Chapter 4
Geodesics

This chapter is a continuation of the idea of parallel transport in Section 3.4 . I debated whether to move that section to this chapter, but ultimately decided linking covariant derivatives and parallel transport was necessary as motivation. But perhaps it would be worthwhile to read that section again now.

In this chapter we develop the theory of geodesics, which generalise the notion of a ‘straight line’. To put the question provocatively: what does it mean to have a straight line in a curved space? Consider this question for ${𝕊}^2$ if somebody asked you to fly an aeroplane in a straight line between two cities. A reasonable definition would be a flight path that did not require steering the plane’s control stick. This is the same as the idea of ‘walking without turning’ that we used previously in our thought experiment. Such paths are called geodesics. On the other hand, this is Riemannian geometry. In euclidean space, the shortest path between two points is a straight line. Perhaps this should be taken as the defining feature of straight lines. It turns out that this length-minimising property is also true of geodesics, so that the two possible definitions coincide.

First we will formalise the definition of geodesic. Although we mostly restrict ourselves to consideration of the Levi-Civita connection, we do examine in the the mutual dependence of geodesics and connections. Using the length of a curve we define a distance function on a Riemannian manifold as the infimum of the length of all smooth curves. We will show that geodesics are critical points of the length functional, using a proof that is reminiscent of the proof that minimal surfaces are critical for area from Section 1.6. Showing that they are locally length minimising, surprisingly, is significantly more difficult. This leads us to construct special coordinates, so-called normal coordinates, in which the geodesic structure of the Riemannian manifold is a little clearer.

4.1 Straight Lines

Adding to the discussion above, we understand that ‘walking without turning’ along a curve $\alpha$ means that the tangent of the curve is parallel. The technicality is that ${\alpha }^{\prime }$ is not a vector field on the manifold, but this is not a problem because it is a vector field along the curve $\alpha$ and we know that that is sufficient to define its covariant derivative.

In a chart we may write $\alpha =({\alpha }^i)$ and use the Christoffel coefficients to describe the covariant derivative:

This is generally known as the geodesic equation. It is of course the parallel transport equation (3.37) with $Y={\alpha }^{\prime }$. But since the vector field in this case is linked the the curve, we now have a second-order nonlinear system of ODEs. However we can use a trick to write this as a system of first order ODEs:

From this we conclude the local existence and uniqueness of geodesics in a neighbourhood of every point and in every direction. Specifically, given a point $p\in M$ and a direction $v\in T_{p}M$ existence implies that there is a curve ${\gamma }_{p,v}:(a,b)\to M$ with ${\gamma }_v(0)=0$ and ${\gamma }_v^{\prime }(0)=v$. We will drop the $p$ when it is unambiguous from context. We assume that ${\gamma }_{p,v}:(a,b)\to M$ is maximal in the sense that any other solution is a restriction of ${\gamma }_{p,v}$ to some smaller domain $(a^{\prime },b^{\prime })\subset (a,b)$. This assumption may always be achieved, because uniqueness makes it possible to glue together any two solutions into a ‘longer’ one. Related to this is the observation that if $\tilde{p}={\gamma }_{p,w}(t_0)$ and $\tilde{w}={\gamma }_{p,w}^{\prime }(t_0)$, then

How do the geodesics of a manifold depend on the choice of covariant derivative? Following Lemma 3.23 we consider the connections $\nabla <!--\; FUNCTION\; APPLICATION\; -->$ and $\widetilde{\nabla <!--\; FUNCTION\; APPLICATION\; -->}=\nabla <!--\; FUNCTION\; APPLICATION\; -->+A$. In a chart we may write $A(\mathit{\partial i},\mathit{\partial j})=A_{ij}^k\mathit{\partial k}$ and since $A$ is $C^{\infty }$-bilinear these functions completely determine $A$. Moreover ${\tilde{\Gamma }}_{ij}^k={\Gamma }_{ij}^k+A_{ij}^k$. Let $\alpha$ be a geodesics of $\nabla <!--\; FUNCTION\; APPLICATION\; -->$. Then the geodesic equation for $\widetilde{\nabla <!--\; FUNCTION\; APPLICATION\; -->}$ reads

