Classification of Einstein metrics on the product of an interval with a three-sphere

TL;DR

This paper classifies all smooth, complete Einstein metrics on the product space of an interval and a three-sphere, revealing a diverse set of geometries including known and new metric families.

Contribution

It provides a comprehensive classification of Einstein metrics on I x S^3 with separate functions for base and fiber, covering all smooth, complete cases.

Findings

Includes many well-known Einstein metrics.

Identifies several one-parameter families of Einstein metrics.

Demonstrates the richness of possible geometries on the space.

Abstract

We present a complete classification of Einstein metrics on the space M = I \times S^3, where I is the interval (0,l) or (0,\infty) or their closures, and we consider separate metric functions f and h (functions of I) for the base and fiber of the Hopf fibration S^1 -> S^3 -> S^2. All such metrics yielding smooth and complete manifolds are included and discussed. The results are surprisingly rich, including many well-known examples and several one-parameter families of metrics with a variety of geometries.