Świątkowski, Jacek Compact 3-manifolds with a flat Carnot-Carathéodory metric. (English) Zbl 0796.53024 Colloq. Math. 63, No. 1, 89-105 (1992). Let \(\Delta\) denote a distribution on a manifold \(M\). A Riemannian metric on \(\Delta\) enables us to measure the length of differentiable curves tangent to \(\Delta\). The metric obtained by minimizing the length of such curves is called the Carnot-Carathéodory (C-C for short) metric. Let \(\Delta\) be the invariant two-dimensional subbundle in the tangent bundle of the Heisenberg group \(H\) generated by two non-central vectors of its Lie algebra. The Riemannian metric on \(\Delta\) for which these generators are orthonormal yields a left invariant C-C metric on \(H\).A C-C metric on a three-dimensional manifold is called flat if it is locally isometric to \(H_ c\), \(H_ c\) being the Heisenberg group with the above described metric. In the reviewed paper, the author investigates compact 3-manifolds with a flat C-C metric. He describes all possible holonomy groups of such manifolds and considering each holonomy group separately, he obtains the full classification. Reviewer: M.Hotloś (Wrocław) MSC: 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) Keywords:Carnot-Carathéodory metric; Heisenberg group; holonomy group PDFBibTeX XMLCite \textit{J. Świątkowski}, Colloq. Math. 63, No. 1, 89--105 (1992; Zbl 0796.53024) Full Text: DOI EuDML