A geometric theory of harmonic and semi-conformal maps

We describe for any Riemannian manifold M a certain scheme ML, lying in between the first and second neighbourhood of the diagonal of M . Semiconformal maps between Riemannian manifolds are then analyzed as those maps that preserve ML; harmonic maps are analyzed as those that preserve the (LeviCivita-) mirror image formation inside ML. Introduction For any Riemannian manifold M , we describe a subscheme ML ⊆ M ×M , which encodes information about as well conformal as harmonic maps out of M in a succinct geometric way. Thus, a submersion φ : M → N between Riemannian manifolds is semi-conformal (=horizontally conformal) iff φ× φ maps ML into NL (Theorem 11); and a map φ : M → N is a harmonic map if it “commutes with mirror image formation for ML”, where mirror image formation is one of the manifestations of the Levi-Civita parallelism (derived from the Riemannian metric). The mirror image preservation property is best expressed in the set theoretic language for schemes, which we elaborate on in Section 1. Then it just becomes the statement: for (x, z) ∈ ML ⊆ M × M , φ(z) = (φ(z)), where the primes denote mirror image formation in x (respectively in φ(x)). In particular, when the codomain is R (the real line with standard metric), this characterization of harmonicity reads φ(z) = 2φ(x)− φ(z), that is, φ(x) equals the average value of φ(z) and φ(z), for any z with (x, z) ∈ ML.