Thus $\alpha$ is also a geodesic of $\widetilde{\nabla <!--\; FUNCTION\; APPLICATION\; -->}$ if and only if this quantity is zero. What is obscured by index notation is the symmetry here. Using a relabelling of summation indices

This can of course be zero for some curve $\alpha$. If $A$ is antisymmetric, $A(X,Y)=-A(X,Y)$, then this quantity is zero for all curves, and in particular the two connections have the same geodesics. Conversely, we know that every vector $w\in T_{p}M$ at every point is the tangent vector to some geodesic of $\nabla <!--\; FUNCTION\; APPLICATION\; -->$. If the two connections have the same geodesics, then this forces $A$ to be antisymmetric. Therefore we have proved

In particular, if we absorb the torsion of a connection $\widetilde{\nabla <!--\; FUNCTION\; APPLICATION\; -->}=\nabla <!--\; FUNCTION\; APPLICATION\; -->-\frac{1}{2}T$, then this does not change the geodesics. This justifies our comment at the end of Chapter 3 that torsion is often unimportant.

The geodesic equation says that the curve is parallel transporting its own tangent vector. If the connection is metric-compatible then this means that the length of the tangent vector is not changing. In Riemannian geometry, we call $\parallel {\alpha }^{\prime }{\parallel }_g$ the speed of the curve $\alpha$.

For any point there is a special geodesic called the constant geodesic. It is the geodesic associated to the zero vector. One can check easily that ${\gamma }_0(t)=p$ for all $t\in ℝ$ solves the geodesic equation with the initial condition ${\gamma }_0(0)=p$ and ${\gamma }_0^{\prime }(0)=0$. The interpretation is that walking in a straight line with initial speed zero means standing still. As a function $ℝ\to M$ they are constant, hence the name. This degenerate example is none-the-less important to include.

Notice that a geodesic is a parameterised curve. To give an example, both $\alpha (t)=(t,0)$ and $\beta (t)=(2t,0)$ are geodesics in ${ℝ}^2$. Their images in ${ℝ}^2$ but they are not equal as parameterised curves because they have different speeds. But our intuition tells us that a straight line is about direction, not speed.

The following lemma tells us how geodesics in the same direction with different initial speeds are related. It says that they have the same image and differ only by a constant rescaling factor. For this reason, geodesics that only differ by speed are often conflated with one another.

A particular case is for $c=-1$. The lemma says ${\gamma }_{-w}(t)={\gamma }_w(-t)$: the geodesic in the reverse direction is the same as walking backwards. If we choose a time $t_0$ and set $\tilde{p}=\gamma (t_0)$ and $\tilde{w}={\gamma }_{p,w}^{\prime }(t_0)$, then we have

This says that if you walk along a geodesic for a certain amount of time $t_0$, turn around and walk back for the same amount of time, then you end back where you started. These properties might seem obvious, but it is important to question whether our intuition carry over to the general setting.

Finally, what can be said about geodesics in an immersed manifold? For this question to be sensible, we should use the tangent connection on $M$ from Definition 3.59. To recap, we have a Riemannian manifold $N$ with the Levi-Civita connection and $M$ is Riemannian immersed in $N$. The tangent connection on $M$ is the projection of ${\nabla <!--\; FUNCTION\; APPLICATION\; -->}^N$ to the tangent space of $M$. A curve $\alpha$ on $M$ is a geodesic iff

Because ${\nabla <!--\; FUNCTION\; APPLICATION\; -->}_{{\alpha }^{\prime }}{\alpha }^{\prime }$ is a second-order derivative of $\alpha$ we interpret it as a type of acceleration. The above equation says that a curve is a geodesic of $M$ when its acceleration in $N$ is always perpendicular to $M$.

