Geodesics and the Exponential Map

A geodesic is a curve that does not turn: its velocity is parallel along itself. On the sphere geodesics are great circles, in the hyperbolic half-plane they are vertical lines and semicircles, on a flat torus they are straight lines that wrap around, closing up or winding densely. Short pieces of geodesics are shortest paths, and conversely every shortest path is a geodesic. That makes geodesics the link between the connection of 9A.2 Connections and the distance of 9A.1 Riemannian Metrics and Model Spaces.

Following the geodesics out from a point gives the exponential map, which wraps the flat tangent space around the manifold. Near the point it is a diffeomorphism, and it gives normal coordinates, the best coordinates a curved space allows: Euclidean to first order at the point, with curvature as the first correction. Further out, geodesics from a point can meet again, and the radius up to which they cannot, the injectivity radius, is a measure of how much room the manifold has. A lower bound on it is exactly what Hamilton's compactness theorem needs (11B.3 Compactness of Ricci Flows) and what Perelman's noncollapsing theorem supplies (12A.4 κ-Noncollapsing).

By the end of this chapter you will be able to:

• write and solve the geodesic equation, and find the geodesics of the model spaces;
• use symmetry (Clairaut's relation) to find geodesics of surfaces of revolution;
• define the exponential map and normal coordinates, and state the Gauss lemma;
• prove that geodesics are locally shortest, and state the Hopf–Rinow theorem;
• define cut points, conjugate points and the injectivity radius, and compute them for spheres and flat tori.

3.1Flying over the pole

3.2The geodesic equation

A curve $\gamma$ is a geodesic if its acceleration vanishes:

This is a second-order system of ODEs, so by 2B.10 Ordinary Differential Equations, for every $p \in M$ and $v \in T_pM$ there is a unique maximal geodesic $\gamma_v$ with $\gamma_v(0) = p$ and $\gamma_v'(0) = v$. Geodesics have constant speed, because parallel transport preserves lengths: $\frac{d}{dt}|\gamma'|^2 = 2\langle D_t\gamma', \gamma'\rangle = 0$. Rescaling the parameter rescales the velocity: $\gamma_{cv}(t) = \gamma_v(ct)$. Isometries take geodesics to geodesics.

For a submanifold of $\mathbb{R}^N$, $D_t\gamma' = (\gamma'')^\top$ (9A.2 Connections), so a geodesic is a curve whose acceleration in space is normal to the submanifold: a particle constrained to the surface and feeling no force but the constraint. On the sphere, great circles have acceleration pointing to the centre, normal to the sphere, so they are geodesics; by uniqueness they are all of them.

Hyperbolic half-plane. For $g = \frac{dx^2 + dy^2}{y^2}$, the geodesics are the vertical half-lines and the semicircles centred on the real axis (Exercise 3.4). In the disc model they are diameters and circular arcs meeting the boundary at right angles. Through a point not on a given geodesic pass infinitely many geodesics that never meet it: Euclid's parallel postulate fails.

Symmetry and Clairaut's relation. If a Killing field $X$ (the generator of a one-parameter group of isometries, 8A.6 Flows and the Lie Derivative) is present, then $\langle\gamma', X\rangle$ is constant along every geodesic $\gamma$ (Exercise 3.5). For a surface of revolution $dr^2 + \varphi(r)^2d\theta^2$, the rotation field $X = \partial_\theta$ has $|X| = \varphi$, and if $\psi$ is the angle between $\gamma'$ and the meridian, $\langle\gamma', \partial_\theta\rangle = |\gamma'|\varphi\sin\psi$. So along a unit-speed geodesic

Clairaut's relation: the radius of the circle of symmetry times the sine of the angle with the meridian is conserved. On a dumbbell, a geodesic entering the neck at an angle must turn steeper as $\varphi$ shrinks, and if $\varphi$ falls below the constant it turns back. The optical rays of 9A.1 Riemannian Metrics and Model Spaces obey the same law, Snell's law.

3.3The exponential map and normal coordinates

The exponential map at $p$ is

defined wherever the geodesic $\gamma_v$ lasts until time $1$. So $\exp_p(tv) = \gamma_v(t)$: it maps lines through the origin to geodesics through $p$. Its differential at $0$ is the identity of $T_pM$ ($d\exp_p(0)v = \gamma_v'(0) = v$), so by the inverse function theorem (2B.9 The Inverse and Implicit Function Theorems), $\exp_p$ maps some ball $B_\varepsilon(0) \subset T_pM$ diffeomorphically onto a neighbourhood of $p$, a normal neighbourhood.

