Frame bundle

About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.

Unified formalism for Palatini gravity

Abstract: The present article is devoted to the construction of a unified formalism for Palatini and unimodular gravity. The basic idea is to employ a relationship between unified formalism for a Griffiths variational problem and its classical Lepage-equivalent variational problem. As a way to understand from an intuitive viewpoint the Griffiths variational problem approach considered here, we may say the variations of the Palatini Lagrangian are performed in such a way that the so called metricity condition, i.e. (part of) the condition ensuring that the connection is the Levi-Civita connection for the metric specified by the vielbein, is preserved. From the same perspective, the classical Lepage-equivalent problem is a geometrical implementation of the Lagrange multipliers trick, so that the metricity condition is incorporated directly into the Palatini Lagrangian. The main geometrical tools involved in these constructions are canonical forms living on the first jet of the frame bundle for the spacetime manifold $M$. These forms play an essential role in providing a global version of the Palatini Lagrangian and expressing the metricity condition in an invariant form; with their help, it was possible to formulate an unified formalism for Palatini gravity in a geometrical fashion. Moreover, we were also able to find the associated equations of motion in invariant terms and, by using previous results from the literature, to prove their involutivity. As a bonus, we showed how this construction can be used to provide a unified formalism for the so called unimodular gravity by employing a reduction of the structure group of the principal bundle $LM$ to the special linear group $SL\left(m\right),m=\mathop{\text{dim}}{M}$.

Causal geometries, null geodesics, and gravity

Abstract: The authors study a generalized notion of null geodesic defined by the Legendrian dynamics of a regular conical subbundle of the tangent bundle on a manifold. A natural extension of the Weyl tensor is shown to exist, and to depend only on this conical subbundle. Given a suitable defining function of the conical bundle, the Raychaudhuri--Sachs equations of general relativity continue to hold, and give rise to the same phenomenon of covergence of null geodesics in regions of positive energy that underlies the theory of gravitation.

A report on functorial connections and differential invariants

Abstract: Let M be an n -dimensional manifold, π:F(M)→M the linear frame bundle, and G a closed subgroup of GL(n,R) . As is known, there is a one-to-one correspondence between the G -structures on M and the sections of the bundle π ¯ :F(M)/G→M . A functorial connection is an assignment of a linear connection ∇(σ) on M to each section σ of the bundle π ¯ which satisfies the following properties: ∇(σ) is reducible to the subbundle P σ ⊂FM corresponding to σ , depends continuously on σ , and for every diffeomorphism f:M→M there holds ∇(f⋅σ)=f⋅∇(σ) . The article is a survey of the authors' recent results concerning functorial connections and their use in constructing differential invariants of G -structures. The most attention is concentrated on the problem of existence of a functorial connection for a given subgroup G⊂GL(n,R) and on the calculation of the number of functionally independent differential invariants of a given order. Special consideration is devoted to the G -structures determined by linear and projective parallelisms and by pseudo-Riemannian metrics.

The direct image of a vector bundle

Abstract: It is well known that if M is a vector bundle over a space Y , and r : X → Y is a map, then there is an ‘induced’ vector bundle r * M over X . It is also known that a vector bundle L over X determines a ‘direct image’ vector bundle r * L over Y provided that r is a finite covering map ((1), § 1).