Manifolds of Mappings for Continuum Mechanics

Michor, Peter W.

doi:10.1007/978-3-030-42683-5_1

Cited by 24 publications

(30 citation statements)

References 72 publications

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“…UPC sets satisfy a strong Markov inequality. All sets in our results fulfil WEP, in particular, they form examples of Whitney manifold germs as introduced in [18]; see also [27] for a related concept. 1.5.…”

Section: Introductionmentioning

confidence: 58%

Arc-smooth functions on closed sets

Rainer

2019

Compositio Math.

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By an influential theorem of Boman, a function f on an open set U in R d is smooth (C ∞ ) if and only if it is arc-smooth, i.e., f •c is smooth for every smooth curve c : R → U . In this paper we investigate the validity of this result on closed sets. Our main focus is on sets which are the closure of their interior, so-called fat sets. We obtain an analogue of Boman's theorem on fat closed sets with Hölder boundary and on fat closed subanalytic sets with the property that every boundary point has a basis of neighborhoods each of which intersects the interior in a connected set. If X ⊆ R d is any such set and f : X → R is arc-smooth, then f extends to a smooth function defined on R d . We also get a version of the Bochnak-Siciak theorem on all closed fat subanalytic and all closed sets with Hölder boundary: if f : X → R is the restriction of a smooth function on R d which is real analytic along all real analytic curves in X, then f extends to a holomorphic function on a neighborhood of X in C d . Similar results hold for non-quasianalytic Denjoy-Carleman classes (of Roumieu type). We will also discuss sharpness and applications of these results.

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Section: Introductionmentioning

confidence: 58%

Arc-smooth functions on closed sets

Rainer

2019

Compositio Math.

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“…Recall that for a closed manifold S , the Grassmannian , i.e., the space of all submanifolds in M which are diffeomorphic to S , is a smooth Fréchet manifold whose tangent space at can be canonically identified as . The Grassmannian is the base of a (locally trivial) smooth principal bundle with structure group ; see [ 2 , 22 , 23 ] and [ 18 , Theorem 44.1]. Recall that the space of embeddings, , is a smooth Fréchet manifold whose tangent space at can be canonically identified as .…”

Section: Manifolds Of Nonlinear Flagsmentioning

confidence: 99%

“…The statement in (a) has been proved by Binz and Fischer [ 2 ] for compact S . The generalization to noncompact S is due to Michor; see [ 2 , 22 , 23 ] and [ 18 , Theorem 44.1]. For a manifold S with nonempty boundary, this bundle has been studied in [ 8 , Theorem 2.2].…”

Section: Manifolds Of Nonlinear Flagsmentioning

confidence: 99%

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Nonlinear flag manifolds as coadjoint orbits

Haller

Vizman

2020

Ann Glob Anal Geom

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“…Manifolds with corners alias quadrantic (orthantic) manifolds. For more information we refer to [5], [8], [7], etc. Let Q = Q m = R m ≥0 be the positive orthant or quadrant.…”

Section: Manifolds With Cornersmentioning

confidence: 99%