Spacetime symmetries

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

The variational problem in lagrange spaces endowed with a special type of (α,β)-metrics

In this paper, we will continue our investigation on the new recently introduced (?,?)-metric F = ? + a?2+?2/? in [12]; where ? is a Riemannian metric; ? is an 1-form and a ? (1/4,+?) is a real positive scalar. We will investigate the variational problem in Lagrange spaces endowed with this type of metrics. Also, we will study the dually local flatness for this type of metric and we will proof that this kind of metric can be reduced to a locally Minkowskian metric. Finally, we will introduce the 2-Killing equation in Finsler spaces.

A cut-off tubular geometry of loop space

Motivated by the computation of loop space quantum mechanics as indicated in [14], here we seek a better understanding of the tubular geometry of loop space [Formula: see text] corresponding to a Riemannian manifold [Formula: see text] around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of [Formula: see text] around the diagonal submanifold, where [Formula: see text] is the Cartesian product of [Formula: see text] copies of [Formula: see text] with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to [Formula: see text] can be obtained by taking the limit [Formula: see text]. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [21] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-[Formula: see text] limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of [Formula: see text] which, for [Formula: see text], is the tangent bundle [Formula: see text].

Dualities and geometrical invariants for static and spherically symmetric spacetimes

In this work, we consider spinless particles in curved spacetime and symmetries related to extended isometries. We search for solutions of a generalized Killing equation whose structure entails a general class of Killing tensors. The conserved quantities along particle’s geodesic are associated with a dual description of the spacetime metric. In the Hamiltonian formalism, some conserved quantities generate a dual description of the metric. The Killing tensors belonging to the conserved objects imply in a nontrivial class of dual metrics even for a Schwarzschild metric in the original spacetime. From these metrics, we construct geometrical invariants for classes of dual spacetimes to explore their singularity structure. A nontrivial singularity behavior is obtained in the dual sector.

HOMOTHETIC KILLING VECTORS IN EXPANDING $\mathcal{HH}$-SPACES WITH Λ

Conformal Killing equations and their integrability conditions for expanding hyperheavenly spaces with Λ in spinorial formalism are studied. It is shown that any conformal Killing vector reduces to homothetic or isometric Killing vector. Reduction of respective Killing equation to one master equation is presented. Classification of homothetic and isometric Killing vectors is given. Type [D] ⊗ [any] is analyzed in detail and some expanding [Formula: see text] complex metrics of types [III, N] ⊗ [III, N] with Λ admitting isometric Killing vectors are found.

CONSERVED QUANTITIES AND DUALITIES FOR PARTICLES IN CURVED SPACE-TIME

We consider particles moving in curved space-time and associated symmetries. We use a generalized Killing equation and search for solutions involving Killing tensors associated with space-time metric. With these tensors some conserved quantities are constructed and they are valid along particles geodesic. In the Hamiltonian formalism for these particles, the conserved quantities can generate a dual description of the metric. We construct nontrivial dual metrics and consider a kind of geometric duality that involves a completely different space-time. From these metrics we calculate geometric invariants and examine the singularity structure of the dual space-time.