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On the properties of coframes

Giovanni Canepa

Abstract

We consider injectivity and surjectivity of some maps on the exterior algebra of isomorphic finite-dimensional vector spaces. We prove the properties of the maps in full generality, for any dimension of the vector space and any subspace. We also draw a connection with the Palatini-Cartan formulation of General Relativity, for which these maps are of crucial importance.

On the properties of coframes

Abstract

We consider injectivity and surjectivity of some maps on the exterior algebra of isomorphic finite-dimensional vector spaces. We prove the properties of the maps in full generality, for any dimension of the vector space and any subspace. We also draw a connection with the Palatini-Cartan formulation of General Relativity, for which these maps are of crucial importance.

Paper Structure

This paper contains 8 sections, 14 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Definitions and conventions
  3. Technical Results
  4. Proof of Theorem \ref{['thm:Main_intro']}
  5. Injectivity
  6. Surjectivity
  7. Corollaries
  8. Coframe of Palatini--Cartan gravity

Key Result

Theorem 1

The map $W_{s}^{l (n,k)}$ defined in e:def_W_intro is surjective if and only if and it is injective if and only if

Theorems & Definitions (31)

  • Theorem 1
  • Corollary 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 21 more