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The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold in its own right. The dimension of is twice the dimension of . Each tangent space of an n-dimensional manifold is an n-dimensional vector space
The definitions of the tangent bundle, the unit tangent bundle and the (oriented orthonormal) frame bundle F can be extended to arbitrary surfaces in the usual way. [7] [15] There is a similar identification between the latter two which again become principal SO(2)-bundles. In other words: The frame bundle is a principal bundle with structure ...
The tangent bundle of projective space over a field can be described in terms of the line bundle (). Namely, there is a short exact sequence, the Euler sequence : 0 → O P n → O ( 1 ) ⊕ n + 1 → T P n → 0. {\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} ^{n}}\to {\mathcal {O}}(1)^{\oplus \;n+1}\to T\mathbb {P} ^{n}\to 0.}
Every vector bundle over a variety X gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way: there is an obstruction in the cohomology group H 2 (X,O*). To see why, recall that a projective bundle comes equipped with transition functions on double intersections of a suitable ...
A similar phenomenon in algebraic geometry is given by a linear system: to give a morphism from a base variety S to a projective space = is equivalent to giving a basepoint-free linear system (or equivalently a line bundle) on S. That is, the projective space X represents the functor which gives all line bundles over S.
Suppose that is a space and that is a line bundle on .A global section of is a function : such that if : is the natural projection, then =.In a small neighborhood in in which is trivial, the total space of the line bundle is the product of and the underlying field , and the section restricts to a function .
Conversely, given such a bundle, an immersion of M with this normal bundle is equivalent to a codimension 0 immersion of the total space of this bundle, which is an open manifold. The stable normal bundle is the class of normal bundles plus trivial bundles, and thus if the stable normal bundle has cohomological dimension k , it cannot come from ...
A line bundle whose base can be embedded in a projective space by such a morphism is called very ample. The group of symmetries of the projective space P k n {\displaystyle \mathbb {P} _{\mathbf {k} }^{n}} is the group of projectivized linear automorphisms P G L n + 1 ( k ) {\displaystyle \mathrm {PGL} _{n+1}(\mathbf {k} )} .