Jacobi Fields and Conjugate Points

Jacobi fields describe the infinitesimal behavior of families of geodesics and provide the key tool for understanding when geodesics cease to be minimizing.

Jacobi Fields

Definition8.5Jacobi field

A vector field $J$ along a geodesic $\gamma$ is a Jacobi field if it satisfies the Jacobi equation:

\nabla_{\gamma'}^2 J + R(J, \gamma')\gamma' = 0,

where $R$ is the Riemann curvature tensor and $\nabla_{\gamma'}$ denotes covariant differentiation along $\gamma$ . Equivalently, $\frac{D^2}{dt^2} J + R(J, \dot\gamma)\dot\gamma = 0$ .

RemarkGeometric interpretation

Jacobi fields arise as variational vector fields of one-parameter families of geodesics. If $\gamma_s(t)$ is a smooth family of geodesics with $\gamma_0 = \gamma$ , then $J(t) = \frac{\partial}{\partial s}\Big|_{s=0} \gamma_s(t)$ is a Jacobi field along $\gamma$ . Thus Jacobi fields measure how nearby geodesics spread or converge.

ExampleJacobi fields in constant curvature

For a geodesic $\gamma$ on a space of constant sectional curvature $K$ , with $J$ orthogonal to $\gamma'$ and initial conditions $J(0) = 0$ , $J'(0) = w$ (a unit vector):

J(t) = \begin{cases} t \cdot w & \text{if } K = 0 \text{ (Euclidean)}, \\ \frac{\sin(\sqrt{K}t)}{\sqrt{K}} \cdot w(t) & \text{if } K > 0 \text{ (sphere)}, \\ \frac{\sinh(\sqrt{|K|}t)}{\sqrt{|K|}} \cdot w(t) & \text{if } K < 0 \text{ (hyperbolic)}, \end{cases}

where $w(t)$ is the parallel transport of $w$ along $\gamma$ .

Conjugate Points

Definition8.6Conjugate point

Points $\gamma(a)$ and $\gamma(b)$ along a geodesic $\gamma$ are conjugate if there exists a non-zero Jacobi field $J$ along $\gamma$ with $J(a) = 0$ and $J(b) = 0$ . The multiplicity of the conjugate point is the dimension of the space of such Jacobi fields.

ExampleConjugate points on the sphere

On $S^n$ with curvature $K = 1$ , the Jacobi fields vanishing at $t = 0$ are proportional to $\sin(t)$ , which vanishes at $t = \pi$ . So the antipodal point is conjugate (with multiplicity $n-1$ ). This reflects the fact that all geodesics from the north pole reconverge at the south pole.

Conjugate Points and Minimality

Theorem8.3Jacobi's theorem on conjugate points

A geodesic $\gamma: [0, b] \to M$ is not a local minimizer of the energy functional if there exists a conjugate point $\gamma(t_0)$ to $\gamma(0)$ with $0 < t_0 < b$ . Conversely, if there are no conjugate points along $\gamma|_{[0,b]}$ , then $\gamma$ is a strict local minimum of energy.

RemarkRelation to the second variation

The second variation of energy along a geodesic $\gamma$ is

\frac{d^2 E}{ds^2}\Big|_{s=0} = \int_0^b \left(|\nabla_{\gamma'} V|^2 - g(R(V, \gamma')\gamma', V)\right) dt,

where $V$ is the variation vector field (vanishing at endpoints). The Jacobi equation is the Euler-Lagrange equation for this quadratic form. Conjugate points are where this quadratic form becomes degenerate, corresponding to the transition from positive definiteness (minimizing) to indefiniteness.

RemarkMorse index theorem

The Morse index of a geodesic (the number of negative eigenvalues of the second variation operator) equals the number of conjugate points counted with multiplicity. This is the Morse index theorem, linking the variational calculus of geodesics to the topology of the path space.