Let be $M$ a Riemannian manifold and $X,Y$ vector fields over $M.$
Now take $p\in M$ arbitrarily, my question is, how construc a variation $f:U\to M,$
$$U\subset \mathbb{R}^2,~~U=\{-\epsilon
$$\dfrac{\partial f}{\partial t}(0,1)=X(p),~~~\dfrac{\partial f}{\partial s}(0,1)= Y(p)~~~~~~ ??$$