Choosing an orthonormal basis $e_1, \dots, e_n$ of $T_pM$ and setting $x \mapsto \exp_p(x^ie_i)$ gives normal coordinates centred at $p$. In them:

• the geodesics through $p$ are the straight lines $t \mapsto tv$;
• $g_{ij}(p) = \delta_{ij}$ and all $\partial_kg_{ij}(p) = 0$, so $\Gamma_{ij}^k(p) = 0$ (Exercise 3.6).

Normal coordinates make the metric Euclidean to first order at one point, and no coordinates can do better: the second derivatives of $g_{ij}$ at $p$ are curvature. On the unit sphere, normal coordinates at the north pole are the azimuthal equidistant projection (the projection of the United Nations emblem), in which $g = dr^2 + \sin^2r\,d\theta^2$; compared with the Euclidean $dr^2 + r^2d\theta^2$, lengths across the radial direction are shrunk by $\frac{\sin r}{r} = 1 - \frac{r^2}{6} + \dots$, a second-order effect. In general, for a vector $v$ perpendicular to the position $x$ in normal coordinates,

where $K(x, v)$ is the sectional curvature of the plane spanned by $x$ and $v$ (9A.4 Curvature and What It Means defines it and states the full tensor expansion).

3.4Completeness

$(M, g)$ is geodesically complete if every geodesic is defined for all time, that is, $\exp_p$ is defined on all of $T_pM$ for every $p$.

The proof (Lee, chapter 6) is a careful use of the local minimising property and of compactness (2B.3 Compactness). Every compact Riemannian manifold is complete. The punctured plane is not: the geodesic heading into the puncture stops, there is no shortest curve between $(-1, 0)$ and $(1, 0)$, and Cauchy sequences converging to the origin have no limit. The Ricci flow is almost always studied on complete manifolds, and the limits of blow-ups (11B.3 Compactness of Ricci Flows) are taken to be complete.

3.5Cut points and the injectivity radius

A geodesic from $p$ stops minimising at some point. On the sphere, the meridian from the north pole minimises up to the south pole, at distance $\pi$, and not beyond: past the south pole the other half of the great circle is shorter. The cut point of $p$ along a unit-speed geodesic $\gamma$ from $p$ is $\gamma(t_0)$, where $t_0$ is the last time at which $\gamma|_{[0, t_0]}$ minimises; the cut locus $\operatorname{Cut}(p)$ is the set of all cut points of $p$. On the round sphere $\operatorname{Cut}(p)$ is the antipode. On a flat torus it is a graph made of the edges of the Voronoi cell of $p$'s lattice of lifts.

A geodesic can stop minimising for two reasons. Either a different geodesic of the same length arrives at the same point (a loop-closing phenomenon, as on the torus), or nearby geodesics from $p$ refocus at the point: it is a conjugate point, where $d\exp_p$ fails to be invertible (9A.7 Jacobi Fields and Curvature versus Topology). The south pole is conjugate to the north pole: the whole family of meridians refocuses there.

The injectivity radius at $p$ is

and $\operatorname{inj}(M) = \inf_p\operatorname{inj}(p)$, which is positive for compact $M$. For the unit sphere it is $\pi$; for $\mathbb{RP}^n$ with the round metric, $\frac{\pi}{2}$; for $\mathbb{R}^n$ and $\mathbb{H}^n$, infinite; for the flat torus $\mathbb{R}^2/(a\mathbb{Z}\times b\mathbb{Z})$, $\frac12\min(a, b)$, half the length of the shortest closed geodesic (Exercise 3.7). A short closed geodesic forces a small injectivity radius, and a very thin flat torus has small injectivity radius everywhere although its curvature is zero.

3.6History

Geodesics on surfaces were studied by Johann Bernoulli and Euler in the eighteenth century as solutions of a variational problem, and Clairaut found his relation for surfaces of revolution in the same period. The exponential map and normal coordinates go back to Riemann's 1854 lecture, and Gauss's lemma to his 1827 Disquisitiones generales circa superficies curvas. Heinz Hopf and Willi Rinow proved their theorem in 1931. Cut loci were studied by Poincaré (1905) for convex surfaces and by J. H. C. Whitehead (1935) in general. Buckminster Fuller's geodesic domes, patented in 1954, take their name from the great circles along which their frames lie.

