is the convex set of minimal volume containing the set, or equivalently, that it is the intersection of all convex sets containing the set.

I’d like to say that the intersection of all metric balls containing the set is the same as the convex hull (not just any convex superset of the set, mind you; specifically metric balls that are supersets of the set in question). I don’t necessarily need this lemma to be true, but it would be nice. The way to show that two sets are equivalent is usually to say that one contains the other, and vice-versa.

It’s quite trivial to show that (in this space) the intersection of all metric balls that contain the set also contains the convex hull. Metric balls are, after all, convex. It’s trickier (to me) to show that the converse would also be true; that is, that the convex hull of a set also contains the intersection of all metric balls that contain the set. Any ideas?

[Update: apparently there’s a construction called a ball hull that is exactly the intersection of all metric balls containing a set. Perhaps it is essentially different from a convex hull.]