Paper · Jan 1, 2024

Clairaut anti-invariant Riemannian maps with Kähler and Ricci soliton structures

Jyoti Yadav, Gauree Shanker

math.DGarXiv:2401.00709v2
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Abstract

The aim of this article is to explore the Clairaut anti-invariant Riemannian maps from/to Kähler manifolds admitting Ricci solitons. We find the curvature relations and calculate the Ricci tensor under different conditions. We discuss the condition under which range space becomes -Ricci soliton. We obtain conditions for the range and kernel spaces of these maps to be Einstein. Next, we find the scalar curvature for range space. Further, we give the relation between Ricci curvature and Lie derivative under these maps. Moreover, we find the condition for a vertical potential vector field on target manifold to be conformal vector field on range space of these maps. Finally, we give non-trivial examples of such maps.

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Submitted
Jan 1, 2024
Last revised
Jun 24, 2024 (v2)
Subjects
math.DG

Comments: 25 pages

Ricci solitonsKähler geometryScalar curvature

Its neighbourhood

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)1Meena 2021 Conformal Submersions Whose Total Manifolds Admit a Ricci Soliton (Kiran Meena and Akhilesh Yadav, 2022)8Meena 2022 Nature of Some Solitons on Almost coKähler Manifolds and Asymptotically Harmonic Manifolds (Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya, 2023)5Ghosh 2023 Clairaut conformal submersions from Ricci solitons (Murat Polat, 2023)3Polat 2023 On the geometry of Riemannian warped product maps (Jyoti Yadav, Harmandeep Kaur and Gauree Shanker, 2025)2Yadav 2025 On the non-existence of symphonic maps (Ahmed Mohammed Cherif and Kaddour Zegga, 2026)7Cherif 2026 Some uniform estimates for Kähler–Ricci Shrinkers (Junsheng Zhang, 2026)6Zhang 2026 Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature (Matheus Andrade Ribeiro de Moura Horácio, 2026)4Horácio 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Clairaut Riemannian maps

    Kiran Meena and Akhilesh Yadav

    Topics Scalar curvature, Kähler geometry, Ricci solitons · Shared terms “Riemannian map”, “anti invariant”, “Clairaut”

  2. On the geometry of Riemannian warped product maps

    Jyoti Yadav, Harmandeep Kaur and Gauree Shanker

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

  3. Clairaut conformal submersions from Ricci solitons

    Murat Polat

    Topics Scalar curvature, Ricci solitons · Shared terms “Clairaut”, “vector field”, “conformal vector”

  4. Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature

    Matheus Andrade Ribeiro de Moura Horácio

    Topics Scalar curvature, Kähler geometry, Ricci solitons · Shared terms “Lie derivative”, “vector field”, “Ricci tensor”

  5. Nature of Some Solitons on Almost coKähler Manifolds and Asymptotically Harmonic Manifolds

    Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya

    Topics Scalar curvature, Kähler geometry, Ricci solitons · Shared terms “manifold admitting”, “Einstein”

  6. Some uniform estimates for Kähler–Ricci Shrinkers

    Junsheng Zhang

    Topics Scalar curvature, Kähler geometry, Ricci solitons

  7. On the non-existence of symphonic maps

    Ahmed Mohammed Cherif and Kaddour Zegga

    Topics Ricci solitons · Shared terms “conformal vector”, “manifold admitting”, “vector field”

  8. Conformal Submersions Whose Total Manifolds Admit a Ricci Soliton

    Kiran Meena and Akhilesh Yadav

    Topics Scalar curvature, Ricci solitons · Shared terms “calculate”, “trivial example”, “vector field”

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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

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.