We can relate this back to normal curvature in Section 1.5. In that situation we have $N={ℝ}^3$ and the connection is just ordinary directional derivatives ${\nabla <!--\; FUNCTION\; APPLICATION\; -->}^{\text{euc}}$. Therefore ${\nabla <!--\; FUNCTION\; APPLICATION\; -->}_{{\alpha }^{\prime }}^N{\alpha }^{\prime }={\alpha }^{″}$. The acceleration ${\alpha }^{″}$ of a curve in ${ℝ}^3$ has components in the $T$ and $N$ directions, Equation (1.28). If we project this onto the tangent plane then we get

Hence we see that $\alpha$ is a geodesic of the surface $M$ if and only if $\alpha$ is parameterised with constant speed and $N$, the normal of the curve, is perpendicular to $M$. Since any regular curve can be reparameterised by arc-length, the first condition is only a technical point. If $N$ is perpendicular to $M$ this means that the angle between $N$ and the surface normal $\nu$ is zero. In other words, the normal curvature is equal to the curvature of $\alpha$. Traditionally, one defines the geodesic curvature of $\alpha$ as

This is a measure of how far an arc-length parameterised curve is from being a geodesic. Alternatively, the observation ${\kappa }_n^2+{\kappa }_g^2={\kappa }^2$ (since the normal and geodesic curvatures are projections of the curvature vector) leads us to say a curve is a geodesic if its curvature is entirely normal.

4.2 The Hyperbolic Plane

So far we have seen the example of ${𝕊}^2$ and ${ℝ}^2$. We have the sense that these spaces are special. There is a another important two-dimensional Riemannian manifold: the hyperbolic plane ${ℍ}^2$. Unlike the sphere, a theorem of Hilbert proves that the hyperbolic plane cannot be isometrically immersed into ${ℝ}^3$.² Therefore we really need to use the tools of manifolds and charts to understand this space, there can be no resorting to geometric tricks in euclidean space.

This space was first discovered in connection to the ‘parallel postulate’ of Euclid. Euclid gave five postulates³, which we would call axioms,

• Let it have been postulated to draw a straight-line from any point to any point.
• And to produce a finite straight-line continuously in a straight-line.
• And to draw a circle with any center and radius.
• And that all right-angles are equal to one another.
• And that if a straight-line falling across two (other) straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side).

Clearly one of these is not like the others. The fifth postulate is called the parallel postulate, because it is equivalent to Playfair’s axiom:

There is at exactly one line that can be drawn parallel to another given one through an external point.

There are several interesting geometries that come from replacing this axiom. If there are no parallel lines then we get projective geometry (one intersection) or spherical geometry (two intersections). If there are more than one parallel line then we get hyperbolic geometry.

For our purposes we will introduce the hyperbolic plane by fiat. Like the euclidean plane it can be covered by a single coordinate chart. There are several ‘models’ of the hyperbolic plane, but we will use the ‘half-plane’ model where the geodesics have the easiest formulas.

The first common misconception to address is the ‘boundary’ at $y=0$. As a Riemannian manifold, there is no boundary. If you lived in the hyperbolic plane, as you tried to approach $y=0$ at constant speed you would find that the coordinates were changing at an every decreasing rate. If we explain this using vectors in coordinates, for $y$ small, a unit-length vector has very small coefficients, and therefore moving at unit-speed makes only a small change in coordinates. Conversely, for large $y$, a unit-length vector has large coefficients. This is analogous to looking at the world in the Mercator map, where a plane on the equator changes its longitude far less than a plane near the poles even at the same speed.

Let us compute the Christoffel coefficients for the Levi-Civita connection. Only two derivatives of the metric are non-zero

and the inverse matrix is

Using Equation (3.55)

and the other four are zero.

This means that we can try to solve the geodesic equation. It is certainly possible to solve it directly, but we will solve it in a special case and then use isometries to obtain the full solution, as this is geometrically more interesting. The special case is the following: let $p=(0,1)$ the the starting point of the geodesic and $v=\partial 2$ be its initial direction. If the geodesic is $\alpha (t)=(x(t),y(t))$ then the geodesic formula is

Notice that if $(x,y)$ is a solution to these ODEs, so too is $(-x,y)$. Moreover, $-x(0)=-0=0$ and $-x^{\prime }(0)=-0=0$, so they have the same initial condition. Therefore they must be equal, in other words $x(t)=-x(t)$, which implies $x(t)\equiv 0$. The second ODE now simplifies to

The initial conditions $y(0)=y^{\prime }(0)=1$ force $C=0$. We continue

again using the initial conditions. Thus the unique geodesic through $(0,1)$ with initial vector $\partial 2$ is the vertical line $\gamma (t)=(0,e^t)$.

