invariant vector field
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2021 ◽  
Vol 11 (18) ◽  
pp. 8763
Author(s):  
Carlos Barceló ◽  
Raúl Carballo-Rubio ◽  
Luis J. Garay ◽  
Gerardo García-Moreno

One of the main problems that emergent-gravity approaches face is explaining how a system that does not contain gauge symmetries ab initio might develop them effectively in some regime. We review a mechanism introduced by some of the authors for the emergence of gauge symmetries in [JHEP 10 (2016) 084] and discuss how it works for interacting Lorentz-invariant vector field theories as a warm-up exercise for the more convoluted problem of gravity. Then, we apply this mechanism to the emergence of linear diffeomorphisms for the most general Lorentz-invariant linear theory of a two-index symmetric tensor field, which constitutes a generalization of the Fierz–Pauli theory describing linearized gravity. Finally we discuss two results, the well-known Weinberg–Witten theorem and a more recent theorem by Marolf, that are often invoked as no-go theorems for emergent gravity. Our analysis illustrates that, although these results pinpoint some of the particularities of gravity with respect to other gauge theories, they do not constitute an impediment for the emergent gravity program if gauge symmetries (diffeomorphisms) are emergent in the sense discussed in this paper.


Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1357
Author(s):  
Sergio Grillo ◽  
Juan Carlos Marrero ◽  
Edith Padrón

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that πU has a manifold structure and πU:U→πU, the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If XU is not vertical with respect to πU, we show that such complete solutions solve the reconstruction equations related to XU and G, i.e., the equations that enable us to write the integral curves of XU in terms of those of its projection on πU. On the other hand, if XU is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of XU up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve expξt, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of expξt is valid for all ξ inside an open dense subset of g.


2020 ◽  
Vol 20 (3) ◽  
pp. 391-400
Author(s):  
Gauree Shanker ◽  
Kirandeep Kaur

AbstractWe prove the existence of an invariant vector field on a homogeneous Finsler space with exponential metric, and we derive an explicit formula for the S-curvature of a homogeneous Finsler space with exponential metric. Using this formula, we obtain a formula for the mean Berwald curvature of such a homogeneous Finsler space.


2020 ◽  
Vol 17 (08) ◽  
pp. 2050112
Author(s):  
Masoumeh Hosseini ◽  
Hamid Reza Salimi Moghaddam

In this paper, we classify all simply connected five-dimensional nilpotent Lie groups which admit [Formula: see text]-metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During this classification, we give the geodesic vectors, Levi-Civita connection, curvature tensor, sectional curvature and [Formula: see text]-curvature.


2020 ◽  
Vol 27 (1) ◽  
pp. 111-120 ◽  
Author(s):  
Mehri Nasehi ◽  
Mansour Aghasi

AbstractIn this paper, we first classify Einstein-like metrics on hypercomplex four-dimensional Lie groups. Then we obtain the exact form of all harmonic maps on these spaces. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and determine all critical points for their energy functional restricted to vector fields of the same length. Furthermore, we give a complete and explicit description of all totally geodesic hypersurfaces of these spaces. The existence of Einstein hypercomplex four-dimensional Lie groups and the non-existence of non-trivial left-invariant Ricci and Yamabe solitons on these spaces are also proved.


2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Mehri Nasehi ◽  
Mansour Aghasi

Abstract In this paper we first classify left-invariant generalized Ricci solitons on four-dimensional hypercomplex Lie groups equipped with three families of left-invariant Lorentzian metrics. Then, on these Lorentzian spaces, we explicitly calculate the energy of an arbitrary left-invariant vector field X and determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. Furthermore, we give a complete and explicit description of all homogeneous structures on these spaces in both Riemannian and Lorentzian cases and determine some of their types. The existence of Einstein four-dimensional hypercomplex Lorentzian Lie groups is proved and it is shown that although the results concerning Einstein-like metrics, conformally flatness and some equations in the Riemannian case are much richer than their Lorentzian analogues, in the Lorentzian case, there exist some new critical points of energy functionals, homogeneous structures and geodesic vectors which do not exist in the Riemannian case.


2009 ◽  
Vol 01 (01) ◽  
pp. 13-27 ◽  
Author(s):  
GABRIEL KATZ

Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula [Formula: see text] which takes its values in A(G). Here Ind G(v) denotes the equivariant index of the field v, [Formula: see text] the v-induced Morse stratification (see [10]) of the boundary ∂X, and [Formula: see text] the class of the (n - k)-manifold [Formula: see text] in A(G). We examine some applications of this formula to the equivariant real algebraic fields v in compact domains X ⊂ ℝn defined via a generic polynomial inequality. Next, we link the above formula with the equivariant degrees of certain Gauss maps. This link is an equivariant generalization of Gottlieb's formulas ([3, 4]).


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