3.7Exercises

Solution

With $g_{xx} = g_{yy} = y^{-2}$, $\partial_yg_{xx} = \partial_yg_{yy} = -2y^{-3}$. The formula of 9A.2 Connections gives $\Gamma_{xy}^x = \frac12y^2(-2y^{-3}) = -\frac1y$, $\Gamma_{xx}^y = -\frac12y^2\partial_yg_{xx} = \frac1y$, $\Gamma_{yy}^y = \frac12y^2\partial_yg_{yy} = -\frac1y$. The equations are $\ddot x - \frac2y\dot x\dot y = 0$ and $\ddot y + \frac1y(\dot x^2 - \dot y^2) = 0$. For $x$ constant, the second is $\ddot y = \dot y^2/y$, solved by $y = e^t$. For $x = \tanh t$, $y = \operatorname{sech}t$: $\dot x = \operatorname{sech}^2t$, $\dot y = -\operatorname{sech}t\tanh t$, and both equations check by differentiation; $x^2 + y^2 = 1$, so it is the unit semicircle, with speed $\frac{\sqrt{\dot x^2 + \dot y^2}}{y} = \frac{\operatorname{sech}t}{\operatorname{sech}t} = 1$.

Solution

$\frac{d}{dt}\langle\gamma', X\rangle = \langle D_t\gamma', X\rangle + \langle\gamma', \nabla_{\gamma'}X\rangle = 0 + 0$, the second by the Killing equation with $Y = Z = \gamma'$. For $X = \partial_\theta$ this is the conservation of $\varphi\sin\psi$, as in the text. Meridians: $\theta$ constant, $r = t$; the equations $\ddot r - \varphi\varphi'\dot\theta^2 = 0$ and $\ddot\theta + 2\frac{\varphi'}{\varphi}\dot r\dot\theta = 0$ hold with $\dot\theta = 0$, $\ddot r = 0$. A circle $r = r_0$ with $\theta = t$ needs $\varphi(r_0)\varphi'(r_0) = 0$, so it is a geodesic exactly where $\varphi'(r_0) = 0$, at the equator or the waist of a neck.

Solution

For $\gamma(t) = tv$, $\ddot\gamma = 0$ and the equation gives $\Gamma_{ij}^k(tv)v^iv^j = 0$; at $t = 0$ this holds for every $v$, and a symmetric bilinear form vanishing on the diagonal vanishes (polarisation), so $\Gamma_{ij}^k(0) = 0$. Then $\partial_kg_{ij}(0) = 0$ from the stated identity.

Solution

(a) The geodesics from $p$ are great circles, minimising up to the antipode at distance $\pi$; $\exp_p$ is a diffeomorphism from the open disc of radius $\pi$ onto $S^2$ minus the antipode, and the whole circle $|v| = \pi$ goes to the antipode. (b) $\exp_p(v) = p + v \bmod$ the lattice; $v \neq w$ have the same image iff $v - w$ is a nonzero lattice vector, of length at least $a$. If $|v|, |w| < \frac a2$ then $|v - w| < a$, so $\exp_p$ is injective; but $v = \frac a2e_1$ and $w = -\frac a2e_1$ have the same image. (It is also a local diffeomorphism everywhere, being a translation.) (c) $\mathbb{RP}^2 = S^2/\pm$: geodesics from $[p]$ are images of great circles, and the geodesics in directions $v$ and $-v$ meet again after length $\frac{\pi}{2}$, at the image of the point on the equator between them; before that the quotient map is injective on the hemisphere's interior.

Solution

The geodesic $t \mapsto (1 - t, 0)$ ends at $t = 1$; the sequence $(\frac1k, 0)$ is Cauchy without a limit; the closed bounded set $\{0 < |x| \leq 1\}$ is not compact. The distance between $(\pm1, 0)$ is $2$ (curves can pass just above the origin), but the only straight segment of length $2$ goes through the origin, and every geodesic in the punctured plane is a straight segment.

Solution

In Cartesian form, $dr^2 + r^2d\theta^2 = |dx|^2$ and $r^2d\theta^2 = |dx|^2 - dr^2$, with $dr = \frac{x_i\,dx^i}{|x|}$. So $g = dr^2 + \frac{\sin^2r}{r^2}(|dx|^2 - dr^2)$, and $\frac{\sin^2r}{r^2} = 1 - \frac{r^2}{3} + O(r^4)$, giving $g = |dx|^2 - \frac{|x|^2}{3}\big(|dx|^2 - \frac{(x_i\,dx^i)^2}{|x|^2}\big) + O(|x|^4)$, the stated formula. For $v \perp x$ it gives $|v|^2(1 - \frac13|x|^2)$, the text's formula with $K = 1$; for $v \parallel x$ there is no correction, as the Gauss lemma requires.