Shear Map

Let α : G × X → X : 𝛼 → 𝐺 𝑋 𝑋 \alpha:G\times X\to X be a Group Action. The shear map of α 𝛼 \alpha is the function

⟨α,π2⟩:G×X→X×X:𝛼subscript𝜋2→𝐺𝑋𝑋𝑋\langle\alpha,\pi_{2}\rangle:G\times X\to X\times X

. This is best thought about as a function between Trivial Bundles, as demonstrated in the following diagram:

G×X𝐺𝑋{{G\times X}}X×X𝑋𝑋{{X\times X}}X𝑋{X}X𝑋{X}⟨α,π2⟩𝛼subscript𝜋2\scriptstyle{\langle\alpha,\pi_{2}\rangle}π2subscript𝜋2\scriptstyle{\pi_{2}}π2subscript𝜋2\scriptstyle{\pi_{2}}idid\scriptstyle{\mathrm{id}}