TITLE: Horizontal distribution of a totally geodesic foliation
QUESTION [7 upvotes]: Let $\mathbb{M}$ be a $n+m$ dimensional manifold. Consider on $\mathbb{M}$ a rank $n$ sub-bundle $\mathcal{H}$ of the tangent bundle. We assume that $\mathcal{H}$ is endowed with a fiber wise inner product $g_\mathcal{H}$.
Under which conditions on $(\mathbb{M}, \mathcal{H}, g_\mathcal{H})$ can we find a totally geodesic Riemannian  foliation $(\mathcal{F},g)$ on $\mathbb{M}$ such that:
1) $g$ is bundle-like, 
2) $g$ restricts to $g_\mathcal{H}$ on $\mathcal{H}$
3) The leaves of the foliation are orthogonal to $\mathcal{H}$ ?

REPLY [2 votes]: Here is a necessary condition. I have not checked if it is sufficient.
If $\mathcal F$ is totally geodesic, then translation along horizontal paths give local isometries, so you have the condition
$$\mathcal L_H g^{\mathcal F}=0\;.$$
Here, $H$ is a horizontal vector field that looks like a horizontal lift of a vector field on the base, so it is parallel along leaves with respect to the Levi-Civita connection, and $g^{\mathcal F}$ is the restriction of the total metric to $\mathcal F$.
We have of course used that the flow of $H$ maps local leaves to local leaves because $g$ is bundle-like.