Adapted connections on metric contact manifolds

In this paper, we describe the space of adapted connections on a metric contact manifold through the space of their torsion tensors. The torsion tensor is an element of the space of TM-valued two-forms, which splits into various subspaces. We study the parts of the torsion tensor according to this splitting to completely describe the space of adapted connections. We use this description to obtain characterizations of the generalized Tanaka-Webster connection and to describe the Dirac operators of adapted connections.Comment: 25 pages; some remarks added, minor correction

A Berger-type theorem for metric connections with skew-symmetric torsion

We prove a Berger-type theorem which asserts that if the orthogonal subgroup generated by the torsion tensor (pulled back to a point by parallel transport) of a metric connection with skew-symmetric torsion is not transitive on the sphere, then the space must be locally isometric to a Lie group with a bi-invariant metric or its symmetric dual (we assume the space to be locally irreducible). We also prove that a (simple) Lie group with a bi-invariant metric admits only two flat metric connections with skew-symmetric torsion: the two flat canonical connections. In particular, we get a refinement of a well-known theorem by Cartan and Schouten. Finally, we show that the holonomy group of a metric connection with skew-symmetric torsion on these spaces generically coincides with the Riemannian holonomy.Comment: 13 pages; we add some new examples and fix minor misprints; final version to appear in Journal of Geometry and Physic

The Gaussian Measure On Algebraic Varieties

We prove that the ring \Aff{\R}{M} of all polynomials defined on a real algebraic variety $M\subset\R^n$ is dense in the Hilbert space L^2(M,e^{-|x|^2}\de\mu), where \de\mu denotes the volume form of $M$ and \de\nu=e^{-|x|^2}\de\mu the Gaussian measure on $M$.Comment: Latex2.09, 6 page