In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle.
The tautological bundle is also called the universal bundle since any vector bundle (over a compact space[1]) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because of this, the tautological bundle is important in the study of characteristic classes.
Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is
the dual of the hyperplane bundle or Serre's twisting sheaf . The hyperplane bundle is the line bundle corresponding to the hyperplane (divisor) in . The tautological line bundle and the hyperplane bundle are exactly the two generators of the Picard group of the projective space.[2]
In Michael Atiyah's "K-theory", the tautological line bundle over a complex projective space is called the standard line bundle. The sphere bundle of the standard bundle is usually called the Hopf bundle. (cf. Bott generator.)
More generally, there are also tautological bundles on a projective bundle of a vector bundle as well as a Grassmann bundle.
The older term canonical bundle has 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.
^Over a noncompact but paracompact base, this remains true provided one uses infinite Grassmannian.
^In literature and textbooks, they are both often called canonical generators.
and 23 Related for: Tautological bundle information
In mathematics, the tautologicalbundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k {\displaystyle...
possible for a bundle to have no non-zero global sections at all; this is the case for the tautological line bundle. When the line bundle is sufficiently...
where it is used as an alternative to canonical: TautologicalbundleTautological line bundleTautological one-form Tautology (grammar), unnecessary repetition...
tangent bundle of the cotangent bundle, the application of the tautological one-form θ to v at (x, ω) is computed by projecting v into the tangent bundle at...
projective space has a natural line bundle over it, called the tautologicalbundle. More precisely, this is called the tautological subbundle, and there is also...
homogeneous space G/H consists of a point in the tautologicalbundle G → G/H. A moving frame is a section of this bundle. It is moving in the sense that as the...
frame bundle associated to the tautologicalbundle on a Grassmannian. When one passes to the n → ∞ {\displaystyle n\to \infty } limit, these bundles become...
In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all...
the projective line, any vector bundle splits in a unique way as a direct sum of the line bundles. The tautologicalbundle, which appears for instance as...
trivial bundle; i.e., there exists a bundle E' such that E ⊕ E' is trivial. This fails if X is not compact: for example, the tautological line bundle over...
for U(n) Chern class tautologicalbundle, a universal bundle for the general linear group. PlanetMath page of universal bundle examples J. J. Duistermaat...
cannot decompose an arbitrary bundle in terms of a Whitney sum of powers of the Tautologicalbundle, or in fact of line bundles in general. Still one can...
variety comes with a natural vector bundle (or locally free sheaf in other terminology) called the tautologicalbundle, which is important in the study of...
algebraic geometry, the tautological ring is the subring of the Chow ring of the moduli space of curves generated by tautological classes. These are classes...
as a principal bundle whose structure group is the orthogonal group O(n). (In fact this principal bundle is just the tautologicalbundle of the homogeneous...
the dual of the tautological line bundle O X ( − 1 ) {\displaystyle {\mathcal {O}}_{X}(-1)} . It is also called the hyperplane bundle. O X ( D ) {\displaystyle...
(k^{n},{\mathcal {R}})} where R {\displaystyle {\mathcal {R}}} is the tautologicalbundle over the Grassmannian. So dim Y r = dim Z r {\displaystyle \dim...
unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M) or simply UTM, is the unit sphere bundle for the tangent bundle T(M). It is...
product. This is not true in the case of complex vector bundles: for example, the tautological line bundle over the Riemann sphere is not isomorphic to its dual...
vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle π : E → B...
the Grassmannian one arrives at the vector bundle E {\displaystyle E} which generalizes the tautologicalbundle of a projective space. Similarly the ( n...
{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1)} is called the tautological line bundle on the projective n {\displaystyle n} -space. A simple example...
S. The cotangent sheaf on a projective space is related to the tautological line bundle O(-1) by the following exact sequence: writing P R n {\displaystyle...
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