Question

hi,

i have the following question: Let $M$ be a n-dimensional manifold (or riemannian or everything thats nice ...) and let $\mathcal{F}$ be a foliation of $M$ by surfaces. Assume, furthermore, that on each leaf $\mathcal{L}$ there is a two form $\alpha_{\mathcal{L}}$. is it possible to construct a globally defined (smooth) 2-form $\alpha$ on $M$ out of the $\alpha_{\mathcal{L}}$? or are there any good books or papers were this is discussed? thanks in advance.

janson

Answer 1

When your manifold is Riemanniann, you can do something obvious: compose $\alpha_{\mathbb{L}}$ with the orthogonal projection from $T_x M$ to $T_x \mathcal{L}$ when $x\in \mathcal{L}$. But you will need some transverse regularity for the $\alpha_{\mathcal{L}}$ to ensure the resulting form to be smooth. This is not really a restriction, since anyway you can easily see that there are families of $\alpha_{\mathcal{L}}$ such that no $2$-forms $\alpha$ on $M$ can be smooth and restrict to $\alpha_{\mathcal{L}}$ along each leaf.

Edit: About the "transverse regularity": if you compute the form $\alpha$ I propose, you'll see that it need not be smooth (or even continuous !); you need to assume that $\alpha_{\mathcal{L}_x}$ (where $\mathcal{L}_x$ is the leaf through $x$), depends smoothly on $x$. This is stronger than asking that each $\alpha_{\mathcal{L}}$ is smooth along $\mathcal{L}$; the difference is mainly about transverse variation of the base point.