Notation for a projection of a differential form

Question

Let $\omega = a_1 dx_1 + a_2 dx_2 + b_1 dy_1 + b_2 dy_2$.

Is there any established notation to denote a mapping that "filters out" the $dy_i$-Terms?

To be more precise, I invent my own one. Assume the notation for the mapping would be $P_{[dy_1, dy_2]}$, then would write:

$P_{[dy_1, dy_2]} (\omega) = b_1 dy_1 + b_2 dy_2$.

In other words, I want a projection of the covectorfield $\omega$ onto the codistribution given by $D = \mathrm {span}(dy_1, dy_2)$.

Answer 1

There is. Let $\pi : \mathbb R^4\to \mathbb R^2$ be the projection to the $y_1, y_2$-plane and $i : \mathbb R^2 \to \mathbb R^4$ be $(y_1, y_2) \mapsto (0,0, y_1, y_2)$. Then your $P_{[dy_1,dy_2]}$ is $\pi^* \circ i^* = (i\circ\pi)^*$.