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General Natural \(\alpha \)-Structures Parallel with Respect to the Schouten–Van Kampen Connection on the Tangent Bundle

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Abstract

We determine the Schouten–Van Kampen connection associated to the Levi-Civita connection of a general natural metric on the total space TM of the tangent bundle of a Riemannian manifold. We provide the necessary and sufficient conditions for the obtained Schouten–Van Kampen connection to be torsion free and then to coincide with the Levi-Civita connection. We characterize the general natural \(\alpha \)-structures on TM, which are parallel with respect to the torsion free Schouten–Van Kampen connection. Finally, we obtain the (para-)Kähler structures of general natural lift type on TM, for which the \(\alpha \)-structure is parallel with respect to both Levi-Civita and Schouten–Van Kampen connection.

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Acknowledgements

The author wants to thank all the referees for their valuable reports.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Technical University “Gheorghe Asachi” of Iaşi, Bd. Carol I, No. 11A, 700506, Iaşi, Romania

    S. L. Druţă-Romaniuc

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  1. S. L. Druţă-Romaniuc
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Correspondence to S. L. Druţă-Romaniuc.

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Dedicated to the memory of Professor Vasile Oproiu (1941–2020)

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Druţă-Romaniuc, S.L. General Natural \(\alpha \)-Structures Parallel with Respect to the Schouten–Van Kampen Connection on the Tangent Bundle. Mediterr. J. Math. 19, 195 (2022). https://doi.org/10.1007/s00009-022-02093-4

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  • DOI: https://doi.org/10.1007/s00009-022-02093-4

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