Search results
Results From The WOW.Com Content Network
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 ...
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} .
The unit tangent bundle is therefore a sphere bundle over M with fiber S n−1. The definition of unit sphere bundle can easily accommodate Finsler manifolds as well. Specifically, if M is a manifold equipped with a Finsler metric F : TM → R, then the unit sphere bundle is the subbundle of the tangent bundle whose fiber at x is the indicatrix ...
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.
The choice of a projective embedding of X, modulo projective transformations is likewise equivalent to the choice of a very ample line bundle on X. A morphism to a projective space : defines a globally generated line bundle by () and a linear system
An ovoid in a 3-dimensional projective space is a set of points, which a) is intersected by lines in 0, 1, or 2 points and b) its tangents at an arbitrary point covers a plane (tangent plane). The geometry of an ovoid in projective 3-space is a Möbius plane, called an ovoidal Möbius plane. The point set of the geometry consists of the points ...