A generalized second-order frame bundle for Fréchet manifolds

Dodson, C. T. J.; Galanis, George; Vassiliou, Efstathios

doi:10.1016/j.geomphys.2004.12.011

Cited by 5 publications

(6 citation statements)

References 19 publications

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“…Also, Christoffel symbols (in the case of vector bundles) or the local forms (in principal bundles) are affected in their representation of linear maps by the fact that continuous linear mappings of a Fréchet space do not remain in the same category. Galanis [25,26] solved the problem for connections that can be obtained as projective limits and we obtain [19] Theorem 4.3. Let ∇ be a linear connection of the second order tangent bundle T 2 M = lim ← − T 2 M i that can be represented as a projective limit of linear connections ∇ i on the (Banach modelled) factors.…”

Section: Fréchet Second Frame Bundlementioning

confidence: 88%

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Some recent work in Frechet geometry

Dodson

2011

Preprint

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Some recent work in Fréchet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Fréchet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Fréchet structure needed for applications. A summary is given of recent results on hypercyclicity of operators on Fréchet spaces. MSC: 58B25 58A05 47A16, 47B37

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Section: Fréchet Second Frame Bundlementioning

confidence: 88%

“…The latter can be thought of also as a generalized Fréchet Lie group by being embedded in H(F) := H(F, F). Then [19],…”

Section: Fréchet Second Frame Bundlementioning

confidence: 99%

Some recent work in Frechet geometry

Dodson

2011

Preprint

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“…Then according to gluing lemma ([6], 5.2.4), LM i is a principal bundle on M i with the structure group H i 0 (E i ). It is easily seen that R i is the natural induces action of H i 0 (E i ) on LM i (see also [8] and [23]).…”

Section: For This Element There Exists a Projective Family Of Chartsmentioning

confidence: 98%

“…It is easily seen that R i is the natural induces action of H i 0 (E i ) on LM i (see also [8] and [23]). Remarks 1.…”

Section: The Fréchet Casementioning

confidence: 98%

Bundle of Frames and Sprays for Fréchet Manifolds

Suri¹,

Moosaei²

2018

International Electronic Journal of Geometry

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First we present a unified theory of connections on bundles necessary for the next studies. For a smooth manifold M , modeled on the Banach space B, we define the bundle of linear frames LM and we endow it with a differentiable structure. Bundle of sprays F M , the pullback of LM via the tangent bundle π : T M −→ M , is a natural bundle which provides us a rich environment to study the geometry of M. Afterward, despite of natural difficulties with Fréchet manifolds and even spaces, we generalize these results to a wide class of Fréchet manifolds, those which can be considered as projective limits of Banach manifolds. As an alternative approach we use pre-Finsler connections on F M and we show that our technique successfully solves ordinary differential equations on these manifolds. As some applications of our results we apply our method to enrich the geometry of two known Fréchet manifolds, i.e. jet of infinite sections and manifold of smooth maps, and we provide a suitable framework for further studies in these areas.

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“…The study of infinite dimensional manifolds has received much interest due to its interaction with bundle structures, fibrations and foliations, jet fields, connections, sprays, Lagrangians and Finsler structures ( [1], [14], [7], [8], [10], [18] and [30]). In particular, non-Banach locally convex modelled manifolds have been studied from different points of view (see for example [2], [4], [11], [12], [19] and [27]). Fréchet spaces of sections arise naturally as configurations of a physical field and the moduli space of inequivalent configurations of a physical field is the quotient of the infinite-dimensional configuration space X by the appropriate symmetry gauge group.…”

Section: Introductionmentioning

confidence: 99%

Second order structures for sprays and connections on Frechet manifolds

Aghasi,

Bahari,

Dodson

et al. 2008

Preprint

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Ambrose, Palais and Singer [6] introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath [23] extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Fréchet manifolds. It is proved that the existence of Christoffel and Hessian structures, connections, sprays and dissections are equivalent on those Fréchet manifolds which can be considered as projective limits of Banach manifolds. These concepts provide also an alternative way for the study of ordinary differential equations on non-Banach infinite dimensional manifolds. Concrete examples of the structures are provided using direct and flat connections.

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A generalized second-order frame bundle for Fréchet manifolds

Cited by 5 publications

References 19 publications

Some recent work in Frechet geometry

Some recent work in Frechet geometry

Bundle of Frames and Sprays for Fréchet Manifolds

Second order structures for sprays and connections on Frechet manifolds

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