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Riemannian Warped Product Maps

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Abstract

In this paper, we introduce Riemannian warped product map as a generalization of warped product isometric immersion and Riemannian warped product submersion with examples and obtain some characterizations. First, we define Riemannian warped product map and find conditions for a Riemannian map to be Riemannian warped product map. We show that Riemannian warped product map is a composition of Riemannian warped product submersion followed by warped product isometric immersion locally. In addition, we show that Riemannian warped product map satisfies the generalized eikonal equation which is a well known bridge between geometrical and physical optics. We also find necessary and sufficient conditions for the fibers, range space of the derivative map of Riemannian warped product map and horizontal distributions to be totally geodesic and minimal. Further, we give some fundamental geometric properties for the study of such smooth maps. Precisely, we construct Gauss formula (second fundamental form), Weingarten formula and tension field. We obtain necessary and sufficient conditions for a Riemannian warped product map to be totally geodesic, harmonic and umbilical. Comparatively, we analyse the obtained results with the existing results for a Riemannian map between Riemannian manifolds.

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Acknowledgements

We are grateful to the referee(s) for his/her efforts to improve the quality of the paper. Kiran Meena gratefully acknowledges the financial support provided by the Department of Atomic Energy, Government of India [Offer Letter No.: HRI/133/1436 Dated 29 November 2022] and research facilities provided by Harish-Chandra Research Institute, Prayagraj, India. The first author Kiran Meena is thankful to her doctoral degree adviser Dr. Akhilesh Yadav for his motivational words.

Funding

Kiran Meena is financial supported by the Department of Atomic Energy, Government of India [Offer Letter No.: HRI/133/1436 Dated 29 November 2022].

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Authors and Affiliations

  1. Department of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Prayagraj, Uttar Pradesh, 211019, India

    Kiran Meena & Hemangi Madhusudan Shah

  2. Department of Mathematics, Ege University, 35100, Izmir, Turkey

    Bayram Şahin

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  1. Kiran Meena
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  2. Bayram Şahin
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  3. Hemangi Madhusudan Shah
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Contributions

(i) Kiran Meena: Conceptualization, Methodology, Investigation, Writing - original draft, Editing of Revision. (ii) Bayram Şahin: Methodology, Investigation, Supervision, Validation, Review. (iii) Hemangi Madhusudan Shah: Methodology, Investigation, Editing. All the authors read and agreed to the manuscript.

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Correspondence to Bayram Şahin.

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Meena, K., Şahin, B. & Shah, H.M. Riemannian Warped Product Maps. Results Math 79, 56 (2024). https://doi.org/10.1007/s00025-023-02084-1

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