5
$\begingroup$

Is there a similar statement to the constant rank theorem for finite dimensional real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dimensional Banach space and $M$ is a finite dimensional real smooth manifold?

Improve this question
$\endgroup$
8
  • 1
    $\begingroup$ There are no countable dimensional Banach spaces. $\endgroup$
    – Jochen Wengenroth
    Commented May 4, 2015 at 6:58
  • 1
    $\begingroup$ What about separable Hilbert spaces? $\endgroup$
    – Benjamin
    Commented May 4, 2015 at 14:53
  • 3
    $\begingroup$ I meant no countable Hamel basis, of course you can have a countable Schauder basis. I believe, that "x-dimensional" usually refers to Hamel bases. Having a countable Schauder basis is equivalent to separability. $\endgroup$
    – Jochen Wengenroth
    Commented May 5, 2015 at 7:34
  • $\begingroup$ Ok, that's clear and you are of course correct, thanks. $\endgroup$
    – Benjamin
    Commented May 5, 2015 at 14:17
  • 1
    $\begingroup$ @Jochen Wengenroth, a separable Banach space need not have a basis so strictly speaking these two things are not equivalent :-) $\endgroup$
    – Tomasz Kania
    Commented May 5, 2015 at 19:34

3 Answers 3

Reset to default
7
$\begingroup$

Yes, there is. The (Constant) Rank Theorem for Banach spaces is Theorem 2.5.15 of the book of R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications (3rd. edition, Springer-Verlag, 2001). There is a demand that the image of $DF[u_0]$ and the kernel of $DF[u_0]$ are closed direct summands for the $u_0\in B$ around which the theorem holds. The first requirement is automatic if $M$ is finite dimensional (edit - march 24th 2024: as well as the second one since in this case, as pointed out by Tobias Diez's comment to Alexander Schmeding's answer, the kernel of $DF[u_0]$ is finite codimensional).

Improve this answer
$\endgroup$
3
$\begingroup$

There is also a version of the constant rank theorem in Glöckner's paper "Fundamentals of submersions and immersions between infinite-dimensional manifolds" (Theorem F of 1) which works specifically if the target is a finite-dimensional manifold (and the source an arbitrary manifold modeled on a locally convex space). The advantage of having a finite-dimensional target is that one can circumvent most of the tedious assumptions one needs for the case of an infinite-dimensional target (i.e. the ones from the version of Abraham, Marsden and Ratiu).

In 1 you can also find some references to constant rank theorems between Banach spaces. They are given after Theorem F.

Improve this answer
$\endgroup$
3
  • $\begingroup$ Interesting. This means that the hypothesis that $M$ is finite-dimensional also removes the requirement that $\ker DF[u_0]$ should be a closed direct summand. That is very useful. Curiously, Glöckner's paper does not quote the book of Abraham et al. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Jun 6, 2015 at 3:44
  • 1
    $\begingroup$ @PedroLauridsenRibeiro: For a finite-dimensional target, the kernel has finite codimension and hence is automatically complemented. $\endgroup$
    – Tobias Diez
    Commented Sep 6, 2018 at 7:28
  • $\begingroup$ @TobiasDiez true (and sorry for acknowledging this so late), I've added your remark to my answer. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Mar 24 at 21:20
3
$\begingroup$

An improvement to the theorem from the book Manifolds, Tensor Analysis and Applications of R. Abraham, J.E. Marsden and T. Ratiu mentioned above was achieved by J. Blot The Rank Theorem in Infinite Dimension. The assumption from Abraham et al. is that for every $u$ in a neighborhood of $u_0$ the subspace $DF[u]$ is closed and $DF[u]|_{B_1}$ is an isomorphism thereon, where $B_1$ is a closed complement of the kernel $ B_2 = \ker DF[u_0]$. It can actually be weekend to the assumption $$ \operatorname{Range}(DF[u]) \cap M_2 = \emptyset $$ where $M_2$ is the closed complement to $M_1 = \operatorname{Range}(DF[u_0]) $.

Note that there is no condition for $DF[u]|_{E_1}$ to be surjective required. Conversely, Abraham et al.'s requirement implies Blot's requirement: Assuming that the intersection is not empty for arbitrarily small neighborhoods, we find $u_n$'s, $y_n$'s and $x_n$'s with $u_n\to u_0$, $y_n \in \operatorname{Range}(DF[u_n]) \cap B_2\setminus\{0\} $, $x_n \in B_1$, and $DF[u_n](x_n) = y_n $. The set $\{x \in M_2 \mid |x| = 1 \}$ is closed. Hence it has positive Hausdorff distance from $M_1$. Hence it's impossible that the distance of $y_n $ to $M_1$ vanishes as $n\to \infty$. This gives a contradiction.

Improve this answer
$\endgroup$

You must log in to answer this question.