Isomorphism classes for Banach vector bundle structures of second tangents

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

doi:10.1017/s0305004106009467

Cited by 7 publications

(11 citation statements)

References 11 publications

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“…Let M and N be two smooth manifolds modeled on the Banach spaces E and E ′ . Motivated by [28], [7] and [25] we state the following two definitions. Definition 2.4.…”

Section: Preliminariesmentioning

confidence: 99%

“…For a differentiable map g : M −→ N , in contrast to T 1 g = T g : T M −→ T N , the tangent map T k g, even for k = 2, is not necessarily a vector bundle morphism [7,25]. In this section first, we investigate under what conditions T k g becomes a vector bundle morphism.…”

Section: T K G As a Vector Bundle Morphismmentioning

confidence: 99%

“…However, as one may have expected, these vector bundle structures depend crucially on the particular connection chosen [6,7,25,26].…”

Section: Introductionmentioning

confidence: 99%

“…More precisely we introduce the higher order differential T k g of a smooth map g : M −→ N between two manifolds M and N and we investigate when T k g is linear on fibres. Linearity of T k x g, x ∈ M , allows us to build a vector bundle morphism T k g : T k M −→ T k N (see [25] and [7] for the special case k = 2). As a consequence, we show that the vector bundle structure on T k M , defined by the aim of a connection map, remains invariant (isomorphic) if it is replaced by a g-related connection map, for some diffeomorphism g : M −→ M .…”

Section: Introductionmentioning

confidence: 99%

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Isomorphism classes for higher order tangent bundles

Suri

2017

Advances in Geometry

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The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. In the previous work of the author he proved that $T^kM$, $1\leq k\leq \infty$, admits a vector bundle structure on $M$ if and only if $M $ is endowed with a linear connection or equivalently a connection map on $T^kM$ is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the notion of the $k$'th order differential $T^kg:T^kM\longrightarrow T^kN$ for a given differentiable map $g$ between manifolds $M$ and $N$. As we shall see, $T^kg$ becomes a vector bundle morphism if the base manifolds are endowed with $g$-related connections. In particular, replacing a connection with a $g$-related one, where $g:M\longrightarrow M$ is a diffeomorphism, follows invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combination of connection maps and manifold of $C^r$ maps we offer three examples to support our theory and reveal its interaction with the known problems such as Sasaki lift of metrics.Comment: 21 page

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“…Let M and N be two smooth manifolds modeled on the Banach spaces E and E ′ . Motivated by [28], [7] and [25] we state the following two definitions. Definition 2.4.…”

Section: Preliminariesmentioning

confidence: 99%

Section: T K G As a Vector Bundle Morphismmentioning

confidence: 99%

“…However, as one may have expected, these vector bundle structures depend crucially on the particular connection chosen [6,7,25,26].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Isomorphism classes for higher order tangent bundles

Suri

2017

Advances in Geometry

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“…They proved that existence of a vector bundle structure on T 2 M is equivalent to the existence on M of a linear connection in the sense of Vilms [16]. By this means, vector bundle structures of T 2 M were classified by Dodson, Galanis and Vassiliou for the Banach case (see [3]). …”

Section: Introductionmentioning

confidence: 99%

Conjugate Connections and Differential Equations on Infinite Dimensional Manifolds

Aghasi

Dodson

Galanis

et al. 2009

Differential Geometry

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Abstract. On a smooth manifold M, the vector bundle structures of the second order tangent bundle, T 2 M bijectively correspond to linear connections. In this paper we classify such structures for those Fréchet manifolds which can be considered as projective limits of Banach manifolds. We investigate also the relation between ordinary differential equations on Fréchet spaces and the linear connections on their trivial bundle; the methodology extends to solve differential equations on those Fréchet manifolds which are obtained as projective limits of Banach manifolds. Such equations arise in theoretical physics.

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Second-Order Time-Dependent Tangent Bundles and Geometric Mechanics

Suri

2017

Mediterr. J. Math.

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Abstract. The aim of this paper is to geometrize time dependent Lagrangian mechanics in a way that the framework of second order tangent bundles plays an essential role. To this end, we first introduce the concepts of time dependent connections and time dependent semisprays on a manifold M and their induced vector bundle structures on the second order time dependent tangent bundle R × T 2 M . Then we turn our attention to regular time Lagrangians and their interaction with R × T 2 M in different situations such as mechanical systems with potential fields, external forces and holonomic constraints. Finally we propose an examples to support our theory.

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Isomorphism classes for Banach vector bundle structures of second tangents

Cited by 7 publications

References 11 publications

Isomorphism classes for higher order tangent bundles

Isomorphism classes for higher order tangent bundles

Conjugate Connections and Differential Equations on Infinite Dimensional Manifolds

Second-Order Time-Dependent Tangent Bundles and Geometric Mechanics

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