 Solution

Let $\gamma :[0,\ell ]\to M$ be a unit-speed geodesic, let $F(s,t)$ be a smooth variation with fixed endpoints, and letbe its variation vector field. Then $V(0)=V(\ell )=0$. If $V^{\perp }=V-⟨V,T⟩T$ is its component normal to $\gamma$, the second variation of Riemannian arc length isHere $D_t={\nabla }_T$ is the covariant derivative along $\gamma$, $R$ is the Riemann curvature tensor, and $I$ is the Riemannian index form. Fixed endpoints remove the boundary term. The normal projection removes a tangential change of parametrization, which does not change length to second order.