nLab circle bundle

Definition

A circle bundle is a principal bundle for the circle group $S^1$. Equivalently this is a $U(1)$-principal bundle, for the unitary group $U(1)$.

Under the canonical representation $\mathbf{B}U(1) \to Vect_{\mathbb{C}}$ the corresponding associated bundle is a complex line bundle.

Examples

• A basic example of a nontrivial circle group-principal bundle is the Hopf fibration $S^3 \to S^2$.

Related concepts

• circle bundle with connection, regular contact manifold

• prequantum circle bundle

• line bundle

• Seifert 3-manifold