Riemannian almost product and para-Hermitian cotangent bundles of general natural lift type

Druţă-Romaniuc, Simona-Luiza

doi:10.1007/s10474-012-0271-y

Cited by 10 publications

(13 citation statements)

References 11 publications

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“…where [[X, Y ]] + is the usual Dorfman bracket (22). Mapping by ρ + then recovers the H-twisted Dorfman bracket on (T ⊕ T * )F + .…”

Section: Interpretation As a Generalized Geometry On (T ⊕ T * )F ±mentioning

confidence: 88%

“…In our framework of para-Hermitian manifolds, instead of understanding the section condition locally as an independence on certain subset of local coordinates, equations (21) and (22) provide an approach to impose the section condition globally.…”

Section: A Comment About the Section Conditionmentioning

confidence: 99%

“…Similarly, ∇ c P−X∂ j = 0, which shows that ∇ c P−X Y + = 0. Recalling (22) yields [[ , ]] − = 0. We comment that this global approach could be a step towards relaxing the section condition partially or altogether, which is deemed desirable in various discussions about the foundations of the DFT (see for example [34][35][36], for a different approach [37]).…”

Section: A Comment About the Section Conditionmentioning

confidence: 99%

“…[22] presents a construction of the almost para-Hermitian structure on T * M and discusses conditions under which this structure is integrable.…”

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confidence: 99%

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Algebroid structures on para-Hermitian manifolds

Svoboda

2018

Journal of Mathematical Physics

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We present a global construction of a so-called D-bracket appearing in the physics literature of Double Field Theory (DFT) and show that if certain integrability criteria are satisfied, it can be seen as a sum of two Courant algebroid brackets. In particular, we show that the local picture of the extended space-time used in DFT fits naturally in the geometrical framework of para-Hermitian manifolds and that the data of an (almost) para-Hermitian manifold is sufficient to construct the D-bracket. Moreover, the twists of the bracket appearing in DFT can be interpreted in this framework geometrically as a consequence of certain deformations of the underlying para-Hermitian structure. *

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“…where [[X, Y ]] + is the usual Dorfman bracket (22). Mapping by ρ + then recovers the H-twisted Dorfman bracket on (T ⊕ T * )F + .…”

Section: Interpretation As a Generalized Geometry On (T ⊕ T * )F ±mentioning

confidence: 88%

Section: A Comment About the Section Conditionmentioning

confidence: 99%

Section: A Comment About the Section Conditionmentioning

confidence: 99%

“…[22] presents a construction of the almost para-Hermitian structure on T * M and discusses conditions under which this structure is integrable.…”

mentioning

confidence: 99%

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Algebroid structures on para-Hermitian manifolds

Svoboda

2018

Journal of Mathematical Physics

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“…A very important feature of any pseudo-Riemannian metric g is that it provides musical isomorphisms g : T M → T * M and g : T * M → T M between the tangent and cotangent bundles. Some properties of geometric structures on cotangent bundles with respect to the musical isomorphisms are proved in [1][2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Musical isomorphisms and problems of lifts

Cakan

Akbulut

Salimov

2016

Chin. Ann. Math. Ser. B

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Para‐Hermitian Geometry, Dualities and Generalized Flux Backgrounds

Marotta

Szabo

2018

Fortschritte der Physik

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We survey physical models which capture the main concepts of double field theory on para-Hermitian manifolds. We show that the geometric theory of Lagrangian and Hamiltonian dynamical systems is an instance of para-Kähler geometry which extends to a natural example of a Born geometry. The corresponding phase space geometry belongs to the family of natural almost para-Kähler structures which we construct explicitly as deformations of the canonical para-Kähler structure by non-linear connections. We extend this framework to a class of non-Lagrangian dynamical systems which naturally encodes the notion of fluxes in para-Hermitian geometry. In this case we describe the emergence of fluxes in terms of weak integrability defined by the D-bracket, and we extend the construction to arbitrary cotangent bundles where we reproduce the standard generalized fluxes of double field theory. We also describe the para-Hermitian geometry of Drinfel'd doubles, which gives an explicit illustration of the interplay between fluxes, D-brackets and different polarizations. The left-invariant para-Hermitian structure on a Drinfel'd double in a Manin triple polarization descends to a doubled twisted torus, which we use to illustrate how changes of polarizations give rise to different fluxes and string backgrounds in para-Hermitian geometry.1 An alternative mathematical approach to double field theory based on graded symplectic geometry is discussed in [21][22][23].

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Riemannian almost product and para-Hermitian cotangent bundles of general natural lift type

Cited by 10 publications

References 11 publications

Algebroid structures on para-Hermitian manifolds

Algebroid structures on para-Hermitian manifolds

Musical isomorphisms and problems of lifts

Para‐Hermitian Geometry, Dualities and Generalized Flux Backgrounds

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