Parallel Transport, Connections, and Covariant Derivatives

Author(s):  
Alexander Effland ◽  
Behrend Heeren ◽  
Martin Rumpf ◽  
Benedikt Wirth

Abstract We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end we extend the variational time discretization of geodesic calculus presented in Rumpf & Wirth (2015, Variational time discretization of geodesic calculus. IMA J. Numer. Anal., 35, 1011–1046), which just requires an approximation of the squared Riemannian distance that is typically easy to compute. First we obtain first-order discrete covariant derivatives via Schild’s ladder-type discretization of parallel transport. Second-order discrete covariant derivatives are then computed as nested first-order discrete covariant derivatives. These finally give rise to an approximation of the curvature tensor. First- and second-order consistency are proven for the approximations of the covariant derivative and the curvature tensor. The findings are experimentally validated on two-dimensional surfaces embedded in ${\mathbb{R}}^3$. Furthermore, as a proof of concept, the method is applied to a space of parametrized curves as well as to a space of shell surfaces, and discrete sectional curvature confusion matrices are computed on low-dimensional vector bundles.


2021 ◽  
pp. 189-212
Author(s):  
Andrew M. Steane

The mathematics of Riemannian curvature is presented. The Riemann curvature tensor and its role in parallel transport, in the metric, and in geodesic deviation are expounded at length. We begin by defining the curvature tensor and the torsion tensor by relating them to covariant derivatives. Then the local metric is obtained up to second order in terms of Minkowski metric and curvature tensor. Geometric issues such as the closure or non-closure of parallelograms are discussed. Next, the relation between curvature and parallel transport around a loop is derived. Then we proceed to geodesic deviation. The influence of global properties of the manifold on parallel transport is briefly expounded. The Lie derivative is then defined, and isometries of spacetime are discussed. Killing’s equation and properties of Killing vectors are obtained. Finally, the Weyl tensor (conformal tensor) is introduced.


Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4865-4873 ◽  
Author(s):  
Milos Petrovic

Generalized m-parabolic K?hler manifolds are defined and holomorphically projective mappings between such manifolds have been considered. Two non-linear systems of PDE?s in covariant derivatives of the first and second kind for the existence of such mappings are given. Also, relations between five linearly independent curvature tensors of generalized m-parabolic K?hler manifolds with respect to these mappings are examined.


Author(s):  
Michael Kachelriess

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.


2014 ◽  
Vol 7 (4) ◽  
pp. 2393-2422 ◽  
Author(s):  
Thomas Batard ◽  
Marcelo Bertalmío

1987 ◽  
Vol 95 ◽  
Author(s):  
J. P. Conde ◽  
S. Aljishi ◽  
D. S. Shen ◽  
V. Chu ◽  
Z E. Smith ◽  
...  

AbstractWe study the dark conductivity σd, dark conductivity activation energy Ea and photoconductivity σph of a-Si:H,F/a-Si,Ge:H,F superlattices both perpendicular and parallel to the plane of the layers. In parallel transport, both the σph and σd are dominated by the alloy layer characteristics with the superposition of carrier confinement quantum effects. In perpendicular transport, the σd shows an interplay of quantum mechanical tunneling through the barriers and of classical thermal emission over the barrier layer and the σph is controlled by the decreasing absorption by the silicon barrier layer as the optical gap Eopt of the structure decreases.We also found that the multilayer structure allows to grow lower gap a-Si,Ge:H,F alloys than achievable under the same deposition conditions for bulk materials. This stabilizing effect allowed us to study low-gap superlattice structures and extract information about these very low gap (<1.2 eV) a- Si,Ge:H,F alloys.


Sign in / Sign up

Export Citation Format

Share Document