A projective vector field (projective) is a smooth vector field on a semi Riemannian manifold (p.ex. spacetime) M {\displaystyle M} whose flow preserves...

concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus...

An affine vector field (sometimes affine collineation or affine) is a projective vector field preserving geodesics and preserving the affine parameter...

means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly...

In physics, a homothetic vector field (sometimes homothetic collineation or homothety) is a projective vector field which satisfies the condition: L X...

the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space...

real projective plane, denoted ⁠ R P 2 {\displaystyle \mathbf {RP} ^{2}} ⁠ or ⁠ P 2 {\displaystyle \mathbb {P} _{2}} ⁠, is a two-dimensional projective space...

two-dimensional K-vector space. This definition is a special instance of the general definition of a projective space. The projective line over the reals...

particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of...

subspaces of a two-dimensional vector space over the reals. The automorphisms of a real projective line are called projective transformations, homographies...

duality and beyond that to duality in any finite-dimensional projective geometry. A projective plane C may be defined axiomatically as an incidence structure...

the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space. Not all projective planes can...

mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel...

In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in...

the infinite real projective space does not have this property. Vector bundles are often given more structure. For instance, vector bundles may be equipped...

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...

In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued...

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 Pn-bundle...

concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles...

and more specifically in projective geometry, a projective frame or projective basis is a tuple of points in a projective space that can be used for...

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces...

nonsingular projective algebraic variety. The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines...

radius vector connecting the origin to the point in question, while ϕ {\displaystyle \phi } is the angle between the projection of the radius vector onto...

hyperplane at infinity of a projective space, the affine transformations are the projective transformations of that projective space that leave the hyperplane...

field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective...

analogy of vector bundles with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over...

vector field preserves geodesics and preserves the affine parameter. The above three vector field types are special cases of projective vector fields...

among them. The word projective comes from the game's relation to projective spaces over the finite field with two elements. Projective Set has been studied...

Affine spaces are contained in projective spaces. For example, an affine plane can be obtained from any projective plane by removing one line and all...

flag variety is defined to mean a projective homogeneous variety, that is, a smooth projective variety X over a field F with a transitive action of a reductive...