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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 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.}
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 ...
When combined with a vanishing theorem, the Riemann–Roch theorem can often be used to determine the dimension of the vector space of sections of a line bundle. Knowing that a line bundle on X has enough sections, in turn, can be used to define a map from X to projective space, perhaps a closed immersion. This approach is essential for ...
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.
A special case of this is the orientable double cover of a differentiable manifold, where the corresponding line bundle is the determinant bundle of the tangent bundle (see below). The Möbius strip corresponds to a double cover of the circle (the θ → 2θ mapping) and by changing the fiber, can also be viewed as having a two-point fiber, the ...
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 ...