Smooth functions tangent to the leaves of a foliation

2012-10-05T22:28:50Z

Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space $$T_f C^\infty(M,N) = \Gamma(M,f^*TN)$$ of all smooth vector fields $\xi(x) \in T_{f(x)}N$ along $f$.

Suppose we are given a foliation $\mathcal{F}$ of codimension $q$ on $N$ and let $C^\infty_{\mathcal{F}}(M,N)$ be the subset of somooth functions $f:M \to N$ tangent to the leaves. Is $C^\infty_{\mathcal{F}}(M,N)$ still a Fréchet manifold? If it is the case, what is its tangent space at a point?

If it is too general to answer, can we at least say something about the case where $\mathcal{F}$ is a foliation of codimension $1$?

Answer by Michael for Smooth functions tangent to the leaves of a foliation

2012-10-07T17:55:57Z

I don't know about the Frechet structure, but can say a few words about the tangent space: if $f_t:M\to N$ is a time dependent family of maps contained in the leaves of $N$ for each $t$, then $\frac{d}{dt} f^*(\alpha)=0$ for any one-form $\alpha \in \Omega^1(N)$ vanishing on the leaves. So a necessary condition for a relative vector field $X$ along a map $f:M\to N$ to be a tangent vector in the space $C^\infty_{\mathcal{L}}(M,N)$ is $L_X(\alpha)=0$ for all one forms $\alpha$ vanishing on the foliation (here $L_X$ is the lie derivative along relative fields which may be defined by $d\circ i_x+i_x\circ d$).

I think (but am not sure) that this condition is also sufficient. But these tangent spaces might change dimension from one point to the other. Think of a moebius strip with the obvious foliations by circles. The middle circle cannot be deformed into any of the others without breaking it up, so in this case the tangent space are only deformations which stay in the same leave, while the other leaves allow small deformations.

Smooth functions tangent to the leaves of a foliation - MathOverflow

Smooth functions tangent to the leaves of a foliation

Answer by Michael for Smooth functions tangent to the leaves of a foliation