High Quality Content by WIKIPEDIA articles! In mathematics, tautological bundle is a term for a particularly natural vector bundle occurring over a Grassmannian, and more specially over projective space. Canonical bundle as a name dropped out of favour, on the grounds that 'canonical' is heavily overloaded as it is, in mathematical terminology, and (worse) confusion with the canonical class in algebraic geometry could scarcely be avoided. Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector space W. If G is a Grassmannian, and Vg is the subspace of W corresponding to g in G, this is already almost the data required for a vector bundle: namely a vector space for each point g, varying continuously. All that can stop the definition of the tautological bundle from this indication, is the (pedantic) difficulty that the Vg are going to intersect. Fixing this up is a routine application of the disjoint union device, so that the bundle projection is from a total space made up of identical copies of the Vg, that now do not intersect. With this, we have the bundle.

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