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An invitation to Cauchy-Riemann and sub-Riemannian geometries

A certain amount of complex analysis, in both one and several variables, combines with some differential geometry to form the basis of research in diverse parts of mathematics. We first describe that foundation and then invite the reader to follow specific geometric paths based upon it. We discuss such topics as the failure of the Riemann mapping theorem in several variables, the geometry of real hypersurfaces in complex Euclidean space, the Heisenberg group, sub-Riemannian manifolds and their metric space structure, the Hopf fibration, sub-Riemannian geodesics on the three-sphere, CR mappings invariant under a finite group, and finite type conditions. All of these topics fall within the branch of mathematics described by the title, but of course they form only a small part of it. We hope that the chosen topics are representative and appealing. The connections among them provide a fertile ground for future research. We modestly hope that this article inspires others to develop these connections further. To this end, our reference list includes a diverse collection of books and accessible articles. Given Riemann’s contributions to geometry, it is not surprising that his name occurs twice in the title. It is more surprising that the uses of his name here come from different parts of mathematics. We find it delightful that contemporary mathematics is forging connections that even Riemann could not have anticipated.