Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection ∇) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of the the triple (M,∇,g) is investi...

We prove that harmonic morphisms preserve the Jacobi operator along harmonic maps. We apply this result to prove infinitesimal and local rigidity (in the sense of Toth) of harmonic morphisms to a sphere. 1. Harmonic morphisms Harmonic maps φ : (M, g) → (N, h) between two smooth Riemannian manifolds are critical points of the energy functional E(φ,Ω) = 1 2 ∫ Ω |dφ| dvg for any compact domain Ω ⊆...

The current paper deals with filtering problems where the observation process, conditioned on the unknown signal, is an elliptic diffusion in a differentiable manifold. Precisely, the observation model is given by a stochastic differential equation in the underlying manifold. The main new idea is to use a Le Jan-Watanabe connection, instead of a usual Levi-Civita connection, in performing the o...

We reinterpret the well known Taub-singularity in terms of a cylinder symmetric geometry. It is shown that a cylindrical analog to the Einstein-Rosen bridge as well as a cosmic string will be present in the geometry. PACS numbers: 02.40.tm , 04.20.Jb Typeset using REVTEX Electronic address: [email protected] Electronic address: [email protected] and [email protected] 1 It is a...

The purpose of this article is to provide an explicit construction for a family of singular instantons on S 4 \ S 2 with arbitrary real holonomy parameter α. This family includes the original α = 1/4, c2 = 3/2 solution discovered by P. Forgács, Z. Horváth, and L. Palla, and our approach is modeled on that of their 1981 paper. Our primary tool is the ansatz due to Corrigan, Fairlie, Wilczek, and...

Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov [34]. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c1 = 0 which are never Kähler; ...

Pseudo-Riemannian geometry is, to a large extent, the study of the Levi-Civita connection, which is the unique torsion-free connection compatible with the metric structure. There are, however, other affine connections which arise in different contexts, such as conformal geometry, contact structures, Weyl structures, and almost Hermitian geometry. In this book, we reverse this point of view and ...

The reparametrization-invariant generating functional for the unitary and causal perturbation theory in general relativity in a finite space-time is obtained. The region of validity of the FaddeevPopov-DeWitt functional is studied. It is shown that the invariant content of general relativity as a constrained system can be covered by two ”equivalent” unconstrained systems: the ”dynamic” (with ”d...

As mentioned in the accompanying biography, Levi-Civita and his teacher Ricci,1 prompted by Felix Klein, wrote an important review of tensor calculus (under the name of ‘absolute differential calculus’) in 1901 [1]. Einstein, advised by Marcel Grossmann, began to study this in 1912 and corresponded with Levi-Civita about it in the period leading up to the publication of the first paper on gener...