We first mention our motivation: For $M=\mathbb{R}$ with usual Riemannian metric the exp map $exp:TM\to M$ is in the form$(x,v)\mapsto x+v$ This map defines a foliation whose leaves are totally geodesic submanifolds with respect to Sasaki metric. They intersect the zero section transversaly. The number of intersections with the zero section is the same for all leaves. They are parallel in the sense that for every two two points $x,y$ in $M\subset TM$ and every geodesic curve joining $x$ to $y$ contained in zero section, the tangent space of the leaf passing $x$ is parallel transported to thats of $y$.

How can one generalize some or all part of this situation in the case of an arbitrary (compact) Riemannian manifold? Under which conditions on Riemannian manifold, all or some part of these situation are true?