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Summary: Geometrical concepts induced by a smooth mapping $f:M\to N$ of manifolds with linear connections are introduced, especially the (higher order) covariant differentials of the mapping tangent to $f$ and the curvature of a corresponding tensor product connection. As an useful and physically meaningful consequence a basis of differential invariants for natural operators of such smooth mappings is obtained for metric connections. A relation to geometry of Riemannian manifolds is discussed.
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