Geodesics | Mathematics - Wyatt's Notes

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:

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

as a rule relativity, this is the equation of geodesic deviation that governs tidal forces. For a congruence of timelike geodesics in the standard model of cosmology, 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.

Mistake 1: Assuming geodesics are always the shortest path Geodesics are locally length-minimizing, but globally they may not be the shortest curve between two points. On a sphere, great circle arcs longer than $\pi$ are geodesics but not shortest paths. The cut locus marks where geodesics cease to be minimizing.

Mistake 2: Confusing the exponential map with a global diffeomorphism The exponential map $\exp_p$ is only a local diffeomorphism near the origin. Beyond the injectivity radius, it may fail to be one-to-one. Students often assume $\exp_p$ gives a global coordinate system, which is only true for spaces like Euclidean space or hyperbolic space.

Mistake 3: Forgetting that the geodesic equation is second-order The equation $\ddot\gamma^k + \Gamma^k_{ij}\dot\gamma^i\dot\gamma^j = 0$ requires both an initial point $p$ and an initial velocity $v$ to determine a unique geodesic. Specifying only the starting point leaves infinitely many geodesics. Students sometimes treat geodesics like integral curves of vector fields, which are first-order and need only an initial point.

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.

flowchart TD
    A[6_Geodesics] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Geodesics are the straightest possible paths on a curved surface. On a flat plane, they are straight lines; on a sphere, they are great circles. The key insight is that a geodesic has zero acceleration as measured along the surface — it does not turn left or right relative to the surface geometry. The exponential map sends a tangent vector to the point you reach by following the geodesic in that direction for unit time, providing local coordinates that straighten out the geometry near a point. On negatively curved surfaces, geodesics diverge rapidly, while on positively curved surfaces they converge — this is why parallel geodesics on a sphere eventually meet.

• Vector Fields and Flows — Geodesics are integral curves of the geodesic spray vector field on the tangent bundle, linking flow theory to metric geometry.

• Differential Geometry Summary — The summary table of Christoffel symbols and curvature tensors provides the computational tools used throughout this chapter.

• Electromagnetic Waves — The wave equation in curved spacetime uses geodesic deviation to describe tidal forces as a rule relativity.

• Special Relativity and Electromagnetism — Minkowski spacetime is a flat Lorentzian manifold whose geodesics are the worldlines of free particles.

• Quantum Mechanics

• Graph Theory

• Classical Mechanics

• Electromagnetism