Constructing Bach Flat Manifolds of signature $(2,2)$ using the modified   Riemannian extension

E. Calvi\~no-Louzao; E. Garc\'ia-R\'io; P. Gilkey; I.; Guti\'errez-Rodr\'iguez; and R. V\'azquez-Lorenzo

arXiv:1807.09218·math.DG·February 20, 2019

Constructing Bach Flat Manifolds of signature $(2,2)$ using the modified Riemannian extension

E. Calvi\~no-Louzao, E. Garc\'ia-R\'io, P. Gilkey, I., Guti\'errez-Rodr\'iguez, and R. V\'azquez-Lorenzo

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TL;DR

This paper constructs Bach flat manifolds of signature (2,2) using modified Riemannian extensions of affine surfaces, highlighting the VSI property and introducing new scalar invariants to distinguish these manifolds.

Contribution

It introduces a method to construct Bach flat manifolds of signature (2,2) via modified Riemannian extensions and develops scalar invariants beyond Weyl type for their distinction.

Findings

01

All constructed examples are VSI manifolds.

02

Scalar invariants not of Weyl type can distinguish these manifolds.

03

Illustrations are provided in the context of homogeneous affine surfaces.

Abstract

We use the modified Riemannian extension of an affine surface to construct Bach flat manifolds. As all these examples are VSI (vanishing scalar invariants), we shall construct scalar invariants which are not of Weyl type to distinguish them. We illustrate this phenomena in the context of homogeneous affine surfaces.

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