AbstractIn this talk we will present applications of the noncommutative residue in Riemannian and CR geometry. It will be divided into 3 main parts.
1. Lower dimensional volumes in Riemannian geometry. Given a Riemannian manifold (M^n,g) a natural geometric is how to define the k'th dimensional volume of $M$ for $k=1,...,n$. Classical Riemannian geometry provides us with an answer for $k=n$ only. We will explain that by extending an idea of A. Connes we can make use of the noncommutative residue for classical PsiDO's and of the framework of noncommutative geometry to define in a purely differential- geometric fashion the k'th dimensional volumes for any k.
2. Lower dimensional volumes in Riemannian geometry. CR structures naturally arise in varous contexts. We also can define lower dimensional volumes in CR geometry. This involves constructing a noncommutative residue trace for the Heisenberg calculus, which is the relevant pseudodifferential calculus at stake in the CR setting.
3. New invariants for CR and contact manifolds. We can define new global invariants of CR and contact structures in terms of noncommutative residues of various geometric projections in the Heisenberg calculus. This allows us to recover recent results of Hirachi and Boutet de Monvel and to answer a question of Fefferman.