Discrete Vector Bundles with Connection and the Bianchi Identity

04/20/2021
by   Daniel Berwick-Evans, et al.
0

We develop a discrete theory of vector bundles with connection that is natural with respect to appropriate mappings of the base space. The main objects are discrete vector bundle valued cochains. The central operators are a discrete exterior covariant derivative and a combinatorial wedge product. We demonstrate the key properties of these operators and show that they are natural with respect to the mappings referred to above. We give a new interpretation in terms of a double averaging of anti-symmetrized cup product which serves as our discrete wedge product. We also formulate a well-behaved definition of metric compatible discrete connections. We characterize when a discrete vector bundle with connection is trivializable or has a trivial lower rank subbundle. This machinery is used to define discrete curvature as linear maps and we show that our formulation satisfies a discrete Bianchi identity.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/08/2022

2D discrete Hodge-Dirac operator on the torus

We discuss a discretisation of the de Rham-Hodge theory in the two-dimen...
research
06/21/2019

Finite Element Systems for vector bundles : elasticity and curvature

We develop a theory of Finite Element Systems, for the purpose of discre...
research
11/06/2017

A Linear Variational Principle for Riemann Mappings and Discrete Conformality

We consider Riemann mappings from bounded Lipschitz domains in the plane...
research
03/08/2020

Discrete Morse Theory, Persistent Homology and Forman-Ricci Curvature

Using Banchoff's discrete Morse Theory, in tandem with Bloch's result on...
research
02/01/2022

Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups

This paper develops the theory of discrete Dirac reduction of discrete L...
research
09/09/2021

Continuum limits for discrete Dirac operators on 2D square lattices

We discuss the continuum limit of discrete Dirac operators on the square...

Please sign up or login with your details

Forgot password? Click here to reset