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Clairaut semi-invariant Riemannian maps to Kaehler manifolds

Murat Polat, Kiran Meena

TL;DR

This work extends the Clairaut Riemannian map (CRM) framework to Clairaut semi-invariant Riemannian maps (CSIRM) from a Riemannian manifold to a Kähler manifold, unifying several RM generalizations. CSIRM is defined with a potential function via $s=e^{f}$, and the authors derive necessary and sufficient conditions for SIRMs to be CSIRM, as well as for CSIRMs to be harmonic or totally geodesic; they also characterize when the $(\ker F_\ast)^{\perp}$-induced foliations are totally geodesic and when the domain and target admit local product decompositions. A nontrivial example demonstrates CSIRM existence, and the results link CRM geometry to Kähler targets, offering criteria that govern geodesic behavior and foliations with potential implications in mathematical physics. Overall, the paper provides a structured framework for analyzing when CSIRMs retain harmonic or geodesic properties and how product structures arise in this setting.

Abstract

In this paper, first, we recall the notion of Clairaut Riemannian map (CRM) ${F}$ using a geodesic curve on the base manifold and give the Ricci equation. We also show that if base manifold of CRM is space form then leaves of $(ker{F}_\ast)^\perp$ become space forms and symmetric as well. Secondly, we define Clairaut semi-invariant Riemannian map (CSIRM) from a Riemannian manifold $(M, g_{M})$ to a Kähler manifold $(N, g_{N}, P)$ with a non-trivial example. We find necessary and sufficient conditions for a curve on the base manifold of semi-invariant Riemannian map (SIRM) to be geodesic. Further, we obtain necessary and sufficient conditions for a SIRM to be CSIRM. Moreover, we find necessary and sufficient condition for CSIRM to be harmonic and totally geodesic. In addition, we find necessary and sufficient condition for the distributions $\bar{D_1}$ and $\bar{D_2}$ of $(ker{F}_\ast)^\bot$ (which are arisen from the definition of CSIRM) to define totally geodesic foliations. Finally, we obtain necessary and sufficient conditions for $(ker{F}_\ast)^\bot$ and base manifold to be locally product manifold $\bar{D_1} \times \bar{D_2}$ and ${(range{F}_\ast)} \times {(range{F}_\ast)^\bot}$, respectively.

Clairaut semi-invariant Riemannian maps to Kaehler manifolds

TL;DR

This work extends the Clairaut Riemannian map (CRM) framework to Clairaut semi-invariant Riemannian maps (CSIRM) from a Riemannian manifold to a Kähler manifold, unifying several RM generalizations. CSIRM is defined with a potential function via , and the authors derive necessary and sufficient conditions for SIRMs to be CSIRM, as well as for CSIRMs to be harmonic or totally geodesic; they also characterize when the -induced foliations are totally geodesic and when the domain and target admit local product decompositions. A nontrivial example demonstrates CSIRM existence, and the results link CRM geometry to Kähler targets, offering criteria that govern geodesic behavior and foliations with potential implications in mathematical physics. Overall, the paper provides a structured framework for analyzing when CSIRMs retain harmonic or geodesic properties and how product structures arise in this setting.

Abstract

In this paper, first, we recall the notion of Clairaut Riemannian map (CRM) using a geodesic curve on the base manifold and give the Ricci equation. We also show that if base manifold of CRM is space form then leaves of become space forms and symmetric as well. Secondly, we define Clairaut semi-invariant Riemannian map (CSIRM) from a Riemannian manifold to a Kähler manifold with a non-trivial example. We find necessary and sufficient conditions for a curve on the base manifold of semi-invariant Riemannian map (SIRM) to be geodesic. Further, we obtain necessary and sufficient conditions for a SIRM to be CSIRM. Moreover, we find necessary and sufficient condition for CSIRM to be harmonic and totally geodesic. In addition, we find necessary and sufficient condition for the distributions and of (which are arisen from the definition of CSIRM) to define totally geodesic foliations. Finally, we obtain necessary and sufficient conditions for and base manifold to be locally product manifold and , respectively.
Paper Structure (4 sections, 15 theorems, 106 equations)

This paper contains 4 sections, 15 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. CSIRMs from Riemannian Manifolds to Kähler Manifolds
  4. Further Studies with Almost Hermitian or Kähler Manifolds

Key Result

Lemma 1

Sahin2 Let ${F}$ be a RM between Riemannian manifolds $(M, g_{M})$ and $(N,g_{N})$. Then, ${F}$ is umbilical RM if and only if for all $X, Y \in \Gamma (ker {F}_{\ast})^{\bot}$ and $H$ is the mean curvature vector field of $range{F}_{\ast}$.

Theorems & Definitions (34)

  • Lemma 1
  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • ...and 24 more