Almost Complex and Hypercomplex Norden Structures Induced by Natural Riemann Extensions

Cornelia-Livia Bejan and Galia Nakova
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Cornelia-Livia Bejan: Department of Mathematics, “Gh. Asachi” Technical University of Iasi, 700506 Iasi, Romania
Galia Nakova: Department of Algebra and Geometry, Faculty of Mathematics and Informatics, “St. Cyril and St. Methodius” University of Veliko Tarnovo, 2 Teodosii Tarnovski Str., 5003 Veliko Tarnovo, Bulgaria

Mathematics, 2022, vol. 10, issue 15, 1-18

Abstract: The Riemann extension, introduced by E. K. Patterson and A. G. Walker, is a semi-Riemannian metric with a neutral signature on the cotangent bundle T ∗ M of a smooth manifold M, induced by a symmetric linear connection ∇ on M . In this paper we deal with a natural Riemann extension g ¯ , which is a generalization (due to M. Sekizawa and O. Kowalski) of the Riemann extension. We construct an almost complex structure J ¯ on the cotangent bundle T ∗ M of an almost complex manifold ( M , J , ∇ ) with a symmetric linear connection ∇ such that ( T ∗ M , J ¯ , g ¯ ) is an almost complex manifold, where the natural Riemann extension g ¯ is a Norden metric. We obtain necessary and sufficient conditions for ( T ∗ M , J ¯ , g ¯ ) to belong to the main classes of the Ganchev–Borisov classification of the almost complex manifolds with Norden metric. We also examine the cases when the base manifold is an almost complex manifold with Norden metric or it is a complex manifold ( M , J , ∇ ′ ) endowed with an almost complex connection ∇ ′ ( ∇ ′ J = 0 ). We investigate the harmonicity with respect to g ¯ of the almost complex structure J ¯ , according to the type of the base manifold. Moreover, we define an almost hypercomplex structure ( J ¯ 1 , J ¯ 2 , J ¯ 3 ) on the cotangent bundle T ∗ M 4 n of an almost hypercomplex manifold ( M 4 n , J 1 , J 2 , J 3 , ∇ ) with a symmetric linear connection ∇. The natural Riemann extension g ¯ is a Hermitian metric with respect to J ¯ 1 and a Norden metric with respect to J ¯ 2 and J ¯ 3 .

Keywords: natural Riemann extension; almost complex manifolds with Norden metric; almost hypercomplex manifolds with Hermitian and Norden metrics; harmonicity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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