Cauchy characteristic vector fields

Definition. Given an exterior differential system $\mathcal{I}$ a Cauchy characteristic vector field is a vector field $X$ such that $X\,\lrcorner\, \omega\in \mathcal{I}$ for every $\omega\in \mathcal{I}$.$\blacksquare$

Proposition. Cauchy characteristic vector fields constitute a Lie subalgebra of $\mathfrak{X}(M)$ .

$\blacksquare$

Proof. It can be shown using formula 4 in formulas for Lie derivative, exterior derivatives, bracket, interior product.

$\blacksquare$

Remarks

If $X$ is a Cauchy characteristic vector field of an EDS, then it is also a symmetry of the EDS. Therefore, in the case of a Pfaffian system, a Cauchy characteristic vector field is a trivial symmetry of the associated distribution.

To check that a vector field is a Cauchy characteristic vector fields we only have to check $X\,\lrcorner\, \omega\in \mathcal{I}$ for a set of forms $\omega$ that generate $\mathcal{I}$ algebraically.

The flow lines of a Cauchy characteristic vector field are called the Cauchy characteristics curves of the EDS. It has to do with method of characteristics...

In @bryant2013exterior and page 30 and Barco thesis page 33 it is defined at a point $x\in M$ the Cauchy characteristic space of an ideal $\mathcal{I}$:

$$ A(\mathcal{I})_x=\{\xi_x \in T_xM: \xi_x\,\lrcorner\,\mathcal{I}_x\subset \mathcal{I}_x\} $$

and

$$ C(\mathcal{I})_x=A(\mathcal{I})_x^{\perp}\subset T^*M= $$ $$ =\{\omega \in \Lambda^1(M)_{x}: X \lrcorner \omega=0 \text{ for all } X\in A(\mathcal{I}) \} $$

This is called the retracting space at $x$, or the Cartan system of $\mathcal{I}$ (Barco, M. A. thesis, page 34).

The differential system defined by $C(\mathcal{I})$ or, what is the same, the distribution $A(\mathcal{I})$ is completely integrable (see @bryant2013exterior Proposition 2.1. page 31, although it can be concluded from Proposition above).

I think that $A(\mathcal{I})$ is a generalization to exterior differential system of the associated distribution to a completely integrable Pfaffian system.

Observe that given a 1-form $\omega \in \mathcal{I}$, for every $\xi\in A(\mathcal{I})$ we have $\xi \,\lrcorner \, \omega \in \mathcal{I}$, so $\xi \,\lrcorner \, \omega=0$ since the contraction is a 0-form, and they are usually not considered in exterior differential systems (if they are present, we can restrict to the manifold defined by its zeros). Therefore $\omega \in C(\mathcal{I})$ and

$$ \mathcal{I} \cap \Omega^1 \subset C(\mathcal{I}) $$

A Cauchy characteristic is an integral manifold of the retracting space. They are a special kind of integral manifold of the original EDS.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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