Pedro M. Gadea

Reductive homogeneous pseudo-Riemannian manifolds

A classification of homogeneous pseudo-Riemannian structures and a characterization of each primitive class are obtained. Several examples are also given.

Homogeneous Lorentzian structures on the oscillator groups

Abstract. We obtain all the homogeneous pseudo-Riemannian structures on the oscillator groups equipped with a family of left-invariant Lorentzian metrics. Moreover, in the 4-dimensional case we determine all the corresponding reductive decompositions and groups of isometries.

Manifolds modelled over free modules over the double numbers

Manifolds over the algebra of double numbers, which include the case of manifolds equipped with a pair of equidimensional supplementary foliations, are studied. To this end, B-holomorphic functions and B-analytic functions on Bn, where B denotes the algebra of double numbers, are defined and studied.

Reductive decompositions and Einstein-Yang-Mills equations associated to the oscillator group

All of the homogeneous Lorentzian structures on the oscillator group equipped with a bi-invariant Lorentzian metric, and then the associated reductive pairs, are obtained. Some of them are solutions of the Einstein–Yang–Mills equations.

The canonical eight-form on manifolds with holonomy group Spin(9)

An explicit expression of the canonical 8-form on a Riemannian manifold with a Spin(9)-structure, in terms of the nine local symmetric involutions involved, is given. The list of explicit expressions of all the canonical forms related to Bergers list of holonomy groups is thus completed. Moreover, some results on Spin(9)-structures as G-structures defined by a tensor and on the curvature tensor of the Cayley planes, are obtained.

Homogeneous Kaehler and Sasakian structures related to complex hyperbolic space

We study homogeneous Kähler structures on a non-compact Hermitian symmetric space and their lifts to homogeneous Sasakian structures on the total space of a principal line bundle over it, and we analyze the case of the complex hyperbolic space.

Homogeneous Riemannian Structures on Berger 3-Spheres

The homogeneous Riemannian structures on the three-dimensional Berger spheres, their corresponding reductive decompositions and the associated groups of isometries are obtained. The Berger 3-spheres are also considered as homogeneous almost contact metric manifolds.

Homogeneous spin Riemannian manifolds with the simplest Dirac operator

Abstract We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds (M, g) which are traceless cyclic with respect to some quotient expression M = G/K and reductive decomposition 𝔤 = 𝔨 ⊕ 𝔪. Using transversally symmetric fibrations of noncompact type, we give a list of them.

The Lorentzian oscillator group as a geodesic orbit space

We prove that the four-dimensional oscillator group Os, endowed with any of its usual left-invariant Lorentzian metrics, is a Lorentzian geodesic (so, in particular, null-geodesic) orbit space with some of its homogeneous descriptions corresponding to certain homogeneous Lorentzian structures. Each time that Os is endowed with a suitable metric and an appropriate homogeneous Lorentzian structure, it is a candidate for constructing solutions in d-dimensional supergravity with at least 24 of the 32 possible supersymmetries.

Some Definitions and Theorems

The present chapter collects some of the main definitions and theorems on which the problems in the previous chapters rely, with special attention to those elements of the theory that sometimes give rise to confusion due to a lack of general agreement in the literature on the precise formulation and/or notation.