Principal Bundle

A principal G 𝐺 G -bundle for G 𝐺 G a Topological Group is a Fibre Bundle π : E → B : 𝜋 → 𝐸 𝐵 \pi:E\to B together with a Continuous Right Group Action ⋅ : E × G → G \cdot:E\times G\to G that preserves the fibres of π 𝜋 \pi (IE: if e ∈ E x 𝑒 subscript 𝐸 𝑥 e\in E_{x} then e ⁢ g ∈ E x 𝑒 𝑔 subscript 𝐸 𝑥 eg\in E_{x} for all g : G : 𝑔 𝐺 g:G ) and acts Freely and Transitively.

More abstractly, a principal G 𝐺 G -bundle is a Continuous G 𝐺 G -Torsor.