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Bi-symphonic maps between Riemannian manifolds

Ahmed Mohammed Cherif, Kaddour Zegga

Abstract

This note introduces an extension to the definition of symphonic maps, denoted as $\varphi:(M,g)\longrightarrow(N,h)$, by exploring variations in the bi-energy functional associated with the pullback metric $\varphi^*h$ between two Riemannian manifolds.

Bi-symphonic maps between Riemannian manifolds

Abstract

This note introduces an extension to the definition of symphonic maps, denoted as , by exploring variations in the bi-energy functional associated with the pullback metric between two Riemannian manifolds.
Paper Structure (4 sections, 6 theorems, 49 equations)

This paper contains 4 sections, 6 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Second variation formula
  4. The first variation of the bi-energy functional

Key Result

Proposition 1

We have for any variation vector field $\upsilon$, where $\sigma_{\varphi}$ is defined by and the divergence of $\sigma_{\varphi}$ is given by here $\nabla d\varphi$ denotes the second fundamental form of $\varphi$.

Theorems & Definitions (15)

  • Proposition 1: First Variation Formula KN
  • Example 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm1']}
  • Theorem 4
  • Lemma 5
  • proof : Proof of Lemma \ref{['lem1']}
  • Lemma 6
  • proof : Proof of Lemma \ref{['Lem2']}
  • Lemma 7
  • ...and 5 more