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An introduction to spectral and differential geometry in Carnot-Carathéorody spaces. (English) Zbl 1105.58012
Slovák, Jan (ed.) et al., The proceedings of the 24th winter school “Geometry and physics”, Srní, Czech Republic, January 17–24, 2004. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 75, 139-196 (2005).
The author, starting by a compact manifold \(M\), considers the large time behaviour of heat on forms on the fundamental cover \(\widetilde{M}\). The large time heat-decay exponents are called the Novikov-Shubin numbers, and they are known to be homotopic-invariants of \(M\). It is also known that the exponent on functions is related to the growth of \(\pi_1(M)\). However, in higher degrees very little is known about the geometric meaning of these numbers.
The author studies in this very nice (and highly self-contained) paper the case where \(\widetilde{M}\) is a graded nilpotent group \(G\) (Carnot group). He shows how the study on \(1\)-forms leads to considering higher-order differential operators, that fit into differential complexes. The author’s construction extends to Carnot-Carathéodory spaces, that is manifolds with a bracket generating distribution in their tangent bundle.
For the entire collection see [Zbl 1074.53001].

58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35K05 Heat equation