Amalgamated Codazzi Raychaudhuri identity for foliation

TL;DR

This paper develops a unified tensor formalism to extend the Raychaudhuri and Codazzi identities, facilitating advanced curvature analysis of embedded surfaces and foliations in physical theories like cosmology.

Contribution

It introduces a flexible tensor framework that generalizes classical identities, enabling comprehensive curvature analysis in complex physical and geometric contexts.

Findings

Unified tensor formalism for curvature analysis

Extension of Raychaudhuri and Codazzi identities

Application to topological defects in cosmology

Abstract

It is shown how a pure background tensor formalism provides a concise but explicit and highly flexible machinery for the generalised curvature analysis of individual embedded surfaces and foliations such as arise in the theory of topological defects in cosmological and other physical contexts. The unified treatment provided here shows how the relevant extension of the Raychaudhuri identity is related to the correspondingly extended Codazzi identity.