Richard Atkins

The Lie Algebra of Local Killing Fields

We present an algebraic procedure that finds the Lie algebra of the local Killing fields of a smooth metric. In particular, we determine the number of independent local Killing fields about a given point on the manifold. Spaces of constant curvature and locally symmetric spaces are also discussed. Furthermore, we obtain a complete classification of the Lie algebra of local Killing fields for surfaces in terms of conditions upon the Gauss curvature.

The Newtonian potential in gravitational cohomology

We examine the Newtonian potential in gravitational cohomology. This is given by a symmetric, two-index tensor field, which satisfies the wave equation in empty space. Furthermore, the associated gravitational field strength, obtained by applying the coboundary operator to the potential, is constructed for the case of a stationary point mass and shown to give the classical result.

Parallel metrics and reducibility of the holonomy group

In this paper we investigate the relationship between the existence of parallel semiRiemannian metrics of a connection and the reducibility of the associated holonomy group. The question as to whether the holonomy group necessarily reduces in the presence of a specified number of independent parallel semi-Riemannian metrics is completely determined by the the signature of the metrics and the dimension d of the manifold, when d / 4. In particular, the existence of two independent, parallel semi-Riemannian metrics, one of which having signature (p, q) with p / q, implies the holonomy group is reducible. The (p,p) cases, however, may allow for more than one parallel metric and yet an irreducible holonomy group: for n = 2m, m ^ 3, there exist connections on R with irreducible infinitesimal holonomy and which have two independent, parallel metrics of signature (m,m). The case of four-dimensional manifolds, however, depends on the topology of the manifold in question: the presence of three parallel metrics always implies reducibility but reducibility in the case of two metrics of signature (2,2) is guaranteed only for simply connected manifolds. The main theorem in the paper is the construction of a topologically non-trivial four-dimensional manifold with a connection that admits two independent metrics of signature (2,2) and yet has irreducible holonomy. We provide a complete solution to the general problem.