2
$\begingroup$

Assume object are smooth at first. If we consider real subbundle, we can define integrability in terms of parameterization or coordinate.

A rank $r$ real subbundle $\mathcal V\le TM$ is called integrable if locally there is a regular smooth parameterization $\Phi(t^1,\dots,t^n)$ such that $\frac{\partial\Phi}{\partial t^1},\dots,\frac{\partial\Phi}{\partial t^r}$ spans $\mathcal V$ in the domain.

Equivalently

A rank $r$ real subbundle $\mathcal V\le TM$ is called integrable if locally there is a smooth coordinate $(x^1,\dots,x^n)$ such that the dual bundle $\mathcal V^\bot$ is spanned by differentials $dx^{r+1},\dots,dx^n$ in the domain.

When we consider complex subbundle, we may not able to use coordinate (or parameterization) language, but we can still use a set of linear independent differential to characterize:

A rank $r$ complex subbundle $\mathcal V\le\mathbb C TM$ is called integrable if locally there are complex-valued functions $f^1,\dots,f^{n-r}$ such that $df^1,\dots,df^{n-r}$ are linear independent at every point in the domain, and $\mathcal V^\bot$ is spanned by $df^1,\dots,df^{n-r}$.

My question is, can we generalize the definition of integrability to $C^\gamma$-complex subbundle for $0<\gamma<1$?

In Holder regularity, we cannot talk about pointwise differential. If the subundle we consider is "coordinate like", then we can define the particular integrable condition of it:

Let $\mathcal V\le TM$ be a $C^\gamma$-real subbundle of rank $r$. We call $\mathcal V$ integrable, if there is a topological parameterization $\Phi(t^1,\dots,t^n)$, such that $\Phi\in C^\gamma$, and $\frac{\partial\Phi}{\partial t^1},\dots,\frac{\partial\Phi}{\partial t^r}$ are continuous and span $\mathcal V$ in the domain.

Here $\frac{\partial\Phi}{\partial t^{r+1}},\dots,\frac{\partial\Phi}{\partial t^n}$ are merely generalized functions, but we don't need to care about them.

It takes some effect, but not too hard, to see that this nonregular parameterization definition coinside with the integrable condition above when $\gamma\ge1$. Indeed we can show that given a $C^\gamma$-nonregular parameterization $\Phi$, where $\frac{\partial\Phi}{\partial t^{r+1}},\dots,\frac{\partial\Phi}{\partial t^n}\in C^{\gamma-1}$ may vanish at some points, we can find a $C^\gamma$-regular parameterization $\Psi$, such that $\partial_1\Psi,\dots,\partial_r\Psi$ do spans $\mathcal V$.

We can put similar condition for almost complex structure, or almost ellpitic structure, or complex Frobenius structure, etc.

But for general complex subbundle, if $\mathcal V+\bar{\mathcal V}$ is not a subbundle, I don't have a good solution.

On the other hand, for involutive condition, which is saying the vector fields on the subbundle is closed under Lie bracket, we can generalize the definition to Holder bundle of regularity $C^\gamma$ for $\gamma>\frac12$. Is that possible to do the similar thing for the integrable condition?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.