Elasticity and conformal tension via the Kaluza-Klein mechanism
The authors consider semi-Riemannian tubes, i.e. warped products \(M\times_f S^1\), where \((M,g)\) is a semi-Riemannian manifold and \(f\) a smooth positive warping function on \(M\). For any \(p\in M\), the fibre \(\gamma_p=\{p\}\times S^1\) provides a circle. The \(r\)-elasticity of fibres is characterized. In particular, a theorem of classification of the triples \((M,g,f)\) which provide a foliation by \(r\)-elastic circles in \(M\times_f S^1\) is proved. Interesting examples are given. For instance, examples of foliated Kaluza-Klein conformal universes with \(G\)-submanifolds leaves are obtained. Among other applications, we mention a characterization of the photon sphere in a Schwarzschild space-time from the point of view of the \(r\)-elasticity.
scientific article; zbMATH DE number 1491092