Geodesic variation

Let $a<b$, let $\epsilon >0$, and let $F:(-\epsilon ,\epsilon )\times [a,b]⟶M$ be a smooth map, smooth up to the two time endpoints. It is a geodesic variation when for every $s\in (-\epsilon ,\epsilon )$ the longitudinal curve $t\mapsto F(s,t)$ is an affinely parametrized geodesic; equivalently, $D_t{\mathrm{\partial }}_{t}F(s,t)=0.$ The central geodesic is $\gamma (t)=F(0,t)$, and its variation field is $V(t)={{\mathrm{\partial }}_{s}F(s,t)\mid }_{s=0}\in T_{\gamma (t)}M\mathrm{.}$ This is the variation-field construction in Smooth variation and variation field of a curve, with the additional condition that every longitudinal curve is geodesic. No fixed-endpoint condition is imposed: $V(a)$ and $V(b)$ may be nonzero. Constant geodesics are allowed, including on zero-dimensional manifolds; dimension one is covered by the same definition. If $M$ is empty, there is no such map on the nonempty parameter interval, so the definition assigns no variation object.