Conjugate Points and the Index Form

A point conjugate to the initial point of a geodesic is detected by a nontrivial Jacobi field that vanishes at both endpoints. Equivalently, the exponential-map differential is singular there. The index form is the quadratic form governing the second variation of energy; a vanishing Jacobi field supplies its null direction, and passage through a conjugate point changes the geodesic’s local variational character under the appropriate Riemannian or causal hypotheses.

Required background. Affine Connections and Covariant Derivatives supplies the general connection and covariant-derivative framework; this page derives the covariant integration-by-parts identity it uses. The Riemann Curvature Tensor fixes the Jacobi sign.

Helpful background. Calculus and Linear Algebra for Relativity supplies bilinear forms and eigenstructure. Presymplectic Potentials and Currents offers an advanced field-space view of degeneracy in variational systems; the index form is developed here independently.

Conjugate points

Let $\gamma:[0,L]\to M$ be an affinely parametrized geodesic. The endpoint $\gamma(L)$ is conjugate to $\gamma(0)$ along $\gamma$ if a nonzero Jacobi field $J$ satisfies

$$J(0)=0, \qquad J(L)=0.$$

Its multiplicity is the dimension of this space of fields. Since Jacobi fields with $J(0)=0$ encode the differential of $\exp_{\gamma(0)}$, conjugacy is equivalent to singularity of that differential at $L\dot\gamma(0)$. It is a property of the chosen geodesic segment, not merely of the two endpoint events: two points can be connected by several geodesics with different conjugacy behavior.

For a submanifold rather than a point as the initial set, focal points replace conjugate points. The initial Jacobi condition then couples the shape operator of the submanifold to $J(0)$ and $DJ/d\lambda(0)$. Conflating focal and point-conjugate conditions loses this boundary geometry.

Second variation and the index form

For a constant-speed Riemannian geodesic and a variation field $V$ vanishing at both endpoints, the second variation of energy is the index form

$$I(V,W)=\int_0^L \left[ g\!\left(\frac{DV}{d\lambda},\frac{DW}{d\lambda}\right) -g(R(V,u)u,W) \right]d\lambda.$$

The curvature sign follows from the Jacobi equation $D^2J/d\lambda^2+R(J,u)u=0$. Integrating by parts gives

$$I(J,W)= \left[g\!\left(\frac{DJ}{d\lambda},W\right)\right]_0^L -\int_0^L g\!\left(\frac{D^2J}{d\lambda^2}+R(J,u)u,W\right)d\lambda.$$

Here both $V$ and $W$ belong to the space of endpoint-vanishing piecewise smooth fields; its $H^1_0$ completion is also suitable. Thus a Jacobi field vanishing at both endpoints lies in the kernel of $I$: $I(J,W)=0$ for every admissible $W$, a stronger statement than $I(J,J)=0$. Before the first conjugate point, the index form is positive definite in the Riemannian fixed-endpoint problem, so the geodesic is a strict local energy minimizer. A conjugate point at the final endpoint produces degeneracy; it does not by itself destroy minimality. Once a conjugate point lies strictly in the interior, a variation with negative index destroys local minimality; see Grant 2009, Proposition 7.14 and Corollary 7.15, pp. 21–22, PDF.

Sphere check

On $S^n_a$ with $n\ge2$, a unit-speed great circle has transverse Jacobi fields

$$J(\lambda)=a\sin(\lambda/a)E(\lambda),$$

where $E$ is parallel and transverse. The first positive zero after the initial point occurs at $L=\pi a$. There are $n-1$ linearly independent choices of $E$, so the antipode has multiplicity $n-1$.

For $0\lt L\lt\pi a$, no nontrivial transverse Jacobi field vanishes at both endpoints and the segment is the unique minimizing geodesic between its endpoints. At $L=\pi a$, the index form has a kernel, but all the great semicircles still minimize length. Beyond $\pi a$, a shorter arc exists, so the original segment is not minimizing. This example distinguishes degeneracy at the endpoint from loss of minimality after passing it.

Lorentzian qualifications

The metric index form is indefinite on arbitrary vector fields in Lorentzian signature. For a unit timelike geodesic and endpoint-vanishing transverse variations, the second variation of proper time is $-I(V,V)$. Before conjugacy it is negative definite, so the geodesic locally maximizes proper time among nearby timelike curves, apart from reparametrizations. A conjugate point strictly before the final endpoint obstructs that local maximality; conjugacy exactly at the endpoint is a degeneracy requiring further analysis. Null geodesics require a transverse quotient and causal endpoint conditions. Spacelike Lorentzian geodesics can already be length saddles in flat space, as the explicit variation example shows. The causal second-variation statements are developed in Grant 2009, §13.2, pp. 37–41, PDF.

A conjugate point is not a cut point by definition. Conjugacy concerns singularity of $d\exp$ and Jacobi fields; the Riemannian cut point marks the endpoint of global minimizing behavior. On a complete Riemannian manifold, the cut time is no later than the first conjugate time. It can be earlier because another minimizing branch arrives, as on a flat cylinder, or equal, as on a round sphere. A first conjugate point cannot precede the cut time, since extending beyond it already destroys local minimality. Neither event is a curvature singularity: the sphere is smooth at its antipode.

The index-form derivation with the Riemannian hypotheses explicit is developed in Calegari 2022, §§7.6–7.8, pp. 61–65. Lorentzian applications require the stated causal restrictions rather than an unqualified sign transfer.

Why the index changes

In the Riemannian setting, suppose a conjugate point occurs at $\lambda_*\lt L$. Let a nonzero Jacobi field $J$ vanish at $0$ and $\lambda_*$, and extend it by zero to a piecewise smooth field $V$ on $[0,L]$. Then $I(V,V)=0$, but $J'(\lambda_*)\ne0$ by uniqueness of the Jacobi initial-value problem. Choose an endpoint-vanishing field $W$ with $W(\lambda_*)=J'(\lambda_*)$. Integration by parts separately across the corner gives

$$I(V,W)=\lVert J'(\lambda_*)\rVert^2\gt0.$$

Consequently,

$$I(V-\epsilon W,V-\epsilon W) =-2\epsilon\lVert J'(\lambda_*)\rVert^2+\epsilon^2I(W,W)\lt0$$

for sufficiently small positive $\epsilon$. Smooth approximations preserve this strict inequality. This constructs the admissible negative variation and explains why an interior conjugate point is decisive.

Conversely, suppose there is no conjugate point for $0\lt\lambda\le L$. In a parallel orthonormal frame, put the Jacobi solutions with initial data $J(0)=0$, $J'(0)=e_i$ into the columns of a matrix $A$. Then

$$A''+\mathcal R A=0, \qquad A(0)=0,\quad A'(0)=I, \qquad \mathcal R(V)=R(V,u)u.$$

Absence of conjugacy makes $A$ invertible for $0\lt\lambda\le L$. The conserved Jacobi Wronskian gives $A^{\mathsf T}A'=A'^{\mathsf T}A$, so $S=A'A^{-1}$ is symmetric and satisfies $S'+S^2+\mathcal R=0$. These identities turn the integrand into an explicit square:

$$\lVert V'\rVert^2-\langle\mathcal R V,V\rangle =\lVert V'-SV\rVert^2 +\frac{d}{d\lambda}\langle V,SV\rangle.$$

For smooth endpoint-vanishing $V$, the boundary term vanishes at $L$ and tends to zero at $0$, because $A=\lambda I+O(\lambda^3)$ and $V=O(\lambda)$. Thus $I(V,V)\ge0$. Equality requires $V'=SV$, so $V=A c$ for a constant vector $c$; $V(L)=0$ and invertibility of $A(L)$ force $c=0$. This proves positive definiteness without assuming a sign for the curvature. Piecewise smooth fields follow by the same integration on their smooth pieces, and the $H^1_0$ statement follows by approximation.

Multiplicity measures how many independent variation directions become degenerate. On $S^2$ the antipode has multiplicity one; on $S^n$ it has multiplicity $n-1$. The Morse index theorem counts conjugate points strictly inside the interval, with multiplicity, to obtain the negative index of the second-variation form along a Riemannian geodesic. Conjugacy at the final endpoint instead contributes to its kernel, or nullity. This theorem is stronger than locating the first loss of minimality and requires its own function-space setting.

For null geodesics, $u$ is itself orthogonal to $u$, so simply restricting to $u^\perp$ leaves the tangential gauge direction. One works with the screen quotient $u^\perp/\operatorname{span}\{u\}$, or fixes an auxiliary null vector, before counting transverse conjugacy. This prevents an affine reparametrization mode from being mistaken for physical focusing.

These distinctions also control numerical searches. Detecting a zero determinant in a Jacobi matrix locates conjugacy only after the initial conditions and transverse quotient are fixed. A change in which geodesic globally minimizes can occur earlier through a competing branch and will not necessarily appear as a determinant zero. Sampling only the curvature scalar cannot decide either event.

Exercises

Evaluate the index form on $J=a\sin(\lambda/a)E$ along a great semicircle of length $\pi a$. Then use $V=\sin(\pi\lambda/L)E$ on an arc of length $L$ to check the change of sign when $L$ crosses $\pi a$.

Solution

Here $DJ/d\lambda=\cos(\lambda/a)E$ and $R(J,u)u=J/a^2$. Therefore

$$I(J,J)=\int_0^{\pi a} \left(\cos^2(\lambda/a)-\sin^2(\lambda/a)\right)d\lambda=0.$$

Equivalently, integration by parts makes both the Jacobi operator and endpoint term vanish. The zero is the variational degeneracy at the conjugate antipode. The endpoint-vanishing trial field on the general arc gives

$$I(V,V)=\frac L2\left(\frac{\pi^2}{L^2}-\frac1{a^2}\right).$$

It is positive before $\pi a$, zero at $\pi a$, and negative afterward. The negative value is an explicit direction in which the energy decreases; positivity for this one trial field alone would not prove positivity for every variation.

References

• Danny Calegari, Introduction to Differential Geometry, preliminary version, May 26, 2022, University of Chicago. PDF.
• James D. E. Grant, Global Lorentzian Geometry, University of Vienna lecture notes, 2008, corrected February 12, 2009. PDF.

Further reading

• John M. Lee, Introduction to Riemannian Manifolds, second edition, Graduate Texts in Mathematics 176, Springer, 2018. DOI.