scholarly journals On Jacobi fields and a canonical connection in sub-Riemannian geometry

2017 ◽  
pp. 77-92 ◽  
Author(s):  
Davide Barilari ◽  
Luca Rizzi
Keyword(s):  
Riemannian Geometry ◽  
Jacobi Fields ◽  
Canonical Connection

Geodesics and Jacobi fields in singular semi-riemannian geometry

1994 ◽  
Vol 446 (1928) ◽  
pp. 441-452 ◽  
Keyword(s):  
Singular Point ◽  
Uniqueness Theorem ◽  
Riemannian Geometry ◽  
Existence And Uniqueness ◽  
Smooth Manifold ◽  
Tensor Field ◽  
Final Section ◽  
Existence And Uniqueness Theorem ◽  
Jacobi Fields ◽  
Neutral Vector

This paper proves an existence and uniqueness theorem for geodesics tangent to a neutral vector at a stable singular point of a smooth symmetric two tensor field g on a smooth manifold M . The final section is devoted to a proof of existence and uniqueness of Jacobi fields along the above mentioned geodesics.

Jacobi fields in nonholonomic mechanics

2021 ◽  
Author(s):  
Alexandre Anahory Simoes ◽  
Juan Carlos Marrero ◽  
David Martin de Diego
Keyword(s):  
Riemannian Geometry ◽  
Nonholonomic System ◽  
Nonholonomic Mechanics ◽  
Jacobi Fields ◽  
Similar Characterization ◽  
Complete Lift ◽  
Curvature And Torsion ◽  
Nonholonomic Connection

Abstract In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi fields in two equivalent ways: the first one, using an appropriate complete lift of the nonholonomic system and, in the second one, using the curvature and torsion of the associated nonholonomic connection.

scholarly journals BRST and Superfield Formalism—A Review

Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 280
Author(s):  
Loriano Bonora ◽  
Rudra Prakash Malik
Keyword(s):  
Riemannian Geometry ◽  
Gauge Theories ◽  
Original Material ◽  
Comprehensive Description ◽  
Higher Spin ◽  
Gauge Anomalies

This article, which is a review with substantial original material, is meant to offer a comprehensive description of the superfield representations of BRST and anti-BRST algebras and their applications to some field-theoretic topics. After a review of the superfield formalism for gauge theories, we present the same formalism for gerbes and diffeomorphism invariant theories. The application to diffeomorphisms leads, in particular, to a horizontal Riemannian geometry in the superspace. We then illustrate the application to the description of consistent gauge anomalies and Wess–Zumino terms for which the formalism seems to be particularly tailor-made. The next subject covered is the higher spin YM-like theories and their anomalies. Finally, we show that the BRST superfield formalism applies as well to the N=1 super-YM theories formulated in the supersymmetric superspace, for the two formalisms go along with each other very well.

scholarly journals Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

2021 ◽  
Author(s):  
Andreas Bernig ◽  
Dmitry Faifman ◽  
Gil Solanes
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Riemannian Space ◽  
Space Forms ◽  
Type Formula ◽  
Isometric Embeddings ◽  
Curvature Measures ◽  
Riemannian Space Forms

AbstractThe recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

scholarly journals A relativistic basis of the quantum theory.—II

1934 ◽  
Vol 145 (855) ◽  
pp. 645-656 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Wave Equation ◽  
Quantum Theory ◽  
General Expression ◽  
Matrix Equation ◽  
Covariant Derivative ◽  
Riemannian Geometry ◽  
Order Equation ◽  
Second Order ◽  
Space Curvature

The paper is a continuation of the last paper communicated to these 'Proceedings.' In that paper, which we shall refer to as the first paper, a more general expression for space curvature was obtained than that which occurs in Riemannian geometry, by a modification of the Riemannian covariant derivative and by the use of a fifth co-ordinate. By means of a particular substitution (∆ μσ σ = 1/ψ ∂ψ/∂x μ ) it was shown that this curvature takes the form of the second order equation of quantum mechanics. It is not a matrix equation, however but one which has the character of the wave equation as it occurred in the earlier form of the quantum theory. But it contains additional terms, all of which can be readily accounted for in physics, expect on which suggested an identification with energy of the spin.

scholarly journals WEYL GEOMETRY AS CHARACTERIZATION OF SPACE-TIME

2011 ◽  
Vol 03 ◽  
pp. 87-97 ◽  
Author(s):  
F. P. POULIS ◽  
J. M. SALIM
Keyword(s):  
General Relativity ◽  
Scalar Field ◽  
Riemannian Geometry ◽  
Axiomatic Approach ◽  
Space Time ◽  
Weyl Geometry ◽  
Test Particles ◽  
Variational Formalism ◽  
Physical Fields

Motivated by an axiomatic approach to characterize space-time it is investigated a reformulation of Einstein's gravity where the pseudo-riemannian geometry is substituted by a Weyl one. It is presented the main properties of the Weyl geometry and it is shown that it gives extra contributions to the trajectories of test particles, serving as one more motivation to study general relativity in Weyl geometry. It is introduced its variational formalism and it is established the coupling with other physical fields in such a way that the theory acquires a gauge symmetry for the geometrical fields. It is shown that this symmetry is still present for the red-shift and it is concluded that for cosmological models it opens the possibility that observations can be fully described by the new geometrical scalar field. It is concluded then that this reformulation, although representing a theoretical advance, still needs a complete description of their objects.

Differential and Riemannian Geometry.

1966 ◽  
Vol 73 (3) ◽  
pp. 327
Author(s):  
G. F. Feeman ◽  
Detlef Laugwitz ◽  
Fritz Steinhardt
Keyword(s):  
Riemannian Geometry
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