Indefinite extrinsic symmetric spaces II

TL;DR

This paper advances the classification of indefinite extrinsic symmetric spaces by using quadratic extensions and cohomology, providing explicit classifications, including all Lorentzian cases, despite the non-semisimple transvection groups.

Contribution

It introduces a systematic method using quadratic extensions and cohomology to classify indefinite extrinsic symmetric spaces, especially when transvection groups are not semisimple.

Findings

Established a one-to-one correspondence between isometry classes and cohomology sets.

Developed explicit classification results for spaces with small index.

Classified all Lorentzian extrinsic symmetric spaces.

Abstract

Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the classification difficult. We use the recently developed method of quadratic extensions for (h,K)-invariant metric Lie algebras to tackle this problem. We obtain a one-to-one correspondence between isometry classes of extrinsic symmetric spaces and a certain cohomology set. This allows a systematic construction of extrinsic symmetric spaces and explicit classification results, e.g., if the metric of the embedded manifold or the ambient space has a small index. We will illustrate this by classifying all Lorentzian extrinsic symmetric spaces.