inverse exponential map atγq,p(t)satisfiesXt(q,p)= (1−t) ̇γq,p(t)
$$ \tilde{\gamma}_{\gamma_{q,p}(t),p}(s) = \gamma_{q,p}(t + s(1-t)), s \in [0,1] $$
$$ \Longrightarrow X_t(q,p) := \dot{\tilde{\gamma}}_{\gamma_{q,p}(t),p}(s)\vert_{s=0} = (1 - t) \dot{\gamma}_{q,p}(t) $$
aRiemannian metricgonM. Given such a metric, we assumeintegrabilityofXand∇, respectively,in the sense that for allpthere exists a functionDpsatisfyingX(q,p) =−gradqDp
pointspandq, one can interpret anyXwith expq(X) =pas a difference vectorXthattranslatesqtop
p−q=−gradqDp(9)Here, the gradient gradqis taken with respect to the canonical inner product onRn
De outra forma, podemos postular que a divergência canônica é a solução da edp: $$ D_p(q) = {1 \over 2 } |grad_q D_p|^2 $$
fixed pointp∈M, we want to define a vector fieldq7→X(q,p), at least in a neighbourhood ofp, thatcorresponds to the difference vector field