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Exploring certain geometric and harmonic properties of the Berger-type metric conformal deformation on the Para-Kähler-Norden manifold

Abderrahim Zagane, Fethi Latti

TL;DR

The paper introduces a Berger-type metric conformal deformation $g^{\alpha}$ on a para-Kähler-Norden manifold, derives the adapted Levi-Civita connection, and computes the deformed curvature, Ricci, and scalar curvatures, including simplifications when $\alpha$ is a Killing potential. It then analyzes harmonic and biharmonic maps in this setting, deriving explicit tension and bitension fields for the identity map in both directions of the deformation and for maps between manifolds, yielding concrete criteria for harmonicity and (proper) biharmonicity. These results extend Berger-type deformations to para-Kähler-Norden geometry and provide tractable formulas to study geometric properties under the deformation. The work has potential implications for understanding geometric flow-like phenomena and energy-minimizing maps in manifolds equipped with para-Kähler-Norden structures under conformal Berger-type deformations.

Abstract

This work presents a novel class of metrics on a para-Kähler-Norden manifold $(M^{2m},F,g)$, derived from a conformal deformation of the Berger-type metric associated with the metric $g$. Initially, we examine the Levi-Civita link associated with this metric. Secondly, we delineate all varieties of curvature for a manifold $M$ equipped with a conformal deformation of Berger-type metric for $g$. Finally, we studied a certain class of harmonic maps.

Exploring certain geometric and harmonic properties of the Berger-type metric conformal deformation on the Para-Kähler-Norden manifold

TL;DR

The paper introduces a Berger-type metric conformal deformation on a para-Kähler-Norden manifold, derives the adapted Levi-Civita connection, and computes the deformed curvature, Ricci, and scalar curvatures, including simplifications when is a Killing potential. It then analyzes harmonic and biharmonic maps in this setting, deriving explicit tension and bitension fields for the identity map in both directions of the deformation and for maps between manifolds, yielding concrete criteria for harmonicity and (proper) biharmonicity. These results extend Berger-type deformations to para-Kähler-Norden geometry and provide tractable formulas to study geometric properties under the deformation. The work has potential implications for understanding geometric flow-like phenomena and energy-minimizing maps in manifolds equipped with para-Kähler-Norden structures under conformal Berger-type deformations.

Abstract

This work presents a novel class of metrics on a para-Kähler-Norden manifold , derived from a conformal deformation of the Berger-type metric associated with the metric . Initially, we examine the Levi-Civita link associated with this metric. Secondly, we delineate all varieties of curvature for a manifold equipped with a conformal deformation of Berger-type metric for . Finally, we studied a certain class of harmonic maps.
Paper Structure (10 sections, 25 theorems, 80 equations)

This paper contains 10 sections, 25 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Fundamental properties of the Berger-type metric conformal deformation
  3. Curvatures of Berger-type metric conformal deformation
  4. Harmonicity concerning the Berger-type metric conformal deformation
  5. The harmonicity of the identity map
  6. Harmonicity of the map $\varphi:(M^{m},g)\rightarrow (N^{2n},h^{\alpha})$
  7. Harmonicity of the map $\phi:(M^{2m},g^{\alpha})\rightarrow (N^{n},h)$
  8. The biharmonicity of identity map
  9. The biharmonicity of identity map $I:(M^{2m},F,g)\rightarrow (M^{2m}, g^{\alpha})$
  10. The biharmonicity of identity map $I:(M^{2m}, g^{\alpha})\rightarrow (M^{2m},F,g)$

Key Result

Lemma 2.2

Given a para-Kähler-Norden manifold $(M^{2m},F,g)$, then we have for each vector fields $X,Y$ and $Z$ on $M$.

Theorems & Definitions (39)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • proof
  • ...and 29 more