nLab
projective bundle

Context

Bundles

Contents

Idea

The concept of projective bundle is the generalization of that of projective space from vector spaces to vector bundles.

Definition

For π:EX\pi \colon E \to X a topological vector bundle over some topological field kk, write EXEE \setminus X \subset E for its complement of the zero section, regarded with its subspace topology.

Then its projective bundle is the fiber bundle P(E)XP(E) \to X whose total space the quotient topological space of EXE \setminus X by the equivalence relation

(v 1v 2)((π(v 1)=π(v) 1)and(ck{0}(v 2=cv 1))) \left( v_1 \sim v_2 \right) \;\Leftrightarrow\; \left( (\pi(v_1) = \pi(v)_1) \,\text{and}\, \left( \underset{c \in k \setminus \{0\}}{\exists} ( v_2 = c v_1) \right) \right)

hence

P(E)(E)/, P(E) \coloneqq (E \setminus )/\sim \,,

and whose bundle projection is

P(E) X [v] π(v). \array{ P(E) &\longrightarrow& X \\ [v] &\mapsto& \pi(v) } \,.

References

Last revised on May 29, 2017 at 03:32:56. See the history of this page for a list of all contributions to it.