Symmetrizing factor

Definition

Let $Z$ be an involutive distribution on an $n$ -dimensional manifold $M$ . Let $X$ be a vector field transverse to $Z$ . A non-vanishing smooth function $f \in C^{\infty} (M)$ is called a symmetrizing factor for $X$ with respect to $Z$ if the vector field $f X$ is a symmetry for $Z$ .

Existence and Non-uniqueness

Given a cinf-symmetry of distribution $X$ , the symmetrizing factor lemma ensures that we can always find a function $f$ such that $f X$ is a true symmetry of a distribution.

In reality, this factor is not unique. Given a common invariant (first integral) $g \in C^{\infty} (M)$ of all generators ${Z_{i}}$ of the distribution, the function $g \cdot f$ is also a symmetrizing factor. This is shown by the relation:

[Z_{i}, g f X] = Z_{i} (g) f X + g [Z_{i}, f X] = 0 + g (\sum c_{i k} Z_{k}) \in Z

Relationship with Integrating Factors

There is an inverse relationship with the integrating factors of 1-forms or Pfaffian equations. This is detailed in the note: integrating factors vs symmetrizing factors.

Connection to Inverse Jacobi Multipliers

In the specific case of distributions generated by a single vector field (corank $n - 1$ ), the symmetrizing factor is closely related to the inverse Jacobi multiplier.