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$HS$-tensional maps and $HM$-tensional maps

Bouazza Kacimi, Ahmed Mohammed Cherif, Mustafa Özkan

Abstract

Let $ψ: (M,g)\longrightarrow (N,h)$ be a smooth map between Riemannian manifolds. The tension field of $ψ$ can be regarded as a map from $(M,g)$ into the Riemannian vector bundle $ψ^{-1}TN$, equipped with the Sasaki metric $G_{S}$. In this paper, we study certain aspects of two types of maps: those whose tension fields are harmonic maps (called $HM$-tensional maps) and those whose tension fields are harmonic sections (called $HS$-tensional maps).

$HS$-tensional maps and $HM$-tensional maps

Abstract

Let be a smooth map between Riemannian manifolds. The tension field of can be regarded as a map from into the Riemannian vector bundle , equipped with the Sasaki metric . In this paper, we study certain aspects of two types of maps: those whose tension fields are harmonic maps (called -tensional maps) and those whose tension fields are harmonic sections (called -tensional maps).

Paper Structure

This paper contains 6 sections, 20 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. The formal definition of the notion "$H$-tensional map" and some examples
  3. Nonexistence results of nonharmonic $H$-tensional maps
  4. A new member of Riemannian's manifold folks: $H$-tensional submanifolds
  5. Nonexistence results of nonminimal $H$-tensional submanifolds
  6. $H$-tensional curves

Key Result

Theorem 3.2

Let $\xi=\sum_{j=1}^m f_{j}\frac{\partial}{\partial x_{j}}$ be a vector field on the Euclidean space $(\mathbb{R}^{m}, <,>)$ with the canonical Euclidean metric $<,>$ and the usual global coordinates $(x_{1},\cdots,x_{m})$, such that $f_{j}=f_j(x)=f_{j}(x_{1},\cdots,x_{m})$. Then, the following sta

Theorems & Definitions (46)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Example 3.4
  • Example 3.5
  • ...and 36 more