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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
In mathematics, complex projective space is the projective space with respect to the field of complex numbers.By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space (see below for an intuitive account).
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a P n -bundle if it is locally a projective n -space; i.e., X × S U ≃ P U n {\displaystyle X\times _{S}U\simeq \mathbb {P} _{U}^{n}} and transition automorphisms are linear.
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 .
In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on ...
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
In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an ( n + 1 ) {\displaystyle (n+1)} -fold sum of the dual of the Serre twisting sheaf .
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 ...