Reconstruction of an affine connection in generalized Fermi coordinates (Q506195)

The reconstruction of a Riemannian metric from its curvature (tensor, Ricci) is a relatively old and interesting problem of Riemannian geometry. As the solution of such problem requires to solve a complicated non-linear system of PDE's, one possibility is to choose a convenient coordinate system with respect to which the system of PDE's is simplified. Such approach, has been successful, for example in the work of \textit{H. H. Hacisalihoglu} and \textit{A. Kh. Amirov} [Sib. Math. J. 39, No. 4, 864--871 (1998); translation from Sib. Mat. Zh. 39, No. 4, 1005--1012 (1998; Zbl 0913.53019)], where they used semigeodesic coordinates to simplify the problem. These are a generalization of the Fermi coordinates. In the present article, the authors introduce pre-semigeodesic charts and apply a version of the Peano-Picard-Cauchy theorem to reconstruct the symmetric affine connection, by the knowledge of its restriction to a fixed \((n-1)\)-dimensional surface and some components of the curvature tensor.