the Morse index theorem
/ MORSS /
When a geodesic is a true shortest path, every small wiggle of it makes it longer — it sits at the bottom of a valley in the space of curves. But once the geodesic is too long, some wiggles make it shorter; it sits on a saddle, and the number of independent 'downhill' directions measures how badly it fails to minimize. The Morse index theorem says that this count of downhill directions equals exactly the number of conjugate points the geodesic has passed, weighted by their multiplicities. It is the precise dictionary between an analytic instability count and a geometric event count.
Make the two sides precise. On one side is the index form I(V, V), the second variation of energy along the geodesic gamma restricted to variation fields V vanishing at both endpoints: I(V, W) = integral of (<V', W'> - <R(V, gamma') gamma', W>) dt. Its index is the dimension of the largest subspace on which I is negative definite — the number of independent directions in which you can shorten gamma. On the other side, count the conjugate points gamma(t) of the starting point p for 0 < t < (the endpoint), each counted with its multiplicity (the dimension of vanishing Jacobi fields). The Morse index theorem states these two numbers are equal, and in particular the index is always finite. The proof compares the infinite-dimensional space of variations with the finite, computable model of broken Jacobi fields.
This is the keystone connecting geodesics to topology. It implies a geodesic minimizes (index 0) exactly until its first conjugate point, recovering the Jacobi criterion. Globally, applied to the loop space of M, it underlies Morse theory of geodesics: critical points are closed geodesics, their indices are conjugate-point counts, and the resulting Morse inequalities relate the number of geodesics to the topology of the loop space — the route by which Bott computed the stable homotopy of the orthogonal and unitary groups. Caveat: the index counts conjugate points strictly inside the interval; whether the endpoint itself is conjugate is a separate, boundary, question (the nullity), and one must fix the convention (sign of R, open vs half-open interval) before the count is meaningful.
Take a geodesic on the unit sphere S^2 starting at the north pole and running down a meridian. While it is shorter than a half great circle (length < pi) it has passed no conjugate point, so its index is 0 — it minimizes. The instant it passes the south pole (length > pi) it has crossed one conjugate point of multiplicity 1, so its index becomes 1: there is exactly one independent way to push it to a strictly shorter curve (slide it off toward the bulge). Run it past the north pole again (length > 2pi) and the index climbs to 2. The index marches up by one (times multiplicity) at each conjugate point.
On the sphere the index of a meridian equals the number of antipodes it has passed — each conjugate point adds one.
The index is the count of conjugate points in the OPEN interval; a conjugate point exactly at the endpoint contributes to the nullity of the index form, not the index. Mixing these up is the most common slip.