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Eells-Sampson type result of Symphonic map

Xiangzhi Cao

TL;DR

The paper extends Eells–Sampson–type rigidity to symphonic maps under a weak curvature condition on the target, encoded by $R^N(X,Y,Z,W)=a(h(X,Z)h(Y,W))-h(X,W)h(Y,Z)$ with $a\le K$ alongside a Ricci bound $\operatorname{Ric}_g \ge (m-1)K f^*h$. It proves that the pullback metric $f^*h$ is parallel and that, under these hypotheses, the map $f$ is constant; in the noncompact setting with finite symphonic energy $E_{sym}(f)$, constancy also holds. The proof combines a Bochner-type formula for $\|f^*h\|^2$, a positive quadratic form argument via matrices $a_{ik}$ and $b_{ij}$, and divergence-theorem techniques to establish rigidity. These results generalize prior compact-manifold symphonic-map theorems to a broader curvature regime, contributing to rigidity phenomena for maps minimizing generalized energy functionals.

Abstract

In this paper, we obtained Eells-Sampson type result of Symphonic map.

Eells-Sampson type result of Symphonic map

TL;DR

The paper extends Eells–Sampson–type rigidity to symphonic maps under a weak curvature condition on the target, encoded by with alongside a Ricci bound . It proves that the pullback metric is parallel and that, under these hypotheses, the map is constant; in the noncompact setting with finite symphonic energy , constancy also holds. The proof combines a Bochner-type formula for , a positive quadratic form argument via matrices and , and divergence-theorem techniques to establish rigidity. These results generalize prior compact-manifold symphonic-map theorems to a broader curvature regime, contributing to rigidity phenomena for maps minimizing generalized energy functionals.

Abstract

In this paper, we obtained Eells-Sampson type result of Symphonic map.

Paper Structure

This paper contains 2 sections, 6 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. the proof of main theorem

Key Result

Theorem 1

Let $f:\left(M^{m \geq 2}, g\right) \rightarrow(N, h)$ be a harmonic map between Riemannian manifolds. Assume that $M$ is closed and that there exists $K>0$ such that Then $f$ is a totally geodesic map. In particular, either i) $f$ is constant, or ii) $f$ is a homothetic immersion, $g$ has positive constant curvature and on $M$ and $f(M)$, respectively for any 2-plane $\Pi$ contained in $\mathrm

Theorems & Definitions (9)

  • Theorem 1: cf.MR4756031
  • Theorem 2: Compact case,MR3940323
  • Theorem 3
  • Remark 1
  • Theorem 4: Noncompact case
  • proof
  • Lemma 1
  • Lemma 2: Formula of Bochner type, cf.MR3940323
  • proof : Proof of Theorem \ref{['thm1']}