Warped products and Spaces of Constant Curvature

TL;DR

This paper generalizes the warped product decompositions of spaces with constant curvature to arbitrary signatures, providing a comprehensive reference with detailed models and a review of circles and spheres in pseudo-Riemannian manifolds.

Contribution

It extends Nolker's results to arbitrary signatures and offers detailed models and a review, serving as a reference in the field.

Findings

Warped product decompositions for arbitrary signature spaces are obtained.

Detailed models for Euclidean and Lorentzian signature spaces are provided.

A review of circles and spheres in pseudo-Riemannian manifolds is included.

Abstract

We will obtain the warped product decompositions of spaces of constant curvature (with arbitrary signature) in their natural models as subsets of pseudo-Euclidean space. This generalizes the corresponding result by S. Nolker to arbitrary signatures, and has a similar level of detail. Although our derivation is complete in some sense, none is proven. Motivated by applications, we will give more information for the spaces with Euclidean and Lorentzian signatures. This is an expository article which is intended to be used as a reference. So we also give a review of the theory of circles and spheres in pseudo-Riemannian manifolds.