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Let $(M, \cal F)$ be a (compact) foliated smooth manifold. I would like to know if there always exists a bundle-like Riemannian metric $g$ for $\cal F$ (i.e. $({\cal L}_U g)(X,Y) = 0$ for all $U \in \Gamma(T\cal F)$ and $X,Y \in \Gamma(T\cal F^{\bot_g})$)?

If not, are there any conditions on $M$ or $\cal F$ (e.g. $\cal F$ is codim. 1) for such a metric to exist?

For instance, in [SC95] it is proven that the set of all bundle-like metrics on a compact foliated manifold is a differentiable infinite dimensional manifold. But I am not sure if this is clearly always non-empty.

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  • $\begingroup$ could you please explain about your notation $\mathcal{L}_{Ug}$? $\endgroup$
    – Ali Taghavi
    Commented Jul 7, 2014 at 19:52
  • $\begingroup$ It is the usual Lie derivative of a tensor $\endgroup$
    – LCC1
    Commented Jul 30, 2014 at 9:38

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It may be empty. Some foliations do not have invariant transversal measure \mu (i.e., such that (LUg)\mu=0), which is a weaker condition. In particular, for codim =1 you may consider the cylinder S^1\times R with a foliation that has the circle S^1\times 0 as a compact leaf while other non-compact leaves (diffeomorphic to R) converge to it.

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  • $\begingroup$ Thank you for your anwer. Fortunately I am interested in codimension 1 foliations on compact manifolds. In light of this, I found the following (maybe classical) result by Morgan: [Mor76]. It remains the question for me (being no expert in foliation theory) if there are maybe geometric or topological requirements on the manifold to have finite holonomy or the stronger holonomy restriction in Morgans paper cited above (Theorem 4 therein). I would appreciate any comment on this. $\endgroup$
    – LCC1
    Commented Sep 26, 2013 at 7:29
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I would have done a comment, but I don't have reputation enough, then this option still block to me.

If $\mathcal{F}$ is a regular foliation on a smooth manifold $M$. We can construct the holonomy groupoid $Hol(\mathcal{F})\rightrightarrows M$, this groupoid is always smooth, but may not be Hausdorff.

There is a recent work of Fernandes and Hoyo about metrics on Lie groupoids, they showed the existence of invariant metrics for proper groupoids, and they showed to that the orbit foliation of this groupoids is a Riemannian Singular Foliation.

From the previous remarks we could set the following result:

Let be $(M,\mathcal{F})$ a regular foliated manifold. If the holonomy groupoid is a proper Hausdorff groupoid, then exist a metric on $M$ such that $\mathcal{F}$ is a Riemannian foliation.

I am not sure, but probably the result holds under the weak hypothesys of finite holonomy.

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