On the metrizability of the bundle space

Etter, David; Griffin, John S.

doi:10.1090/s0002-9939-1954-0063023-2

Cited by 4 publications

(7 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then ϕ(x) ∈ ∂ϕ(U ϕ ) if and only if ψ(x) ∈ ∂ψ(U ψ ). (6) (6) A full proof is contained in the forthcoming [15,Section 3]. However here is a rough sketch: Argue by contradiction.…”

Section: Manifolds Of Mappings For Manifolds With Rough Boundarymentioning

confidence: 99%

See 1 more Smart Citation

Extending Whitney’s extension theorem: nonlinear function spaces

Roberts¹,

Schmeding²

2022

Annales De l'Institut Fourier

View full text Add to dashboard Cite

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains C, with non-smooth boundary, in possibly non-compact manifolds. Assuming C is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where C only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator.Résumé. -Nous considérons une version du problème de l'extension de Whitney, globale et non linéaire, pour les fonctions lisses à valeurs dans des variétés et définies sur des domaines fermés C à bords non-lisses dans des variétés possiblement non compactes. Supposant que C est une sous-variété à bord anguleux, ou qu'elle est compacte et localement convexe à bords non-lisses, nous montrons que l'opérateur de restriction, à partir des fonctions définies partout, est une submersion de variétés localement convexes, et donc possède des scindages linéaires locaux sur les cartes. Nous considérons à cet effet l'opérateur de restriction correspondant pour les espaces localement convexes de sections de fibrés vectoriels à support compact, permettant aussi de tariter le cas plus général où C n'a que des restrictions légères sur les cusps vers l'intérieur et l'extérieur, et montrons l'existence d'un opérateur de prolongement.

show abstract

Section: Manifolds Of Mappings For Manifolds With Rough Boundarymentioning

confidence: 99%

“…Since N is metrisable and modelled on a metrisable space, T k N is metrisable by [6]. Thus the spaces C co (T k M, T k N ) are metrisable by [33, Proposition 5.10 (e)] whence the embedding from Lemma 6.7 indentifies C ∞ co (M, N ) as a subspace of a metrisable space.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Extending Whitney’s extension theorem: nonlinear function spaces

Roberts¹,

Schmeding²

2022

Annales De l'Institut Fourier

View full text Add to dashboard Cite

show abstract

“…Since N is metrisable and modelled on a metrisable space, T k N is metrisable by [7]. Thus the spaces C co (T k M, T k N ) are metrisable by [34, Proposition 5.10 e)] whence the embedding from Lemma 5.7 indentifies C ∞ co (M, N ) as a subspace of a metrisable space.…”

Section: Lemmamentioning

confidence: 99%

“…Then ϕ(x) ∈ ∂ϕ(U ϕ ) if and only if ψ(x) ∈ ∂ψ(U ψ ). 7 In essence Lemma 5.2 shows that the formal boundary of M arises from the topological boundary of the images of charts in the model space.…”

Section: Lemmamentioning

confidence: 99%

See 1 more Smart Citation

Extending Whitney's extension theorem: nonlinear function spaces

Roberts,

Schmeding

2018

Preprint

View full text Add to dashboard Cite

This article shows that there is a continuous extension operator for compactly-supported smooth sections of vector bundles on possibly noncompact smooth manifolds, where the closed set to which sections are restricted satisfy a mild restriction on possible boundary cusps. These function spaces are locally convex but in general not Fréchet and so one cannot use the existing theory for Fréchet spaces of smooth functions. Further, in the global, nonlinear case of all smooth functions between possibly noncompact smooth manifolds, we prove that the analogous restriction map between locally convex manifolds of smooth functions is a submersion in the sense of admitting continuous local splittings on charts, under the assumption the closed set is either a submanifold with corners, or is locally convex with rough boundary and compact. In all these results the spaces of functions are spaces of smooth functions, and not spaces of Whitney jets.

show abstract