Paper · May 3, 2025

On the geometry of Riemannian warped product maps

Jyoti Yadav, Harmandeep Kaur, Gauree Shanker

math.DGarXiv:2505.01865
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Abstract

In this paper, we begin by introducing Clairaut Riemannian warped product maps and establish the condition under which a regular curve becomes a geodesic. We obtain the conditions for a Riemannian warped product map to be Clairaut Riemannian warped product map followed by Ricci curvature. Further, we study the Ricci soliton structure on a Riemannian warped product manifold using curvature tensor. We examine the Bochner type formulae for Clairaut Riemannian warped product map and construct a supporting example. Furthermore, we extend the study to introduce and examine some geometric aspects of conformal Riemannian warped product maps. We derive the integral formula for scalar curvature of conformal Riemannian warped product map. Finally, we construct an example for conformal Riemannian warped product map.

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Submitted
May 3, 2025
Subjects
math.DG

Comments: 23 Pages

Ricci solitonsScalar curvatureRotational symmetry & warped products

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Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Clairaut Riemannian maps (Kiran Meena and Akhilesh Yadav, 2021)6Meena 2021 A note on gradient Ricci soliton warped metrics (José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro, 2021)4Gomes 2021 On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization (Abbas M Sherif, Peter K S Dunsby and Rituparno Goswami, 2021)7Sherif 2021 On noncompact warped product Ricci solitons (Valter Borges, 2022)8Borges 2022 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)5Kotschwar 2023 Clairaut anti-invariant Riemannian maps with Kähler and Ricci soliton structures (Jyoti Yadav and Gauree Shanker, 2024)1Yadav 2024 Geometric properties of second Ricci solitons (Masoumeh Khalili, Ghodratallah Fasihi-Ramandi and Shahroud Azami, 2025)2Khalili 2025 On Ricci solitons whose level hypersurfaces have parallel second fundamental form (Matheus Andrade Ribeiro de Moura Horácio, 2026)3Horácio 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Clairaut anti-invariant Riemannian maps with Kähler and Ricci soliton structures

    Jyoti Yadav and Gauree Shanker

    Topics Scalar curvature, Ricci solitons · Same authors Jyoti Yadav, Gauree Shanker · Shared terms “Clairaut”, “soliton structure”

  2. Geometric properties of second Ricci solitons

    Masoumeh Khalili, Ghodratallah Fasihi-Ramandi and Shahroud Azami

    Topics Rotational symmetry & warped products, Ricci solitons

  3. On Ricci solitons whose level hypersurfaces have parallel second fundamental form

    Matheus Andrade Ribeiro de Moura Horácio

    Topics Rotational symmetry & warped products, Scalar curvature, Ricci solitons

  4. A note on gradient Ricci soliton warped metrics

    José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro

    Topics Rotational symmetry & warped products, Ricci solitons

  5. Backward propagation of warped product structures and asymptotically conical shrinkers

    Brett Kotschwar

    Topics Rotational symmetry & warped products, Ricci solitons

  6. Clairaut Riemannian maps

    Kiran Meena and Akhilesh Yadav

    Topics Scalar curvature, Ricci solitons · Shared terms “Clairaut Riemannian”

  7. On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization

    Abbas M Sherif, Peter K S Dunsby and Rituparno Goswami

    Topics Rotational symmetry & warped products, Scalar curvature, Ricci solitons · Shared terms “soliton structure”

  8. On noncompact warped product Ricci solitons

    Valter Borges

    Topics Rotational symmetry & warped products, Ricci solitons

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.