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We first mention our motivation: For $M=\mathbb{R}$ with usual Riemannian metric the exp map $exp:TM\to M$ is in the form$(x,v)\mapsto x+v$ The level sets of this map define a foliation whose leaves are totally geodesic submanifolds with respect to Sasaki metric of $TM$. 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 points $x,y$ in $M\subset TM$ and every $M$- geodesic curve $\gamma $ joining $x$ to $y$, the tangent space of the leaf passing $x$ is parallel transported along $\gamma$ 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 parts of these situations are the case?

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I think that the appropriate generalization is that of a tubular neighborhood of a codimension 1 hypersurface in $M$. If the hypersurface $P$ of $M$ is compact, then there exists an open set $U$ in $M$ which contains $P$ and the tangent bundle admits an orthogonal decomposition $$TU=TP \oplus NP$$ where $NP$ is the normal bundle of $P$. An analagous statement to the Gauss lemma shows that this induces a foliation of $U$ whose leaves are hypersurfaces "parallel" to $P$. For a deeper look, I would suggest looking at Lee's new edition of Riemannian manifolds, specifically the discussion of semigeodesic coordinates.

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    $\begingroup$ Thank you very much for your answer, your reference to Lee's book and your attention to my question. $\endgroup$ – Ali Taghavi Feb 26 at 15:34

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