Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $M$ be a smooth manifold and $G_k(M)$ be the $k$-dimensional Grassmian bundle of $M$. Let $K\subset M$ be a compact subset and $E:K\to G_k(M)$ be a continuous distribution on $K$.

We say $E$ is integrable on $K$ if there exists a foliation $\mathcal{F}$ (or lamination, since it may only foliates a subset of $M$), such that $T_x\mathcal{F}(x)=E_x$.

My quesion is: if $(E,K)$ is integrable, will there exist an open neighborhood $U\supset K$ that admits an integrable extension $\widetilde{E}:U\to G_k(M)$?


For example $K$ is a hyperbolic invariant set of a diffeomorphism $f:M\to M$. It is known that there are stable and unstable foliations (manifolds) through $K$. I donot know if we can extend the foliations to an open neighborhood of $K$.

share|improve this question

2 Answers 2

up vote 2 down vote accepted

The answer in general is no.

If $K$ is a submanifold of $M$ then tangent bundle of $K$ defines an integrable distribution on $K$. To wit we are talking about the foliation with just one leaf: $K$.

If we can extend this foliation to a neighborhood of $K$ then the restriction of Bott's connection to $K$ induces a flat connection on the normal bundle of $K$. This imposes restrictions on $K$. For instance $K$ cannot be a projective curve in $\mathbb P^2(\mathbb C)$.


Recall that Bott's connection is the partial connection on the normal bundle of $\mathcal F $ defined as follows. If $T\mathcal F$ is the tangent bundle of a foliation and $N\mathcal F$ is its normal bundle then they fit into the exact sequence: $$ 0 \to T \mathcal F \to TM \stackrel{\pi}{\to} N\mathcal F \to 0 . $$ Now, if $v$ is a local section of $T\mathcal F$ and $w$ is a local section of $N\mathcal F$ then Bott's partial connection $\nabla : T\mathcal F \times N\mathcal F \to N \mathcal F$ is defined by the formula
$$ \nabla_{v}(w) = \pi ([v, \pi^{-1}(w)]) $$ where $\pi^{-1}(w)$ is an arbitrary lifting of $w$ to $TM$. The involutiviness of $T\mathcal F$ implies that $\nabla$ is well-defined. It is not hard to check that $\nabla^2=0$.

share|improve this answer

Goggle my paper "on extending local foliations" in the quarterly journal in 1977. I addressed the following general question: given a compact smooth manifold X and a finite collection of codimension q embedded submanifolds each with trivial normal bundles when could the local product foliations be extended to a global foliation. While in general the answer is no, there are a number of positive results.

share|improve this answer
2  
Here is a link: qjmath.oxfordjournals.org/content/28/2/163.extract Dr. Golbus, could you name a specific result in your paper that answers or otherwise addresses the question of the poster? –  Todd Trimble May 12 at 0:39

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.