Reconstruction of an Affine Connection in Generalized Fermi Coordinates

On a manifold with affine connection, we introduce special pre-semigeodesic charts which generalize Fermi coordinates. We use a version of the Peano’s–Picard’s-Cauchy-like Theorem on the initial values problem for systems of ODSs. In a fixed pre-semigeodesic chart of a manifold with a symmetric affine connection, we reconstruct, or construct, the connection in some neighborhood from the knowledge of the “initial values”, namely the restriction of the components of connection to a fixed surface S and from some of the components of the curvature tensor R in the full coordinate domain. In Riemannian space, analogous methods are used to retrieve (or construct) the metric tensor of a pseudo-Riemannian manifold in a domain of semigeodesic coordinates from the known restriction of the metric to some non-isotropic hypersurface and some of the components of the curvature tensor of type (0, 4) in the ambient space.