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Suppose that a Lie group $G$ acts smoothly on a manifold $X$. We can easily prove the following theorem by using the constant rank theorem, which is a stronger theorem than the implicit function theorem.

Theorem A. Let $x$ be a point of $X$ and define $F:G \to X$ by $F(g)=gx$. Then there are open neighbourhoods $U \subset G$ of the identity and $V \subset X$ of $x$ such that $F(U)$ is a closed submanifold of $V$.

Questions:

(i) Is it possible to prove Theorem A by using the implicit function theorem (without applying the constant rank theorem)?

(ii) Atiyah-Bott use an infinite-dimensional version of Theorem A in §14: Fix a complex (smooth) vector bundle $E$ over a closed Riemann surface $M$ with a Hermitian metric. Let $G$ be the group $(\mathcal{G}^c)^k$ of complex gauge transformations of class $L^2_k$ and $X$ the space $\mathcal{A}^{k-1}$ of unitary connections of class $L^2_{k-1}$ ($k \geq 2$). According to Atiyah-Bott, we can prove Theorem A in this case by using the implicit function theorem for Banach manifolds. (We may need the Fredholm property of the derivative $dF$.)

How do we prove Theorem A in this case?

(iii) Do we have an infinite-dimensional version of the constant rank theorem (which can be applied to Question (ii))?

H. Shindoh
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  • for $iii)$, I hope you get more answers than http://math.stackexchange.com/questions/377157/infinite-dimensional-constant-rank-theorem – Albert May 27 '13 at 10:06
  • So far I can not find any helpful discussion in the book by Lang. – H. Shindoh May 27 '13 at 15:43
  • I think there is an infinite dimensional version of the constant rank theorem, under assumption of the existence of suitable topological "supplementaries" (unsure of the english terminology, sorry ; it's related to what you say about the Fredholm property). it is not mentionned in Lang's book, but I believe the proof of the classical constant rank theorem can be carried just the same (it uses the immersion and submersion lemmas for Banach spaces, which are proved in Lang). I have not thought about how to apply it to your problem. – Albert May 27 '13 at 16:07

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Just a partial answer to your third question:

There are versions of the constant rank theorem which could be applicable to your situation:

Some versions of the constant rank theorem between Banach spaces can be found in

Margalef-Roig, J. and E. Outerelo Dominguez, “Differential Topology,” North-Holland, Amsterdam, 1992.

Also a version of the constant rank theorem is contained as Theorem F in https://arxiv.org/pdf/1502.05795.pdf

Alexander Schmeding
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