A note on gradient Einstein-type manifolds

TL;DR

This paper proves that certain gradient Einstein-type manifolds with constant scalar curvature are isometric to spheres or Einstein manifolds under specific conditions, extending results to some noncompact cases.

Contribution

It establishes classification results for gradient Einstein-type manifolds, showing they are isometric to spheres or Einstein manifolds under various geometric assumptions.

Findings

Compact gradient Einstein-type manifolds with constant scalar curvature are isometric to the sphere.

Noncompact homogeneous gradient Einstein-type manifolds are Einstein.

Results extend to degenerate and nondegenerate cases under certain conditions.

Abstract

In this note, we show that a nontrivial, compact, degenerate or nondegenerate, gradient Einstein-type manifold of constant scalar curvature is isometric to the standard sphere with a well defined potential function. Moreover, under some geometric assumptions, the noncompact case is also treated. In this case, the main result is that a homogeneous, proper, noncompact, nondegenerate, gradient Einstein-type manifold is an Einstein manifold.