the second variation formula
Once the first variation is zero you have a critical point, but is it a true bottom of a valley or just a flat spot on a ridge? The second variation answers this. It is the second derivative of the energy (or area) functional along a variation — the Hessian of the functional at a critical point. Where the first variation produced the equation 'be a geodesic / be minimal', the second variation produces a quadratic form on variation fields whose sign decides stability: positive means a strict local minimum in that direction, negative means you can lower the functional, zero is the borderline.
For a geodesic gamma the second variation of energy is the index form I(V, V) = integral over gamma of ( |nabla_{gamma'} V|^2 - g(R(V, gamma') gamma', V) ) dt, for variation fields V vanishing at the endpoints. Read it as a tug-of-war: the first term, the squared covariant derivative, penalizes wiggling and wants positivity; the second term involves the curvature tensor, and where sectional curvature is positive it can make I negative, allowing nearby shorter curves. The associated Euler-Lagrange equation of this quadratic form is the Jacobi equation nabla^2 V + R(V, gamma') gamma' = 0, whose solutions are Jacobi fields; conjugate points are exactly where a nontrivial Jacobi field vanishes again, and past the first conjugate point the geodesic stops being a local minimum. The Morse index theorem then equates the index of I (the dimension of the maximal subspace on which I is negative) with the number of conjugate points counted with multiplicity. For a minimal submanifold the analogous second variation of area is integral ( |nabla V|^2 - |A|^2 |V|^2 - Ric(V, V) ) and its operator is the Jacobi/stability operator.
This is the bridge from variational geometry to spectral theory: stability is governed by an eigenvalue problem, and 'index' is a count of negative eigenvalues. Two honest cautions. First, the curvature convention matters — the sign in front of the curvature term depends on whether your text writes R(X,Y)Z with one sign or the other, so always state your convention before quoting a stability inequality. Second, vanishing second variation (a zero of the index form) does not by itself prove a minimum; degenerate critical points need a third-order or a global argument, and a geodesic can be a local but not global minimizer even with positive second variation.
On a unit sphere, an arc of a great circle of length exactly pi has its endpoints conjugate: a Jacobi field that is zero at both ends exists (push the arc to a neighbouring great circle), the index form is zero on it, and slightly longer great-circle arcs acquire negative index — they are no longer minimizing.
The first conjugate point at distance pi on the unit sphere is exactly where the second variation loses definiteness.
A common slip is to think positive second variation proves a global minimum; it only proves a local one — global minimization is a separate, harder question requiring no shorter competitor anywhere, not just nearby.