Analytic torsion of generic rank two distributions in dimension five (Q2159436)

Generic rank two distributions are geometric structures on 5-manifolds; the local model is provided by a certain nilpotent Lie group \(G\) whose Lie algebra is explicitely given in terms of brackets. Explicitely, they are given by plane fields satisfying certain conditions on Lie brackets. To such a distribution one associates a so-called Rumin complex, which can be used to recover the de Rham cohomology of the underlying manifold [\textit{M. Rumin}, C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 11, 985--990 (1999; Zbl 0982.53029)]. The choice of a sub-Riemannian metric on the distribution allows to define an adjoint for the differentials of the Rumin complex, hence hypoelliptic (Laplacian-like) operators and finally an analytic torsion following the method of Ray-Singer (all analytical details in this context are checked in the paper under review). More generally one can consider complexes and torsion with coefficients in a flat bundle with a Hermitian metric. In dimension 3 the analogous setting is that of contact structures and for these the same construction was performed by Rumin-Seshadri who also showed that the resulting torsion can be related to the classical Ray-Singer torsion [\textit{M. Rumin} and \textit{N. Seshadri}, Ann. Inst. Fourier 62, No. 2, 727--782 (2012; Zbl 1264.58027)]. The main tool for this is a variational formula (computing variation in the torsion under variation of the metrics). Here the author establishes a similar result here for rank two distributions. Moreover he shows that this variation can be computed by integrating a local density (an ``anomaly formula''). Finally, he computes his torsion in terms of Ray-Singer torsion for 5-manifolds admitting locally flat generic rank two distributions (i.e. quotients of \(G\) by lattices).