Some canonical metrics {\em via} Aubin's local deformations

TL;DR

This paper uses Aubin's local metric deformations to construct Riemannian metrics with specific Weyl tensor properties on all compact manifolds, showing in dimension four that no topological obstructions prevent metrics with non-vanishing Bach tensor.

Contribution

It introduces a method based on Aubin's deformations to produce metrics with prescribed Weyl tensor conditions, especially demonstrating the absence of topological obstructions in four dimensions.

Findings

Existence of metrics with non-vanishing Weyl tensor on all compact manifolds.

In dimension four, no topological obstructions to metrics with non-zero Bach tensor.

Application of Aubin's deformations to control curvature properties.

Abstract

In this paper, using special metric deformations introduced by Aubin, we construct Riemannian metrics satisfying non-vanishing conditions concerning the Weyl tensor, on every compact manifold. In particular, in dimension four, we show that there are no topological obstructions for the existence of metrics with non-vanishing Bach tensor.