Closed generalized Einstein manifolds with radially flat Ricci curvature

TL;DR

This work establishes rigidity results for closed generalized $(\lambda, n+m)$-Einstein manifolds with constant scalar curvature under a radially flat Ricci curvature condition $\omega= df \wedge i_{\nabla f} z = 0$. By developing a network of tensorial identities involving the Cotton, Weyl, and a new 3-tensor $T$, the authors derive structural constraints that force sphere-like geometries: either ${\mathbb S}^n$ or a product with a circle in the zero-set-disconnectivity case, and, under positive isotropic curvature, the product case reduces to ${\mathbb S}^1 \times {\mathbb S}^{n-1}$. A stronger condition $z(\nabla f,\nabla f)=0$ yields the sphere, and when the zero set is connected, a warped-product analysis combined with conformal geometry shows $M$ is ${\mathbb S}^n$ up to finite cover. The results connect static-vacuum-like equations, warped-product Einstein structures, and curvature conditions to achieve sphere rigidity in a generalized Einstein setting.

Abstract

In this paper, we show that a closed $n$-dimensional generalized $(λ, n+m)$-Einstein manifold of constant scalar curvature with weakly radially zero Ricci curvature is isometric to either a sphere ${\Bbb S}^n$, or a product ${\Bbb S}^{1} \times Σ^{n-1}$ of a circle with an $(n-1)$-dimensional Einstein manifold of positive Ricci curvature, up to finite cover and rescaling. Furthermore, if we assume $(M, g)$ has positive isotropic curvature, $M$ must be isometric to either a sphere ${\Bbb S}^n$, or a product ${\Bbb S}^{1} \times {\Bbb S}^{n-1}$ of a circle with an $(n-1)$-sphere.

Theorems & Definitions (72)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof