Lie Groups as Four-dimensional Complex Manifolds with Norden Metric

Gribachev, Kostadin; Teofilova, Marta

doi:10.1007/s00022-008-2053-9

Cited by 4 publications

(4 citation statements)

References 4 publications

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“…An explicit example of a Riemannian almost product manifold from the main class is proposed in [4]. Similar investigations on Lie groups with additional tensor structures are made in [3] and [6].…”

Section: Introductionmentioning

confidence: 90%

Invariant Tensors under the Twin Interchange of Norden Metrics on Almost Complex Manifolds

Manev

2015

Results. Math.

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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 4-dimensional manifold of the studied type and the mentioned invariant objects are found in an explicit form.2010 Mathematics Subject Classification. Primary 53C15; Secondary 53C25.

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Section: Introductionmentioning

confidence: 90%

Invariant Tensors under the Twin Interchange of Norden Metrics on Almost Complex Manifolds

Manev

2015

Results. Math.

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“…Several examples of Kähler-Norden manifolds are given in Ref. [4][5][6]8] and other papers. Theorem 3 allows us to construct many new examples of Kähler-Norden manifolds as the total spaces of the cotangent bundles of some Kähler-Norden manifolds.…”

Section: Cotangent Bundles With Natural Riemann Extensions As Almost ...mentioning

confidence: 99%

“…Beside Riemannian and Lorentzian geometry, a special role is played by manifolds with a metric of neutral signature, among which almost complex manifolds with Norden metric constitute a particular class. These manifolds are investigated by many authors and several examples are given in the literature (e.g., [4][5][6][7][8][9] and the references therein). Several papers constructed almost complex Norden structures on the total space of the tangent bundle (see [4]); however, such structures on the total space of the cotangent bundle are not so rich.…”

Section: Introductionmentioning

confidence: 99%

Almost Complex and Hypercomplex Norden Structures Induced by Natural Riemann Extensions

Bejan

Nakova

2022

Mathematics

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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∗M4n of an almost hypercomplex manifold (M4n,J1,J2,J3,∇) 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.

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“…(45) J 1 X 1 = X 2 , J 1 X 2 = −X 1 , J 1 X 3 = −X 4 , J 1 X 4 = X 3 ; J 2 X 1 = X 3 , J 2 X 2 = X 4 , J 2 X 3 = −X 1 , J 2 X 4 = −X 2 ; J 3 X 1 = −X 4 , J 3 X 2 = X 3 , J 3 X 3 = −X 2 , J 3 X 4 = X 1 and then (11) are valid. In [10], it is constructed the manifold (G, J 2 , g) as an example of a 4-dimensional quasi-Kähler manifold with Norden metric.…”

Section: A 4-dimensional Examplementioning

confidence: 99%

Associated Nijenhuis Tensors on Manifolds with Almost Hypercomplex Structures and Metrics of Hermitian–Norden Type

Manev¹

2016

Results Math

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Abstract. An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian-Norden type. There are studied relations between the six associated Nijenhuis tensors as well as their vanishing. It is given a geometric interpretation of the vanishing of these tensors as a necessary and sufficient condition for the existence of a unique connection with totally skew-symmetric torsion tensor. Similar idea is used in the paper of T. Friedrich and S. Ivanov in Asian J. Math. (2002) for some other structures. Finally, an example of a 4-dimensional manifold of the considered type with vanishing associated Nijenhuis tensors is given.

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Lie Groups as Four-dimensional Complex Manifolds with Norden Metric

Cited by 4 publications

References 4 publications

Invariant Tensors under the Twin Interchange of Norden Metrics on Almost Complex Manifolds

Invariant Tensors under the Twin Interchange of Norden Metrics on Almost Complex Manifolds

Almost Complex and Hypercomplex Norden Structures Induced by Natural Riemann Extensions

Associated Nijenhuis Tensors on Manifolds with Almost Hypercomplex Structures and Metrics of Hermitian–Norden Type

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