The projective bundle of a real or complex vector bundle has fiber the space of one-dimensional subspaces of . It carries a tautological bundle ; in the real case this is the projectivization of a real vector bundle.
For a rank- complex vector bundle and , the cohomology of its projective bundle is free over the cohomology of the base on . Consequently pullback is injective, which yields the splitting principle for complex vector bundles by iteration.
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In algebraic geometry and differential geometry, a projective bundle is a space that parametrizes lines (or higher-dimensional projective subspaces) in a vector bundle. More formally, given a vector bundle \( E \) over a topological space (or algebraic variety) \( X \), the projective bundle associated with \( E \) is denoted by \( \mathbb{P}(E) \) and consists of the projectivization of the fibers of \( E \).