Some Frobenius theorems in global analysis

Leslie, Joshua A.

doi:10.4310/jdg/1214428441

Cited by 30 publications

(17 citation statements)

References 6 publications

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“…is continuous and the evaluation map ev 1 : A slightly weaker version of the preceding theorem can already be found in [Les68]. Note that it applies in particular to gauge groups of G-bundles over M. We have added the assumption that G is locally exponential because this is needed for the standard constructions of charts of the group C r (M, G) (cf.…”

Section: (G1) G N Is a Smooth Banach Manifold Modeled On E N (G2) Gmentioning

confidence: 95%

Towards a Lie theory of locally convex groups

2006

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In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.

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Section: (G1) G N Is a Smooth Banach Manifold Modeled On E N (G2) Gmentioning

confidence: 95%

Towards a Lie theory of locally convex groups

2006

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“…Here we adopt the definition of Leslie [5], [6] for the differentiability of mappings between Fréchet spaces. However, the differentiability proposed by Kriegl and P. Michor [4] is also suited to our study.…”

Section: Preliminariesmentioning

confidence: 99%

Second order tangent bundles of infinite dimensional manifolds

Dodson

Galanis

2004

Journal of Geometry and Physics

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Abstract. The second order tangent bundle T 2 M of a smooth manifold M consists of the equivalent classes of curves on M that agree up to their acceleration. It is known [1] that in the case of a finite n-dimensional manifold M , T 2 M becomes a vector bundle over M if and only if M is endowed with a linear connection. Here we extend this result to M modeled on an arbitrarily chosen Banach space and more generally to those Fréchet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite-dimensional dynamical systems.

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“…This new connection coincides with the original one ω having the same local forms with respect to the sections {s a } a∈A (for details and the relevant proofs see [2]). Note also that for the differentiability of mappings between Fréchet spaces we adopt the definition of J. Leslie ([6], [7]). However, the differentiability proposed by A. Kriegl and P. Michor in [8] fits also to our case.…”

Section: Preliminariesmentioning

confidence: 99%

Universal connections in Fréchet principal bundles

Galanis

2007

Period Math Hung

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A new methodology leading to the construction of a universal connection for Fréchet principal bundles is proposed in this paper. The classical theory, applied successfully so far for finite dimensional and Banach modelled bundles, collapses within the framework of Fréchet manifolds. However, based on the replacement of the space of continuous linear mappings by an appropriate topological vector space, we endow the bundle J 1 P of 1-jets of the sections of a Fréchet principal bundle P with a connection form by means of which we may "reproduce" every connection of P .

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Some Frobenius theorems in global analysis

Cited by 30 publications

References 6 publications

Towards a Lie theory of locally convex groups

Towards a Lie theory of locally convex groups

Second order tangent bundles of infinite dimensional manifolds

Universal connections in Fréchet principal bundles

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