Variation vector field ($V$)

For a variation $F(s,t)$ of a curve $\gamma (t)=F(0,t)$, the variation vector field is $V(t)={\partial }_{s}F(0,t)$. A fixed-endpoint variation has $V=0$ at both endpoints.

Along a geodesic $\gamma$ with tangent $T$, the index form on endpoint-vanishing vector fields is

For a unit-speed geodesic and a fixed-endpoint variation, the second variation of length is the index form of the normal component of its variation vector field.