Invariant Tensors under the Twin Interchange of the Pairs of the Associated Metrics on Almost Paracomplex Pseudo-Riemann
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Invariant Tensors under the Twin Interchange of the Pairs of the Associated Metrics on Almost Paracomplex Pseudo-Riemannian Manifolds Mancho Manev Abstract. The object of study are almost paracomplex pseudo-Riemannian manifolds with a pair of metrics associated each other by the almost paracomplex structure. A torsion-free connection and tensors with geometric interpretation are found which are invariant under the twin interchange, i.e. the swap of the counterparts of the pair of associated metrics and the corresponding Levi-Civita connections. A Lie group depending on two real parameters is constructed as an example of a fourdimensional manifold of the studied type and the mentioned invariant objects are found in an explicit form. Mathematics Subject Classification. Primary 53C15, Secondary 53C25. Keywords. Invariant tensor, Affine connection, Almost paracomplex manifold, Pseudo-Riemannian metric.
Introduction Manifolds with almost product structure and Riemannian metric are well known [13]. Usually, the almost product structure acts as an isometry with respect to the metric, i.e. it is said that the metric is compatible with the structure. A special and remarkable case is when the almost product structure is traceless and then it is called an almost paracomplex structure. In this case, the eigenvalues +1 and −1 of the structure have one and the same multiplicity, thus the dimension of such a manifold is even. An almost paracomplex manifold is a counterpart of an almost complex manifold. The compatible metric with an almost complex structure is a Hermitian metric. The requirement that the metric be Riemannian on an almost paracomplex manifold is not necessarily and thus we suppose here that the metric is pseudoRiemannian. 0123456789().: V,-vol
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The associated (0, 2)-tensor of a Hermitian metric is a two-form while the associated (0, 2)-tensor of any compatible metric on almost paracomplex manifold is also a compatible metric. Therefore, in this case, we dispose of a pair of mutually associated compatible metrics with respect to the almost paracomplex structure, known also as twin metrics. Such almost paracomplex manifolds are studied in the latter three decades by a lot of authors (e.g. [1– 3,5,6,10–12,14–17]), including under the name Riemannian almost product manifolds. An interesting problem on almost paracomplex (pseudo-)Riemannian manifolds is the presence of tensors with some geometric interpretation which are invariant or anti-invariant under the so-called twin interchange. This is the swap of the counterparts of the pair of compatible metrics and their LeviCivita connections. The aim of the present work is to solve this problem in the general case and to illustrate the invariant objects by example from a significant class of the considered manifolds. Invariant connection and invariant tensors under twin interchange on Riemannian almost product manifolds with non-integrable structure are found in [10]. A similar investigation for almost complex manifolds with Norden metrics
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