Geodesics

A geodesic is a curve $\gamma(t)$ whose acceleration is zero: $\nabla_{\dot{\gamma}} \dot{\gamma} = 0$.

In local coordinates, the geodesic equation is:

$\ddot{\gamma}^k + \Gamma^k_{ij}\, \dot{\gamma}^i \dot{\gamma}^j = 0$

This is a second-order ODE, so geodesics exist and are unique given an initial point and velocity.

Proposition 6.1. Geodesics are locally distance-minimizing: for sufficiently small $t$, the geodesic $\gamma$ from $p$ to $\gamma(t)$ has length equal to the Riemannian distance $d(p, \gamma(t))$.

For $p \in M$, the exponential map $\exp_p : T_p M \to M$ is defined by $\exp_p(v) = \gamma_v(1)$, where $\gamma_v$ is the geodesic with $\gamma_v(0) = p$ and $\dot{\gamma}_v(0) = v$.

Proposition 6.2. The exponential map is a local diffeomorphism near the origin: there exists $\varepsilon > 0$ such that $\exp_p$ is a diffeomorphism from $\{v \in T_p M : \|v\| < \varepsilon\}$ onto an open neighborhood of $p$.

A Riemannian manifold $(M, g)$ is geodesically complete if every maximal geodesic is defined for all time (i.e., on $\mathbb{R}$).

Theorem 6.3 (Hopf-Rinow). For a connected Riemannian manifold, the following are equivalent:

1. $M$ is geodesically complete.
2. $(M, d)$ is a complete metric space (where $d$ is the Riemannian distance).
3. Every closed and bounded subset of $M$ is compact.
4. There exists $p \in M$ such that $\exp_p$ is defined on all of $T_p M$.

Corollary. Every compact Riemannian manifold is geodesically complete.

A Jacobi field $J(t)$ along a geodesic $\gamma$ is a vector field that satisfies the Jacobi equation:

$\frac{D^2}{dt^2} J(t) + R(J(t), \dot{\gamma}(t)) \dot{\gamma}(t) = 0$

Jacobi fields describe the variation of nearby geodesics. They measure how geodesics spread apart or come together under the influence of curvature.

Proposition 6.4. A vector field $J$ along $\gamma$ is a Jacobi field if and only if it arises as the variation field of a one-parameter family of geodesics $\gamma_s(t)$ with $\gamma_0 = \gamma$ and $J(t) = \partial_s \gamma_s(t)|_{s=0}$.

Two points $p, q \in M$ are conjugate along a geodesic $\gamma$ if there exists a non-zero Jacobi field along $\gamma$ vanishing at both $p$ and $q$.

Theorem 6.5. A geodesic $\gamma$ ceases to be length-minimizing past its first conjugate point.

Proof sketch. A non-zero Jacobi field vanishing at the endpoints gives a variation that shortens the curve, demonstrating that $\gamma$ is not a local minimum of length. $\blacksquare$

The geodesic deviation equation describes the relative acceleration of nearby geodesics:

$\frac{D^2}{dt^2} J^i = -R^i_{\,jkl} \dot{\gamma}^j J^k \dot{\gamma}^l$

In general relativity, this is the equation of geodesic deviation that governs tidal forces. For a congruence of timelike geodesics, it gives the relative acceleration of nearby test particles.

Example. On $S^2$, geodesics are great circles. Jacobi fields along the equator show that all geodesics starting at the north pole reconverge at the south pole (the antipodal point is conjugate).

Riemannian normal coordinates at $p$ are given by the inverse of the exponential map: $\varphi = \exp_p^{-1} : U \to T_p M \cong \mathbb{R}^n$. In these coordinates:

• The metric at $p$ is Euclidean: $g_{ij}(p) = \delta_{ij}$.
• The Christoffel symbols vanish at $p$: $\Gamma^k_{ij}(p) = 0$.
• Geodesics through $p$ are straight lines through the origin.

