 Projectivization of copies of the real tautological line bundle

ID: projectivization-of-copies-of-the-real-tautological-line-bundle

Tensoring a fixed line with

R^{k}

does not change its projective directions, so

P (k γ_{R}^{1, n + 1}) ≅ RP^{n} \times RP^{k - 1} .

x

and

v

are the degree-one generators from the two factors, then the tautological line over this projective bundle has first Stiefel–Whitney class

x + v

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