# Category: Vector bundles

Lie algebroid
In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union of the tangent spaces o
Complex vector bundle
In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. C
Normal bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Kobayashi–Hitchin correspondence
In differential geometry, algebraic geometry, and gauge theory, the Kobayashi–Hitchin correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einst
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The de
Kosmann lift
In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field on a Riemannian manifold is the canonical projection on the orthonormal frame bundle of its natura
In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle L at a point P on an algebraic variety. It was introduced by Demailly to measure a certain rate of growth, of the ten
Tensor bundle
In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connecti
Zariski geometry
In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve,
Connection (vector bundle)
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect"
In linear and homological algebra, a monad is a 3-term complex A → B → C of objects in some abelian category whose middle term B is projective and whose first map A → B is injective and whose second m
Vector bundles on algebraic curves
In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach, or as locally free sheaves on algebraic cu
Vector-valued differential form
In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bund
Quillen determinant line bundle
In mathematics, the Quillen determinant line bundle is a line bundle over the space of Cauchy–Riemann operators of a vector bundle over a Riemann surface, introduced by Quillen. Quillen proved the exi
Coherent sheaf cohomology
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questio
Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex. The general line
Nonabelian Hodge correspondence
In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after and Carlos Simpson) is a correspondence between Higgs bundles and r
Lie algebra bundle
In mathematics, a weak Lie algebra bundle is a vector bundle over a base space X together with a morphism which induces a Lie algebra structure on each fibre . A Lie algebra bundle is a vector bundle
Parallelizable manifold
In mathematics, a differentiable manifold of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at every point of the tangent vectorsprovide a basis of
Tractor bundle
In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an effective representation of the conformal group (see associated bundle)
Cone (algebraic geometry)
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec of a quasi-coherent graded OX-algebra R is called the cone or affine cone of R.
Horrocks–Mumford bundle
In algebraic geometry, the Horrocks–Mumford bundle is an indecomposable rank 2 vector bundle on 4-dimensional projective space P4 introduced by Geoffrey Horrocks and David Mumford. It is the only such
Ample line bundle
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The m
Algebra bundle
In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure. It follows that the transition functions are algebra isomorphisms
Higgs bundle
In mathematics, a Higgs bundle is a pair consisting of a holomorphic vector bundle E and a Higgs field , a holomorphic 1-form taking values in the bundle of endomorphisms of E such that . Such pairs w
Banach bundle
In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.
Iitaka dimension
In algebraic geometry, the Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by L. This is 1 less than the
Inverse bundle
In mathematics, the inverse bundle of a fibre bundle is its inverse with respect to the Whitney sum operation. Let be a fibre bundle. A bundle is called the inverse bundle of if their Whitney sum is a
Parallelization (mathematics)
In mathematics, a parallelization of a manifold of dimension n is a set of n global smooth linearly independent vector fields.
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as
Tango bundle
In algebraic geometry, a Tango bundle is one of the indecomposable vector bundles of rank n − 1 constructed on n-dimensional projective space Pn by
Beauville–Laszlo theorem
In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebra
Tautological bundle
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmann
Horrocks bundle
In algebraic geometry, Horrocks bundles are certain indecomposable rank 3 vector bundles (locally free sheaves) on 5-dimensional projective space, found by .
Bochner–Kodaira–Nakano identity
In mathematics, the Bochner–Kodaira–Nakano identity is an analogue of the Weitzenböck identity for hermitian manifolds, giving an expression for the antiholomorphic Laplacian of a vector bundle over a
Canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety of dimension over a field is the line bundle , which is the nth exterior power of the cotangent bundle Ω on V. Over the complex
Clifford module bundle
In differential geometry, a Clifford module bundle, a bundle of Clifford modules or just Clifford module is a vector bundle whose fibers are Clifford modules, the representations of Clifford algebras.
Horrocks construction
In mathematics, the Horrocks construction is a method for constructing vector bundles, especially over projective spaces, introduced by Geoffrey Horrocks . His original construction gave an example of
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space (for example could be a topological space, a manif
Hermitian Yang–Mills connection
In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomo
S-equivalence
S-equivalence is an equivalence relation on the families of semistable vector bundles on an algebraic curve.
In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nona
Plumbing (mathematics)
In the mathematical field of geometric topology, among the techniques known as surgery theory, the process of plumbing is a way to create new manifolds out of . It was first described by John Milnor a
Principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous
Symbol of a differential operator
In mathematics, the symbol of a linear differential operator is a polynomial representing a differential operator, which is obtained, roughly speaking, by replacing each partial derivative by a new va
Holomorphic vector bundle
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : E → X is holomorphic. Fund
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying lin
Lange's conjecture
In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by and proved by Montserrat Teixidor i Bigas and in 1999.
Flat vector bundle
In mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection.
Jumping line
In mathematics, a jumping line or exceptional line of a vector bundle over projective space is a projective line in projective space where the vector bundle has exceptional behavior, in other words th
Clifford bundle
In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford
Birkhoff–Grothendieck theorem
In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over is a direct sum of holomorp
Splitting principle
In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computat
Dual bundle
In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces.