On a sphere of radius 1 (constant curvature K = 1), the Jacobi equation along a geodesic becomes J'' + J = 0. The solutions are J(t) = A sin(t) + B cos(t). This means Jacobi fields vanish (have conjugate points) at t = π. What does this imply geometrically?
AGeodesics on the sphere are undefined past t = π
BAll geodesics from a point refocus at the antipodal point (distance π), and geodesics beyond that distance are no longer minimizing
CThe curvature changes sign at t = π
DThe exponential map becomes complex-valued at t = π
The Jacobi field J(t) = sin(t) vanishes at t = 0 and t = π. Geometrically, this means a one-parameter family of geodesics from a point p (say, lines of longitude from the north pole) all meet again at the antipodal point (south pole), which is the conjugate point at distance π. Beyond the conjugate point, the geodesic is no longer minimizing — there are shorter paths going 'the other way.' This is why great-circle arcs longer than half the circumference are not shortest paths.
Question 2 True / False
In a Riemannian manifold with non-positive sectional curvature (K ≤ 0), there are no conjugate points along any geodesic.
TTrue
FFalse
Answer: True
The Jacobi equation J'' + R(J,γ')γ' = 0 has the curvature acting as a 'restoring force.' When K ≤ 0, the force pushes Jacobi fields AWAY from zero (the term R(J,γ')γ' has the 'wrong sign' for oscillation). A Jacobi field that starts at zero grows monotonically and never returns to zero — so there are no conjugate points. By the Cartan-Hadamard theorem, this implies the exponential map is a covering map, and the universal cover of M is diffeomorphic to ℝⁿ.
Question 3 Short Answer
Why are Jacobi fields important for understanding the second variation of arc length?
Think about your answer, then reveal below.
Model answer: The second variation of the energy/length functional of a geodesic involves the Jacobi operator (the linear differential operator J ↦ J'' + R(J,γ')γ'). A geodesic is a local minimum of length if and only if the second variation is non-negative, which happens if and only if there are no conjugate points along the geodesic. Jacobi fields that vanish at both endpoints (conjugate points) are the 'zero modes' of the second variation — they represent directions in which the geodesic can be varied without changing the length to first order but where the second-order change is zero or negative.
This is the Riemannian analogue of the second-derivative test in calculus. A geodesic is a critical point of the length functional (first variation = 0). The Jacobi fields determine whether it is a local minimum (no conjugate points), saddle point (conjugate points exist), or degenerate critical point. The Morse index theorem counts the number of conjugate points (with multiplicity) along a geodesic segment, which equals the number of negative eigenvalues of the second variation.