The geometry of a bi-Lagrangian manifold

Etayo, Fernando; Santamaría, Rafael; Trías, Ujué R.

doi:10.1016/j.difgeo.2005.07.002

Cited by 45 publications

(55 citation statements)

References 31 publications

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“…Furthermore, the tensor ω := ηK is skew It follows that the eigenbundles L andL are isotropic with respect to η and Lagrangian with respect to ω. When ω is closed, the pair (ω, K) defines an almost bi-Lagrangian structure [25].…”

Section: )mentioning

confidence: 99%

A Unique Connection for Born Geometry

Freidel

Rudolph

Svoboda

2019

Commun. Math. Phys.

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It has been known for a while that the effective geometrical description of compactified strings on d-dimensional target spaces implies a generalization of geometry with a doubling of the sets of tangent space directions. This generalized geometry involves an O(d, d) pairing η and an O(2d) generalized metric H. More recently it has been shown that in order to include T-duality as an effective symmetry, the generalized geometry also needs to carry a phase space structure or more generally a para-Hermitian structure encoded into a skew-symmetric pairing ω. The consistency of string dynamics requires this geometry to satisfy a set of compatibility relations that form what we call a Born geometry. In this work we prove an analogue of the fundamental theorem of Riemannian geometry for Born geometry. We show that there exists a unique connection which preserves the Born structure (η, ω, H) and which is torsionless in a generalized sense. This resolves a fundamental ambiguity that is present in the double field theory formulation of effective string dynamics.

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Section: )mentioning

confidence: 99%

A Unique Connection for Born Geometry

Freidel

Rudolph

Svoboda

2019

Commun. Math. Phys.

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“…In the terminology of [13] this structure would be an almost Kähler D-manifold. If additionally the eigendistributions of r are integrable, we have a bi-Lagrangian foliation [9].…”

Section: Geometric Structures Compatible With a Pseudo Riemannian Metricmentioning

confidence: 99%

Interpolation of geometric structures compatible with a pseudo Riemannian metric

2016

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Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a product structure and symmetric with respect to g, then r induces a pseudo Riemannian product structure on M . Sometimes the integrability condition is expressed by the closedness of an associated two-form: if j is almost complex on M and ω(x, y) = g(jx, y) is symplectic, then M is almost pseudo Kähler. Now, product, complex and symplectic structures on M are trivial examples of generalized (para)complex structures in the sense of Hitchin. We use the latter in order to define the notion of interpolation of geometric structures compatible with g. We also compute the typical fibers of the twistor bundles of the new structures and give examples for M a Lie group with a left invariant metric.

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“…In particular, we discuss almost para-Kähler manifolds (or almost bi-Lagrangian manifolds) and explain different classes of examples of such manifolds. For more information we invite the interested reader to consult the survey articles [1,3,7].…”

Section: Almost Para-kähler Manifoldsmentioning

confidence: 99%

Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds

2010

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Given an almost para-Kähler manifold equipped with a metric and para-complex connection, we define a generalized second fundamental form and generalized mean curvature vector of space-like Lagrangian submanifolds. We then show that the deformation induced by this variant of the mean curvature vector field preserves the Lagrangian condition, if the connection satisfies also some Einstein condition. In case the almost para-Kähler structure is integrable, the flow coincides with the classical mean curvature flow in the pseudo-Riemannian context.

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The geometry of a bi-Lagrangian manifold

Cited by 45 publications

References 31 publications

A Unique Connection for Born Geometry

A Unique Connection for Born Geometry

Interpolation of geometric structures compatible with a pseudo Riemannian metric

Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds

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