The embedding of the spacetime in five-dimensional spaces with arbitrary non-degenerate Ricci tensor

TL;DR

This paper extends embedding theorems for semi-Riemannian manifolds, showing they can be embedded in higher-dimensional spaces with arbitrary non-degenerate Ricci tensors, which is significant for higher-dimensional spacetime theories.

Contribution

It proves a new theorem extending the Campbell-Magaard theorem, allowing embeddings with prescribed Ricci tensors in higher-dimensional spaces.

Findings

Any n-dimensional semi-Riemannian manifold can be locally embedded in an (n+1)-dimensional space with a specified Ricci tensor.

The theorem generalizes previous embedding results to include spaces with arbitrary non-degenerate Ricci tensors.

The embedding of Ricci-flat spacetimes is explicitly demonstrated.

Abstract

We discuss and prove a theorem which asserts that any n-dimensional semi-Riemannian manifold can be locally embedded in a (n+1)-dimensional space with a non-degenerate Ricci tensor which is equal, up to a local analytic diffeomorphism, to the Ricci tensor of an arbitrary specified space. This may be regarded as a further extension of the Campbell-Magaard theorem. We highlight the significance of embedding theorems of increasing degrees of generality in the context of higher dimensional spacetimes theories and illustrate the new theorem by establishing the embedding of a general class of Ricci-flat spacetimes.