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In a neighborhood of a (positive definite) Riemannian space in which special,semigeodesic, coordinates are given, the metric tensor can be calculated fromits values on a suitable hypersurface and some of components of the curvaturetensor of type $(1,3)$ in the coordinate domain. Semigeodesic coordinates are ageneralization of the well-known Fermi coordinates, that play an important rolein mechanics and physics, are widely used in Minkowskian space, and indifferential geometry of Riemannian spaces in general. In the present paper, weconsider a more general situation. We introduce special pre-semigeodesic chartscharacterized both geometrically and in terms of the connection, formulate aversion of the Peano's-Picard's-Cauchy-like Theorem on existence and uniquenessof solutions of the initial values problems for systems of first-order ordinarydifferential equations. Then we use the apparatus in a fixed pre-semigeodesicchart of a manifold equipped with the linear symmetric connection. Our aim isto reconstruct, or construct, the symmetric linear connection in someneighborhood from the knowledge of the "initial conditions": the restriction ofthe connection to a fixed $(n-1)$-dimensional surface $S$ and some of thecomponents of the curvature tensor $R$ in the "volume" (coordinate domain).  By analogous methods, we retrieve (or construct) the metric tensor of type$(0,4)$ of a pseudo-Riemannian manifold in a domain of semigeodesic coordinatesfrom the known restriction of the metric to some non-isotropic hypersurface andsome of the components of the curvature tensor in the volume. In comparison toother authors, we give shorter proofs of constructive character based onclassical results on first order ODEs (ordinary differential equations).

Reconstruction of the connection or metric from some partial information