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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 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.}
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 ...
The tangent space of at , denoted by , is then defined as the set of all tangent vectors at ; it does not depend on the choice of coordinate chart :. The tangent space T x M {\displaystyle T_{x}M} and a tangent vector v ∈ T x M {\displaystyle v\in T_{x}M} , along a curve traveling through x ∈ M {\displaystyle x\in M} .
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.
For example, when f is a smooth morphism, is simply a vector bundle, known as the tangent bundle along the fibers of f. Using A 1 -homotopy theory , the Grothendieck–Riemann–Roch theorem has been extended by Navarro & Navarro (2017) to the situation where f is a proper map between two smooth schemes.
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 ...