On a theorem of Fermi. (English) Zbl 0888.53030

Slavskii, V. V.

Commentat. Math. Univ. Carol. 37, No. 4, 867-872 (1996).

Summary: A conformally flat metric \(\bar g\) is said to be Ricci superosculating with \(g\) at the point \(x_0\) if \(g_{ij}(x_0)=\bar g_{ij} (x_0),\;\Gamma^k_{ij} (x_0)=\overline\Gamma^k_{ij}(x_0),\;R^k_{ij} (x_0)=\overline R^k_{ij}(x_0)\), where \(R_{ij}\) is the Ricci tensor. In this paper the following theorem is proved:
If \(\gamma\) is a smooth curve in the Riemannian manifold \(M\) (without self-intersections), then there is a neighbourhood of \(\gamma\) and a conformally flat metric \(\bar g\) which is Ricci superosculating with \(g\) along the curve \(\gamma\).

MSC:

53C20	Global Riemannian geometry, including pinching
53A30	Conformal differential geometry (MSC2010)

Keywords:

conformal connection; development