Differential Geometry

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both L 2 combinatorial and L 2 analytic torsion invariants associated to flat Hilbertian bundles over compact polyhedra and manifolds; we view them as volume forms on the reduced L 2 homology and cohomology. These torsion invariants specialize to the the classical Reidemeister-Franz torsion and the Ray-Singer torsion in the finite dimensional case. Under the assumption that the L 2 homology vanishes, the determinant line can be canonically identified with $\R$, and our L 2 torsion invariants specialize to the L 2 torsion invariants previously constructed by A.Carey, V.Mathai and J.Lott. We also show that a recent theorem of Burghelea et al. can be reformulated as stating equality between two volume forms (the combinatorial and the analytic) on the reduced L 2 cohomology.

We prove an index theorem concerning the pushforward of flat B-vector bundles, where B is an appropriate algebra. We construct the associated analytic torsion form T. If Z is a smooth closed aspherical manifold, we show that T gives invariants of the homotopy groups of Diff(Z).

The usual formulations of time-dependent mechanics start from a given splitting Y=R×M of the coordinate bundle Y→R . From physical viewpoint, this splitting means that a reference frame has been chosen. Obviously, such a splitting is broken under reference frame transformations and time-dependent canonical transformations. Our goal is to formulate time-dependent mechanics in gauge-invariant form, i.e., independently of any reference frame. The main ingredient in this formulation is a connection on the bundle Y→R which describes an arbitrary reference frame. We emphasize the following peculiarities of this approach to time-dependent mechanics. A phase space does not admit any canonical contact or presymplectic structure which would be preserved under reference frame transformations, whereas the canonical Poisson structure is degenerate. A Hamiltonian fails to be a function on a phase space. In particular, it can not participate in a Poisson bracket so that the evolution equation is not reduced to the Poisson bracket. This fact becomes relevant to the quantization procedure. Hamiltonian and Lagrangian formulations of time-dependent mechanics are not equivalent. A degenerate Lagrangian admits a set of associated Hamiltonians, none of which describes the whole mechanical system given by this Lagrangian.

The fibre bundles adjoint to generalized almost quaternionic structures are studied. The most important classes of generalized almost quaternionic manifolds are considered.

Notions of self-dual and anti self-dual almost quaternionic structures are introduced. The complete classification of self-dual and anti self-dual generalized Kaehler manifolds is obtained.

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces. The main new technical ingredient in the proofs is the use of a proper "non-stationary" generalization of the classical theory of normal forms for local contractions.

This text is a revised version of the authors Habilitationsschrift which was submitted to the University of Augsburg, 1993. Fuchs type differential operators are used to model the analysis on manifolds with cone--like singularities, or more general, stratified spaces. This book provides a self--contained treatment of the analysis of the heat equation and index theory for these operators. Major topics are short--time asymptotics, η -- and ζ --functions, (relative) index theorems. Another chapter is devoted to the discussion of deficiency indices and Dirac Schrödinger operators.

We define for arbitrary modules over a finite von Neumann algebra $\cala$ a dimension taking values in [0,∞] which extends the classical notion of von Neumann dimension for finitely generated projective $\cala$-modules and inherits all its useful properties such as additivity, cofinality and continuity. This allows to define L 2 -Betti numbers for arbitrary topological spaces with an action of a discrete group Γ extending the well-known definition for regular coverings of compact manifolds. We show for an amenable group Γ that the p -th L 2 -Betti number depends only on the $\cc\Gamma$-module given by the p -th singular homology. Using the generalized dimension function we detect elements in $G_0(\cc\Gamma)$, provided that Γ is amenable. We investigate the class of groups for which the zero-th and first L 2 -Betti numbers resp. all L 2 -Betti numbers vanish. We study L 2 -Euler characteristics and introduce for a discrete group Γ its Burnside group extending the classical notions of Burnside ring and Burnside ring congruences for finite Γ . Keywords: Dimension functions for finite von Neumann algebras, L 2 -Betti numbers, amenable groups, Grothendieck groups, Burnside groups

Poisson homogeneous spaces for Poisson groupoids are classfied in terms of Dirac structures for the corresponding Lie bialgebroids. Applications include Drinfel'd's classification in the case of Poisson groups and a description of leaf spaces of foliations as homogeneous spaces of pair groupoids.

We translate a classification scheme for periodic CMC surfaces developed by J. Dorfmeister and the author to discrete CMC surfaces in the sense of A. Bobenko and U. Pinkall. The scheme uses the dressing action on discrete CMC surfaces to arrive at a classification for periodic discrete CMC surfaces.