[Paper Review] Tame Fréchet submanifolds

Walter Freyn|arXiv (Cornell University)|May 14, 2013

Advanced Numerical Analysis Techniques5 references3 citations

TL;DR

This paper introduces tame Fréchet submanifolds of co-Banach type in tame Fréchet manifolds, establishing that inverse images of regular values under certain tame maps are naturally tame Fréchet submanifolds. The key contribution is a functional analytic framework enabling the construction of such submanifolds via an implicit function theorem in the tame Fréchet setting, with applications to Kac-Moody geometry and isoparametric submanifolds.

ABSTRACT

We introduce the new class of submanifolds of co-Banach type in tame Fréchet manifolds and construct tame Fréchet submanifolds as inverse images of regular values of certain tame maps. Our method furnishes an easy way to construct tame Fréchet manifolds. The results presented are key ingredients in the construction of affine Kac-Moody symmetric spaces; they have also important applications in the study of isoparametric submanifolds in tame Fréchet spaces.

Motivation & Objective

• To develop a foundational framework for submanifolds in tame Fréchet manifolds, particularly in non-Banach ambient spaces.
• To address the lack of a viable inverse function theorem in general Fréchet spaces by restricting to tame Fréchet structures.
• To provide a constructive method for generating tame Fréchet submanifolds using regular values of tame maps.
• To establish the existence of tame Fréchet submanifolds in settings relevant to affine Kac-Moody symmetric spaces and infinite-dimensional differential geometry.
• To generalize finite-dimensional submanifold constructions—such as inverse image of regular values—to the tame Fréchet setting.

Proposed method

• Introduce the notion of a tame Fréchet regular point, requiring the differential to be surjective and the kernel to be complemented in a Banach space.
• Define a tame Fréchet regular value as a point in the target space such that all preimages are tame regular points.
• Use the implicit function theorem in the tame Fréchet category, relying on the existence of a tame Fréchet isomorphism between local charts.
• Construct a decomposition of the ambient tame Fréchet space into the kernel of the differential and a complemented Banach subspace.
• Apply the implicit function theorem to the map $\Phi: F_0 \times \overline{B} \to F_0 \times B $, defined by $ (x,y) \mapsto (x, \varphi(x,y)) $, to show that the preimage is a tame Fréchet submanifold.
• Verify that the resulting submanifold inherits a tame Fréchet structure via local charts with quasi-isometric transition functions.

Experimental results

Research questions

• RQ1 Can the inverse image of a regular value under a tame Fréchet map be shown to be a tame Fréchet submanifold?
• RQ2 What conditions ensure that the kernel of the differential of a tame map is complemented in a Banach subspace?
• RQ3 How can the implicit function theorem be adapted to the tame Fréchet setting to construct submanifolds?
• RQ4 In what geometric contexts do tame Fréchet submanifolds of co-Banach type naturally arise?
• RQ5 Can isoparametric submanifolds in Hilbert spaces be extended to tame Fréchet spaces via intersection with the Fréchet space?

Key findings

• The inverse image $ \varphi^{-1}(g) $ of a tame Fréchet regular value $ g \in N_B $ under a tame Fréchet map $ \varphi: N_F \to N_B $ is a tame Fréchet submanifold of co-Banach type.
• The kernel of the differential $ d\varphi_p $ at a regular point is a complemented subspace in the tangent space, ensuring a well-defined splitting of the ambient space.
• The implicit function theorem in the tame Fréchet category allows the construction of local charts for the submanifold via a tame Fréchet isomorphism $ \Phi $.
• Spheres defined by $ \|x\|_n = 1 $ in a tame Fréchet space that is an inverse limit of Hilbert spaces are tame Fréchet submanifolds of co-finite type.
• Intersections of finitely many such unit spheres are also tame Fréchet submanifolds of co-finite type.
• Isoparametric submanifolds $ M_F = M \cap F $ in a tame Fréchet space $ F $, where $ M \subset H_k $ is a proper Fredholm isoparametric submanifold in a Hilbert space, are tame Fréchet submanifolds of co-finite type.