(local) coframe

Overview

A coframe is the dual of a (local) frame so it is a basis for the dual of a vector bundle (namely $T^{\ast }M$).

Definition

A (local) coframe is a set of covector fields $({\epsilon }_1,\dots ,{\epsilon }_n)$ defined on $U\subseteq M$ that is linearly independent and spans $T^{\ast }U$.

Coframe dual to frame

Given a local frame of $TM$ $(E_1,\dots ,E_n)$, a (local) coframe dual to $(E_i)$ is a set of covector fields $({\epsilon }_1,\dots ,{\epsilon }_n)$ such that $(({\epsilon }_1{)}_p,\dots ,({\epsilon }_n{)}_p)$ is the dual basis to $((E_1{)}_p,\dots ,(E_n{)}_p)$.