44,696 research outputs found
Extended Derdzinski-Shen theorem for the Riemann tensor
We extend a classical result by Derdzinski and Shen, on the restrictions
imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor.
The new conditions of the theorem include Codazzi tensors (i.e. closed 1-forms)
as well as tensors with gauged Codazzi condition (i.e. "recurrent 1-forms"),
typical of some well known differential structures.Comment: 5 page
An introduction to inflation after Planck: from theory to observations
These lecture notes have been written for a short introductory course on the
status of inflation after Planck and BICEP2, given at the Xth Modave School of
Mathematical Physics. The first objective is to give an overview of the theory
of inflation: motivations, homogeneous scalar field dynamics, slow-roll
approximation, linear theory of cosmological perturbations, classification of
single field potentials and their observable predictions. This includes a
pedagogical derivation of the primordial scalar and tensor power spectra for
any effective single field potential. The second goal is to present the most
recent results of Planck and BICEP2 and to discuss their implications for
inflation. Bayesian statistical methods are introduced as a tool for model
analysis and comparison. Based on the recent work of J. Martin et al., the best
inflationary models after Planck and BICEP2 are presented. Finally a series of
open questions and issues related to inflation are mentioned and briefly
discussed.Comment: 39 pages, 9 figures, to be published in the proceedings of the Xth
Modave School in Mathematical Physics. Important parts of those lecture notes
draw from chapters 1 and 2 of the author's PhD thesis, arXiv:1109.557
E. Cartan's attempt at bridge-building between Einstein and the Cosserats -- or how translational curvature became to be known as {\em torsion}
\'Elie Cartan's "g\'en\'eralisation de la notion de courbure" (1922) arose
from a creative evaluation of the geometrical structures underlying both,
Einstein's theory of gravity and the Cosserat brothers generalized theory of
elasticity. In both theories groups operating in the infinitesimal played a
crucial role. To judge from his publications in 1922--24, Cartan developed his
concept of generalized spaces with the dual context of general relativity and
non-standard elasticity in mind. In this context it seemed natural to express
the translational curvature of his new spaces by a rotational quantity (via a
kind of Grassmann dualization). So Cartan called his translational curvature
"torsion" and coupled it to a hypothetical rotational momentum of matter
several years before spin was encountered in quantum mechanics.Comment: 36 p
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