On a theorem of Fermi

Slavskii, V. V.

Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 4, pp. 867-872.

Résumé

Conformally flat metric $\bar g$ is said to be Ricci superosculating with $g$ at the point $x_0$ if $g_{ij}(x_0)=\bar g_{ij}(x_0)$, $\Gamma _{ij}^k(x_0)=\bar \Gamma _{ij}^k(x_0)$, $R_{ij}^k(x_0)=\bar R_{ij}^k(x_0)$, where $R_{ij}$ is the Ricci tensor. In this paper the following theorem is proved: \medskip {\sl If $\,\gamma $ is a smooth curve of the Riemannian manifold $M$ {\rm (}without self-crossing{\rm (}, then there is a neighbourhood of $\,\gamma $ and a conformally flat metric $\bar g$ which is the Ricci superosculating with $g$ along the curve $\gamma $.\/}

Détail

Classification : 53A30, 53B20, 53C20
Keywords: conformal connection; development