Some Remarks on Geodesic and Curvature Preserving Mappings

Abstract

We ask for the converse of Gauss’ theorema egregium. Because in general isocurved manifolds are not isometric we ask stronger for isocurved, geodesic equivalent manifolds. For these we give a local criterion from which there follows that two-dimensional manifolds ${\overline{\mathcal{M}}}^2$ and of that type essentially are isometric, or both are Euclidean with an affine mapping in the ordinary sense.