Para-hyperhermitian structures on tangent bundles

Abstract

In this paper we construct a family of almost para-hyperhermitian structures on the tangent bundle of an almost parahermitian manifold and study its integrability. Also, the necessary and sufficient conditions are provided for these structures to become para-hyper-Kähler.Key words: para-hyperhermitian structure, tangent bundle, paracomplex space form. PRELIMINARIESAn almost product structure on a smooth manifold M is a tensor field P of type (1,1) on M, P = ±Id, such thatwhere Id is the identity tensor field of type (1,1) on M.An almost para-hermitian structure on a differentiable manifold M is a pair (P, g), where P is an almost product structure on M and g is a semi-Riemannian metric on M satisfyingIn this case, (M, P, g) is said to be an almost para-hermitian manifold. It is easy to see that the dimension of M is even. Moreover, if ∇P = 0, then (M, P, g) is said to be a para-Kähler manifold.An almost complex structure on a smooth manifold M is a tensor field J of type (1,1) on M such thatAn almost para-hypercomplex structure on a smooth manifold M is a triple H = (J α ) α=1,3 , where J 1 is an almost complex structure on M and J 2 , J 3 are almost product structures on M, satisfyingIn this case (M, H) is said to be an almost para-hypercomplex manifold.A semi-Riemannian metric g on (M, H) is said to be compatible or adapted to the almost parahypercomplex structure H = (J α ) α=1,3 if it satisfies Remark 2.3. If (M, P, g) is an almost para-hermitian manifold, then we can define three tensor fields J 1 , J 2 , J 3 on T M by the equalities:It is easy to see that J 1 is an almost complex structure and J 2 , J 3 are almost product structures. We also have the following result (see [15]).

Generalised kinematics for double field theory

“…Before doing so, we present a key example. Example: A canonical example (see also [39,54]) that illustrates the usefulness of the notion of L-integrability is when P = T M is the total space of the tangent bundle of a d-dimensional manifold (M, g, D) equipped with a metric g and a compatible connection D. We denote by π : P → M the canonical projection, by dπ : T P → T M its differential and by L = Ker(dπ) the vertical bundle. In physical terms the vertical bundle is the space of velocities and it is canonically identified with the tangent bundle since the vertical fiber is π −1 (x) = T x M. The affine connection D can be viewed as an Ehresmann connection, i.e.…”

Algebroid structures on para-Hermitian manifolds

“…For more details see [26] and references therein. Canonical examples of (almost) para-Hermitian manifolds are given by the total space of the tangent and cotangent bundle of a manifold [22,27]. Further examples of para-Hermitian and para-Kähler manifolds can be found for example in [16,28], and a classification of almost para-Hermitian manifolds is given in [29].…”

Para-Quaternionic Structures on the 3-Jet Bundle

“…The main purpose of this paper is to construct a para-quaternionic structure or para-hyperhermitian structure on the 3-jet bundle which is the generalization of this construction on tangent bundle (see [13], [20]), we also investigate its integrability, we obtain the necessary and sufficient conditions for these structures to become para-hyper-Kähler and finally we prove that the 3-jet bundle can not be a para-quaternionic Kähler manifold.…”