We describe for any Riemannian manifold M a certain scheme M L, lying in between the first and second neighbourhood of the diagonal of M. Semi-conformal maps between Riemannian manifolds are then analyzed as those maps that preserve M L; harmonic maps are analyzed as those that preserve the (Levi-Civita-) mirror image formation inside M L.
We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction.