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Compact 3-manifolds with a flat Carnot-Carathéodory metric. (English) Zbl 0796.53024

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.

MSC:

53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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