In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k {\displaystyle...

where it is used as an alternative to canonical: Tautological bundle Tautological line bundle Tautological one-form Tautology (grammar), unnecessary repetition...

get a line bundle on P ( V ) {\displaystyle \mathbf {P} (V)} . This line bundle is called the tautological line bundle. This line bundle is sometimes...

In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T ∗ Q {\displaystyle T^{*}Q} of a manifold Q . {\displaystyle...

homogeneous space G/H consists of a point in the tautological bundle G → G/H. A moving frame is a section of this bundle. It is moving in the sense that as the...

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 tautological bundle. More precisely, this is called the tautological subbundle, and there is also...

may view the Maurer–Cartan form as a 1-form defined on the tautological principal bundle associated with a homogeneous space. If H is a closed subgroup...

frame bundle associated to the tautological bundle on a Grassmannian. When one passes to the n → ∞ {\displaystyle n\to \infty } limit, these bundles become...

for U(n) Chern class tautological bundle, 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 Tautological bundle, or in fact of line bundles in general. Still one can...

realizes Cn+1\{0} as a complex line bundle over the base space CPn. (In fact this is the so-called tautological bundle over CPn.) A point of CPn is thus...

that it is equipped with the tautological bundle γ n → G r n , {\displaystyle \gamma ^{n}\to Gr_{n},} a rank n vector bundle that can be defined as the...

In mathematics, a frame bundle is a principal fiber bundle F ( E ) {\displaystyle F(E)} associated with any vector bundle E {\displaystyle E} . The fiber...

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...

{S} ^{2})=\mathbb {Z} [H]/(H-1)^{2}} where H is the class of the tautological bundle on S 2 = P 1 ( C ) , {\displaystyle \mathbb {S} ^{2}=\mathbb {P}...

(k^{n},{\mathcal {R}})} where R {\displaystyle {\mathcal {R}}} is the tautological bundle over the Grassmannian. So dim ⁡ Y r = dim ⁡ Z r {\displaystyle \dim...

variety comes with a natural vector bundle (or locally free sheaf in other terminology) called the tautological bundle, which is important in the study of...

the projective line, any vector bundle splits in a unique way as a direct sum of the line bundles. The tautological bundle, which appears for instance as...

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...

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...

{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1)} is called the tautological line bundle on the projective n {\displaystyle n} -space. A simple example...

the Grassmannian one arrives at the vector bundle E {\displaystyle E} which generalizes the tautological bundle of a projective space. Similarly the ( n...

as a principal bundle whose structure group is the orthogonal group O(n). (In fact this principal bundle is just the tautological bundle of the homogeneous...

bundle of a G-structure. If Q is realized as a reduction of the frame bundle of M, then the solder form is given by the pullback of the tautological form...

T\to {\underline {V}}\to Q\to 0} where T {\displaystyle T} is the tautological bundle whose fiber, over any element w ∈ G r ( k , V ) {\displaystyle w\in...

as a subring of H∗(BU(1); Q) = Q[w], where w is element dual to tautological bundle. For the n-torus, K0(BTn) is numerical polynomials in n variables...

bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M), UTM, or SM is the unit sphere bundle for the tangent bundle T(M). It is a fiber bundle...

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...

the associated bundle P ×H Rn. This is furthermore required to be a bundle isomorphism. Frame bundles have a (canonical or tautological) solder form which...