Talk:Tautological bundle

Correct definition

The definition of the tautological bundle is missing the subset $\lbrace (x,0):x\in \mathbb {CP} ^{n}\rbrace$ since the vector $0$ don't belong to any class. — Preceding unsigned comment added by NeoBeowulf (talk • contribs) 17:48, 26 September 2014 (UTC)

Not well written

This sentence appears in the section Formal definition:

"The total space of the bundle is the set of all pairs (V, v) consisting of a point V of the Grassmannian and a vector v in V; it is given the subspace topology of the Cartesian product <math>G_n(\R^{n+k}) \times \R^{n+k}."

This is a truly terrible choice of notationm especially since "V" was already used n the opening paragraph to denote the vector space of which the elements of the Grassmannian are subspaces!

Also: The Grassmannian is not a "parameter space" — that is a misuse of the word "parameter". It is the space of all subspaces of a vector space.

The section Facts contains this passage:

"*The tautological line bundle γ_1, k is locally trivial but not trivial, for k ≥ 1. This remains true over other fields."

But the previous section uses the identical character "k" to denote an arbitrary field! This kind of tone-deaf writing does not help the reader. 2601:200:C000:1A0:88EC:91B3:837D:2DEF (talk) 08:53, 1 August 2021 (UTC)