Already in this argument we already saw a glimpse to the method we will use to find all other geodesics. We used the reflection $R:(x,y)\mapsto (-x,y)$. This preserved the geodesic equations because this is in fact an isometry of ${ℍ}^2$. After we strip away the terminology of Definitions 3.4 and 3.9, what we see is that an isometry is a diffeomorphism, in particular a bijective map, such that the pushforward of vector doesn’t change the inner product. The pushforward of $v\in T_{(x,y)}{ℍ}^2$ by reflection is

for $p=(x,y)$ and $q=R(p)=(-x,y)$, and

because the components of the metric are independent of $x$.

Another obvious isometry is horizontal translation $(x,y)\mapsto (x+a,y)$. From this we can conclude that the geodesic through $(a,1)$ with initial direction $\partial 2$ is $\gamma (t)=(a,e^t)$. These are simply all the vertical lines.

The translation $(x,y)\mapsto (x,y+a)$ is not an isometry. For one thing, it is not even a well-defined map on ${ℍ}^2$, which only has the points of the upper half-plane. But even between points where it is well defined, we see that the metric is changed because $g_{11}(x,y+a)\ne g_{11}(x,y)$.

It turns out that simple scaling of the points $D_r:(x,y)\mapsto (\mathit{rx},\mathit{ry})$ is an isometry. This is called dilation. We compute

Unfortunately, this does tell us any new geodesics, because it takes vertical lines to vertical lines.

We need a different idea. Notice that because the matrix of the metric is a scalar of the euclidean metric (conformal), hyperbolic angles and euclidean angles are equal:

Therefore we should look for transformations of the upper half-plane that preserve euclidean angles (conformal). Perhaps you have encountered circle inversion before. It restricts to give a bijective map on the upper half-plane $S(x,y)={(x^2+y^2)}^{-1}(x,y)$, because $(0,0)\notin {ℍ}^2$. We see that, for $p=(x,y)$, $q=S(x,y)$,

The calculation for $g\left(\mathit{TS}(\partial 2|p),\mathit{TS}(\partial 2|p)\right)$ is exactly the same. That leaves

Therefore $S(x,y)={(x^2+y^2)}^{-1}(x,y)$ is an isometry, not just conformal.

Remarkably circle inversion and the horizontal translations are all the isometries we need to get all the geodesics. By applying $S$ to the geodesic $\gamma (t)=(a,e^t)$ we get the geodesic

Looking on a graphing tool we see that they are semicircles centered on the $x$-axis. This is easy to verify algebraically

In particular we see that we have semicircles of every radius. By horizontal translation then, the set of geodesics includes every vertical line as well as every semicircle centered on the $x$-axis (from now on we will just say semicircle, leaving the centering implicit).

But in fact this accounts for every unit-speed geodesic. Choose any point $p$ and any direction $v$ with $\parallel v{\parallel }_g$. If $v$ is $\pm \partial 2$, then the desired geodesic is the vertical line. For any other $v$, by straightedge and compass (for coolness factor) one can construct a semicircle through that point with the direction as tangent. This must be the unique geodesic. For non-unit-speed geodesics, one can simply rescale $t$, using Lemma 4.6.

These isometries are not only transitive on the set of geodesics, but also on the set of points. This is easily achieved by dilating $(0,1)$ to $(0,y)$ and then a horizontal translation to $(x,y)$. Every point can therefore be mapped to any other by first bringing it to $(0,1)$ and then sending it on its way. Spaces with a transitive set of isometries are called homogeneous. But the hyperbolic plane is also isotropic, it looks the same in every direction. This means that there is a full set of isometric rotations. Begin with $\partial 2|(0,1)$. Dilate and translate it to any point on a different vertical line as above. We know that this vertical line is transformed by circle inversion to a semicircle. A semicircle has every possible tangent direction except $\pm \partial 2$. You can now translate and dilate it back to $(0,1)$. This give every direction except $-\partial 2$. But this can be achieved through two $\pi ∕2$ rotations.

