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Affine connections on the algebra of differential forms

Yong Wang, Shuang Wang

Abstract

In this paper, we define the semi-symmetric metric connection on the algebra of differential forms. We compute some special semi-symmetric metric connections and their curvature tensor and their Ricci tensor on the algebra of differential forms. We study the distribution on the algebra of differential forms and we get its Gauss-Codazzi-Ricci equations associated to the semi-symmetric metric connection. We also study the Lie derivative of the distribution on the algebra of differential forms. We define the canonical connection and the Schouten connection and the Vrancreanu connection on the algebra of differential forms and get some properties of these connections.

Affine connections on the algebra of differential forms

Abstract

In this paper, we define the semi-symmetric metric connection on the algebra of differential forms. We compute some special semi-symmetric metric connections and their curvature tensor and their Ricci tensor on the algebra of differential forms. We study the distribution on the algebra of differential forms and we get its Gauss-Codazzi-Ricci equations associated to the semi-symmetric metric connection. We also study the Lie derivative of the distribution on the algebra of differential forms. We define the canonical connection and the Schouten connection and the Vrancreanu connection on the algebra of differential forms and get some properties of these connections.
Paper Structure (6 sections, 32 theorems, 83 equations)

This paper contains 6 sections, 32 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. A semi-symmetric metric connection on on the algebra of differential forms
  3. The distribution on the algebra of differential forms and a semi-symmetric metric connection
  4. The Lie derivative of connections for distributions on the algebra of differential forms
  5. Several other affine connections on the algebra of differential forms
  6. Acknowledgements

Key Result

Theorem 2.5

(Theorem 4.2 in MS)There is a unique symmetric (torsionless) and metric compatible affine graded connection $\nabla^L$ which satisfies the Koszul formula for all homogeneous $X,Y,Z\in {\rm Der}\Omega(M)$.

Theorems & Definitions (53)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Proposition 2.10
  • ...and 43 more