Lemma 6.6 (Gauss Lemma). For $v \in T_p M$, $\exp_p$ is a radial isometry: for any $w \in T_v(T_p M) \cong T_p M$:

$\langle d(\exp_p)_v(v), d(\exp_p)_v(w) \rangle = \langle v, w \rangle$

Corollary. Geodesics are locally length-minimizing: for $v$ sufficiently small, $\gamma(t) = \exp_p(tv)$ is the unique shortest curve from $p$ to $\exp_p(v)$.

Problem 1. Find the geodesics of the Poincaré half-plane $\mathbb{H}^2$ with metric $ds^2 = (dx^2 + dy^2)/y^2$.

Solution. The geodesic equations give circles centered on the $x$-axis and vertical lines. These are the paths of minimal length in hyperbolic geometry. $\blacksquare$

Problem 2. Show that geodesics on $S^n$ are great circles.

Problem 3. Compute the Jacobi fields along a geodesic in $\mathbb{R}^n$ with the Euclidean metric. Explain the result in terms of geodesic spread.

Problem 4. Prove that if $M$ is complete and has non-positive sectional curvature, then the exponential map $\exp_p$ is a covering map for every $p \in M$ (Cartan-Hadamard theorem).

Problem 5. Show that on a compact Riemannian manifold, every geodesic is defined for all time.

Geodesics can also be characterized as critical points of the energy functional:

$E(\gamma) = \frac{1}{2} \int_a^b \|\dot{\gamma}(t)\|^2\, dt$

The Euler-Lagrange equations for $E$ give the geodesic equation. The length functional $L(\gamma) = \int_a^b \|\dot{\gamma}(t)\|\, dt$ has the same critical points but is parametrization-independent.

First variation formula:

$\left.\frac{d}{ds}\right|_{s=0} E(\gamma_s) = -\int_a^b \langle V(t), \nabla_{\dot{\gamma}}\dot{\gamma}\rangle\, dt - \sum_i \langle V(t_i), \Delta\dot{\gamma}(t_i)\rangle$

where $V(t)$ is the variation field and $\Delta\dot{\gamma}$ is the jump discontinuity at break points.

Second variation formula:

$\left.\frac{d^2}{ds^2}\right|_{s=0} E(\gamma_s) = \int_a^b \left(\left\|\frac{DV}{dt}\right\|^2 - \langle R(V, \dot{\gamma})\dot{\gamma}, V\rangle\right) dt + \text{boundary terms}$

The second variation is used to study stability of geodesics and to prove that conjugate points indicate loss of minimizing property.

Theorem 6.7 (Morse Index Theorem). The index of a geodesic $\gamma$ (the number of linearly independent Jacobi fields vanishing at the endpoints with a conjugate point in between) equals the number of conjugate points along $\gamma$, counted with multiplicity.

This theorem connects the calculus of variations to the topology of the loop space of a manifold.

On a surface of constant curvature $K$, the area of a geodesic triangle with interior angles $\alpha, \beta, \gamma$ is:

$K \cdot \text{Area} = \alpha + \beta + \gamma - \pi$

• On $S^2$ ($K = 1$): sum of angles $> \pi$, area $= \alpha + \beta + \gamma - \pi$.
• On $\mathbb{H}^2$ ($K = -1$): sum of angles $< \pi$, area $= \pi - (\alpha + \beta + \gamma)$.
• On $\mathbb{R}^2$ ($K = 0$): sum of angles $= \pi$, area arbitrary.

Problem 6. Show that the exponential map $\exp_p$ is a radial isometry near the origin (Gauss lemma). Use this to prove that geodesics are locally length-minimizing.

Problem 7. Prove that on a complete Riemannian manifold with non-positive sectional curvature, no two points are conjugate.

Problem 8. Show that the geodesic flow on the unit tangent bundle of a compact Riemannian manifold is a Hamiltonian flow with respect to the natural symplectic structure.