On natural Riemann extensions

Kowalski, Oldřich; Sekizawa, Masami

doi:10.5486/pmd.2011.4992

Cited by 20 publications

(24 citation statements)

References 11 publications

(23 reference statements)

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“…. , e n } be its dual basis in T x M. As in [8] we denote by the same letter e i the parallel extension of each e i (along geodesics starting at x) to a normal neighborhood of x in M, for i = 1, n. Therefore {e 1 , . .…”

Section: The Natural Riemann Extensionḡ Is Defined In Terms Of Classimentioning

confidence: 99%

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On some differential operators on natural Riemann extensions

Bejan

Kowalski

2015

Ann Glob Anal Geom

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Natural Riemann extensions are pseudo-Riemannian metrics (introduced by Sekizawa and studied then by Kowalski-Sekizawa), which generalize the classical Riemann extension defined by Patterson-Walker. Let M be a manifold with an affine connection and let T * M be the total space of its cotangent bundle. On T * M endowed with a natural Riemann extension, we study here the Laplacian and give necessary and sufficient conditions for the harmonicity of a certain family of (local) functions. We also prove a gradient formula for natural Riemann extensions.

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Section: The Natural Riemann Extensionḡ Is Defined In Terms Of Classimentioning

confidence: 99%

“…For the Levi-Civita connection∇ of the Riemann extensionḡ, we get the formulas (see e.g., [8]): Moreover, for X ∈ χ (M), ∇ X is a (1, 1)-tensor field of M defined by…”

Section: Notations 31 Ifmentioning

confidence: 99%

On some differential operators on natural Riemann extensions

Bejan

Kowalski

2015

Ann Glob Anal Geom

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“…For the Levi-Civita connection ∇ of the proper natural Riemann extension g we get the formulas (see [5]):…”

Section: Almost Para-hermitian Structures Induced By Natural Riemann mentioning

confidence: 99%

Almost Para-Hermitian and Almost Paracontact Metric Structures Induced by Natural Riemann Extensions

Bejan

Nakova

2018

Results Math

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In this paper we consider a manifold (M, ∇) with a symmetric linear connection ∇ which induces on the cotangent bundle T * M of M a semi-Riemannian metric g with a neutral signature. The metric g is called natural Riemann extension and it is a generalization (made by M. Sekizawa and O. Kowalski) of the Riemann extension, introduced by E. K. Patterson and A. G. Walker (1952). We construct two almost para-Hermitian structures on (T * M, g) which are almost para-Kähler or para-Kähler and prove that the defined almost para-complex structures are harmonic. On certain hypersurfaces of T * M we construct almost paracontact metric structures, induced by the obtained almost para-Hermitian structures. We determine the classes of the corresponding almost paracontact metric manifolds according to the classification given by S. Zamkovoy and G. Nakova (2018). We obtain a necessary and sufficient condition the considered manifolds to be paracontact metric, K-paracontact metric or para-Sasakian.Date: 9th November 2018.

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“…For further references relation to Riemannian extensions, see [1,6,10,15,21,22,23]. Classical Riemannian extensions have been generalized in several ways, see, as an example [13]. In [3,4], the authors introduced another generalization which is called modified Riemannian extension.…”

Section: Introductionmentioning

confidence: 99%

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

Bilen

Gezer

2018

J. Geom.

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Let M be an n−dimensional differentiable manifold equipped with a torsion-free linear connection ∇ and T * M its cotangent bundle. The present paper aims to study a metric connection ∇ with nonvanishing torsion on T * M with modified Riemannian extension g ∇,c . First, we give a characterization of fibre-preserving projective vector fields on (T * M, g ∇,c ) with respect to the metric connection ∇. Secondly, we study conditions for (T * M, g ∇,c ) to be semi-symmetric, Ricci semi-symmetric, Z semi-symmetric or locally conharmonically flat with respect to the metric connection ∇. Finally, we present some results concerning the Schouten-Van Kampen connection associated to the Levi-Civita connection ∇ of the modified Riemannian extension g ∇,c .Mathematics subject classification 2010. 53C07, 53C35, 53A45.

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On natural Riemann extensions

Cited by 20 publications

References 11 publications

On some differential operators on natural Riemann extensions

On some differential operators on natural Riemann extensions

Almost Para-Hermitian and Almost Paracontact Metric Structures Induced by Natural Riemann Extensions

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

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