Invariants of conformal and projective structures
Given a non-degenerate affine hypersurface \(M^n\) of \(\mathbb{R}^{n+ 1}\), then any so-called relative normalization (that is, the choice of a transversal vector field with vanishing transversal connection form) induces a semi-Riemannian metric \(h\) and a torsion-free Ricci-symmetric connection \(\nabla\) on \(M^n\). Then \(h\) and \(\nabla\) determine the conjugate connection \(\overline\nabla\), which is again torsion-free and Ricci-symmetric (in fact, the author always tacitly assumes connections to be torsion-free and Ricci-symmetric). When changing the relative normalization, these data obviously change, \(h\) by a conformal transformation, \(\overline\nabla\) by a strongly projective transformation and \(\nabla\) by a so-called conprojective tarnsformation. In this way one has a natural class of conjugate triples on any non-degenerate hypersurface of \(\mathbb{R}^{n+1}\). The paper under review studies the converse question: given a class of such conjugate triples on a simply connected manifold \(M^n\), when does there exist an immersion of \(M^n\) into \(\mathbb{R}^{n+ 1}\) inducing this class on \(M^n\). The answer is very elegant: if and only if one of the connections in the class is flat. The next question is to determine necessary and sufficient conditions for such classes to contain a flat connection. For this purpose, the author studies some invariant curvature tensors which vanish in case the class contains a flat connection.
scientific article; zbMATH DE number 785207