Now that we have determined the geodesics, we can comment on the similarities and differences to other geometries. The first observation is that between any two points there is a unique geodesic. Secondly any two geodesics can intersect at most once. Both of these properties are similar to the euclidean plane but are in contrast to spherical geometry, where antipodal points have infinitely many geodesics between them and every pair of geodesics intersects twice.

We can also consider the parallel postulate. Choose a geodesic and a point $p$ external to it. We know that there is an isometry that will transform the geodesic into the $y$-axis, so we assume this without loss of generality. What are the geodesics through the point that do not cross the $y$-axis? Not only is there the vertical line through $p$, there are infinitely many semicircles. In particular, there is one semicircle through $p$ which is tangent to the $y$-axis. This is called a limiting parallel. The vertical line through $p$ is also a limiting parallel, based on our earlier explanation that as points increase in $y$ value for constant $x$ value, the distance between them shrinks.

A triangle in the hyperbolic plane is of course a shape with three geodesic sides, none of which are parallel. Just as for euclidean geometry, we can try to classify triangles up to isometry, which is called congruence in elementary geometry. By isometry, we may bring any edge of the triangle to the $y$-axis, so we assume that one vertex is at $(0,1)$ and the other $(0,b)$. Let $\alpha$ be the angle at $(0,1)$ and $\beta$ the angle at $(0,b)$. We see that this information already uniquely determines a triangle, thus we have the ‘angle-side-angle’ rule of triangle congruence.

But more is true. There are restrictions on the angles for two other sides to intersect be able to intersect, namely $\alpha +\beta <\pi$, just as for euclidean space. But unlike the classical situation, even if $\alpha +\beta <\pi$ this is no guarantee that the two sides meet. For any $\alpha ,\beta$, there is some $B>1$ such that for $b=B$ the sides are limiting parallels. As $b$ ranges from $B$ down to $1$, the third angle ranges from $0$ (limiting parallel) to $\pi -\alpha -\beta$ (all three corners very close to $(0,1)$, so we can approximate the circles by their tangents). Therefore the sum of angles in a hyperbolic triangle is always less that $\pi$. Also note that the angle increases monotonically. This means there is a 1-to-1 correspondence between $b$ and the third angle. In other words, two hyperbolic triangle are congruent if and only if they have corresponding angles: the ‘angle-angle-angle’ rule!

We close this section with a useful calculation method. It is often profitable to use complex numbers $x+\mathit{\iota y}$ to describe the points of ${ℍ}^2$. The chief advantage is the ease of writing isometries. Clearly horizontal translation and dilations are $z\mapsto z+a$ and $z\mapsto \mathit{rz}$ for $a,r\in ℝ$ respectively. But inversion in the unit circle is $z\mapsto {\overline{z}}^{-1}$ and reflection in the $y$-axis is $z\mapsto -\overline{z}$. As you can see by the presence of complex conjugation, these later two isometries are orientation reversing; they are both reflections after all. Therefore it is common to combine them to an isometry $z\mapsto -z^{-1}$. This along with translation and dilation generate all orientation preserving isometries. A general orientation preserving isometry is therefore

for $a,b,c,d\in ℝ$ with $\mathit{ad}-\mathit{bc}=1$. Transformations of this form are called Möbius transformations. The set of isometries is three-dimensional so it was necessary to normalise the four constants, and using the ‘determinant’ makes the formula for the inverse transform simple.

4.3 Length and Distance

In the above discussion of hyperbolic triangle, we were careful not to speak of the distance between the vertices of triangle because we simply have not yet defined distance on a Riemannian manifold. Let us remedy that situation. In euclidean space, we had a distance function already and in Theorem 1.6 were able to show that the length of a path as in Definition 1.4 was the integral of its speed. But here we have no prior distance function, but the Riemannian metric does give allows us to determine the speed. Therefore we ‘reverse’ these theorems:

There are (at least) two equivalent definitions of the distance function on a Riemannian manifold in the literature, with the difference being over which set of paths the infimum is taken. We have chosen ‘smooth curves’. Perhaps more common is to choose ‘piecewise smooth curves’. A curve is piecewise smooth if it is continuous and the set of non-smooth points is finite. These non-smooth points are called corners. Smooth functions are clearly nicer to work with than piecewise smooth function, if you work with the latter you are forever splitting the curve into its smooth intervals and you must provide correction terms for the corners. The advantage of the piecewise approach is clear if you try to prove the triangle inequality for the distance function. In the piecewise approach it is immediate because the concatenation of a piecewise smooth curve from $p$ to $p^{\prime }$ and from $p^{\prime }$ to $p^{″}$ is again piecewise smooth. However the concatenation of smooth paths will in general not be a smooth path, and some smoothing will be required.

We truly need to take the infimum, even for subsets of euclidean space. Consider $U={ℝ}^2\setminus \{0\}$ and two points $p,-p$. The distance between them is $2\parallel p\parallel$. The unique path in the full plane that achieves this is the straight line, but this is not a path in the manifold, because it would pass through the removed origin. However, by taking paths that pass arbitrarily close to the origin, we see that the distance between these points is still $2\parallel p\parallel$

This leads to an important point: even for manifolds that are Riemannian immersed in euclidean space the distance function of Definition 4.15 is not the restriction of the euclidean distance function. Consider $U={ℝ}^2\setminus \overline{B(0,1)}$. Now the (limiting) shortest path between $p$ and $-p$ skims the boundary of the unit disk An easy estimate shows that any such path is longer than $2\sqrt{1+\parallel p{\parallel }^2}$, which is strictly greater than $2\parallel p\parallel$. Therefore the distance function coming from the Riemannian metric and the restriction of the euclidean distance function to $U$ are different.

These considerations also apply to a general metric space. Given a distance function, we can define the length of a continuous curve as in Definition 1.4. Then we can define a second distance function, called the intrinsic distance function, as in Definition 4.15, as the infimum of the length of paths. The intrinsic distance between points is easily proved to be greater than or equal to their distance. Metric spaces for which the distance function is equal to the intrinsic distance are called length spaces. For euclidean space, one can calculate that the length of a line segment is equal to the distance between the end points, which proves that euclidean space is a length space.

Now that we have enough examples to make us cautious, let us begin to prove that the Riemannian distance function is reasonably behaved. First we show that locally it is equivalent to the euclidean metric of a chart. Then we show that $d^g$ has the three properties of a distance function. The proof that $d^g$ is symmetric is trivial and the proof that it has positivity will be addressed in Corollary 4.20. The triangle inequality Corollary 4.27 requires Corollary 4.26, the ‘corner rounding’ lemma.

Hopefully this is illustrative of the general strategy in this topic: We try to reduce the situation to a single chart, and then in the chart we can compare our metric $g$ with the easily understood euclidean metric on the chart.

We now prepare to prove the triangle inequality for $d^g$. First we give an example of blending two smooth curves.

This formula for $\gamma$ also works for smooth curves $\alpha$ and $\beta$ in euclidean space of any dimension, so long as $\alpha$ is defined for $t<𝜀$ and $\beta$ is defined for $t>-𝜀$. In fact, it is not even necessary for the two curves to meet at $t=0$.

Let us calculate how much the blending increases the length. For convenience, assume that the curves $\alpha ,\beta$ are parameterised by arc-length. The speed can be bounded from above:

which gives

Considering that $\alpha ,\beta$ are unit length parameterised, so the concatenated curve has length $2𝜀$, the length of the blended curve is only slightly larger than the distance between the curves at $t=0$. We can give a more concrete bound in terms of $𝜀$ and that distance:

Together then

It is important to note again that the formula for blending requires that $\alpha$ is defined for $t<𝜀$ and $\beta$ is defined for $t>-𝜀$. So then how can we know that two smooth curves meeting at a point can both be extended past the corner? Well one answer is that Definition 1.2 defines a smooth path as the restriction of a smooth curve to a closed interval. However we give a stronger result that applies to piecewise smooth curves. This demonstrates that our definition is equivalent to the other common one in the literature. The idea is to not try to extend the curves at the corner, we allow ourselves to ‘graft on’ at a nearby point.

Now we can use three blends to remove a corner from a piecewise smooth curve. In fact, performing this operation on smaller pieces of the curve ever closer to the corner results in curves that have almost the same length.