Contents

Contents

Idea

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

Definition

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

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

$\left( v_1 \sim v_2 \right) \;\Leftrightarrow\; \left( (\pi(v_1) = \pi(v_2)) \,\text{and}\, \left( \underset{c \in k \setminus \{0\}}{\exists} ( v_2 = c v_1) \right) \right)$

hence

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

and whose bundle projection is

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

References

Last revised on April 3, 2019 at 06:53:39. See the history of this page for a list of all contributions to it.