AI & ChatGPT searches , social queries for GRASSMANN BUNDLE
Search references for GRASSMANN BUNDLE. Phrases containing GRASSMANN BUNDLE
See searches and references containing GRASSMANN BUNDLE!
Search references for GRASSMANN BUNDLE. Phrases containing GRASSMANN BUNDLE
See searches and references containing GRASSMANN BUNDLE!GRASSMANN BUNDLE
In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X: p : G d ( E ) → X {\displaystyle
Grassmann_bundle
Vector bundle existing over a Grassmannian
also tautological bundles on a projective bundle of a vector bundle, as well as a Grassmann bundle. The older term canonical bundle has dropped out of
Tautological_bundle
Short exact sequence of sheaves on projective space
The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.) Let P A
Euler_sequence
Mathematical space
Grassmannian G r k ( V ) {\displaystyle \mathbf {Gr} _{k}(V)} , also known as a Grassmann manifold, is a differentiable manifold that parameterizes the set of all
Grassmannian
Fiber bundle whose fibers are projective spaces
projective bundle of a vector bundle E is the same thing as the Grassmann bundle G 1 ( E ) {\displaystyle G_{1}(E)} of 1-planes in E. The projective bundle P(E)
Projective_bundle
Differential geometry topic
tangent k-planes in the tangent bundle TM. The target space for the Gauss map N is a Grassmann bundle built on the tangent bundle TM. In the case where M =
Gauss_map
Hermann Grassmann: Grassmann's laws Grassmann algebra Grassmann bundle Grassmann dimensions Grassmann graph Grassmann integral Grassmann number Grassmann variables
List of things named after Hermann Grassmann
List_of_things_named_after_Hermann_Grassmann
Continuous surjection satisfying a local triviality condition
In mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is locally a product space, but globally
Fiber_bundle
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
Bundle of linear subspaces of the tangent bundle
the contact bundle is obtained by combining Grassmannians of the tangent spaces at each point, it is a special case of the Grassmann bundle and of the
Contact_bundle
Scheme parametrizing flags in the fibers of a vector bundle
is a point, the flag bundle is the usual flag variety of a vector space. If r = 1 {\displaystyle r=1} , it is the Grassmann bundle Gr d 1 ( E ) {\displaystyle
Flag_bundle
Concept in mathematics
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
Tensor_bundle
projective space P 3 ˘ {\displaystyle {\breve {\mathbb {P} ^{3}}}} as the Grassmann bundle p : P 3 ˘ → ∗ {\displaystyle p:{\breve {\mathbb {P} ^{3}}}\to *} parametrizing
Segre_class
Isomorphism between the tangent and cotangent bundles of a manifold
isomorphism) is an isomorphism between the tangent bundle T M {\displaystyle \mathrm {T} M} and the cotangent bundle T ∗ M {\displaystyle \mathrm {T} ^{*}M} of
Musical_isomorphism
Assignment of a tensor continuously varying across a region of space
language of vector bundles, the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles w times. While locally
Tensor_field
Concept in geometry
tautological bundle of the Grassmann bundle G i {\displaystyle G_{i}} of rank rk E i {\displaystyle \operatorname {rk} E_{i}} sub-bundles of E i ⊗ E i
Localized_Chern_class
Mathematical operation on vector spaces
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Tensor_product
Mathematical function, in linear algebra
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Linear_map
Algebraic operation on coordinate vectors
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Dot_product
System of moving vectors in differential geometry
affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along
Parallel_transport
Specification of a derivative along a tangent vector of a manifold
contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically
Covariant_derivative
Algebraic structure in linear algebra
et fils Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (in German), O. Wigand, reprint: Grassmann, Hermann
Vector_space
Type of unphysical field in quantum field theory which provides mathematical consistency
gauge-field fiber bundle.) Used in the above identity for the determinant, these fields become the Fadeev-Popov ghost fields. Because Grassmann numbers anti-commute
Faddeev–Popov_ghost
Exterior algebraic map taking tensors from p forms to n-p forms
can play a role in differential geometry when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms. This
Hodge_star_operator
Geometric structure
g ) , {\displaystyle (M,g),\,} one defines the spinor bundle to be the complex vector bundle π S : S → M {\displaystyle \pi _{\mathbf {S} }\colon {\mathbf
Spinor_bundle
Array of numbers describing a metric connection
frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is
Christoffel_symbols
Expression that may be integrated over a region
aspects of the exterior algebra of differential forms appears in Hermann Grassmann's 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik
Differential_form
Straight path on a curved surface or a Riemannian manifold
double tangent bundle TTM into horizontal and vertical bundles: T T M = H ⊕ V . {\displaystyle TTM=H\oplus V.} The double tangent bundle can be visualized
Geodesic
Method for specifying point positions
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Coordinate_system
Operation that pairs a left and a right R-module into an abelian group
bundles T M , T ∗ M {\displaystyle TM,T^{*}M} are viewed as locally free sheaves on M. The exterior bundle on M is the subbundle of the tensor bundle
Tensor_product_of_modules
Concept in differential geometry
differentiable principal bundle or vector bundle with a connection. Let G be a Lie group and P → M be a principal G-bundle on a smooth manifold M. Suppose
Exterior_covariant_derivative
Construct allowing differentiation of tangent vector fields of manifolds
simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry
Affine_connection
Branch of mathematics
and applications involve single vectors, mathematicians such as Hermann Grassmann considered structures involving pairs, triplets, and multivectors that
Multilinear_algebra
Matrix operation which flips a matrix over its diagonal
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Transpose
Math/physics concept
formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection
Connection_form
Universal construction in multilinear algebra
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Tensor_algebra
Structure defining distance on a manifold
Sg defines a section of the bundle Hom(TM, T*M) of vector bundle isomorphisms of the tangent bundle to the cotangent bundle. This section has the same
Metric_tensor
Set of vectors used to define coordinates
père et fils Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (in German), reprint: Hermann Grassmann. Translated
Basis_(linear_algebra)
Manifold with supersymmetry structure
{\displaystyle A} is a C Z ∞ {\displaystyle C_{Z}^{\infty }} -sheaf of Grassmann algebras of rank m {\displaystyle m} where C Z ∞ {\displaystyle C_{Z}^{\infty
Graded_manifold
Type of derivative in differential geometry
principal bundle. Now, if we're given a vector field Y over M (but not the principal bundle) but we also have a connection over the principal bundle, we can
Lie_derivative
Tensor in differential geometry
curvature form of the canonical line bundle. The canonical line bundle is the top exterior power of the bundle of holomorphic Kähler differentials: κ
Ricci_curvature
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Kronecker_delta
Shorthand notation for tensor operations
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Einstein_notation
Branch of mathematics
considerable interest in physics. The apparatus of vector bundles, principal bundles, and connections on bundles plays an extraordinarily important role in modern
Differential_geometry
Affine connection on the tangent bundle of a manifold
the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free
Levi-Civita_connection
Topological space that locally resembles Euclidean space
and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle. The n-sphere Sn is a generalisation of the
Manifold
Study of curves from a differential point of view
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Differentiable_curve
Operation in mathematics
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Tensor_contraction
Construct in differenital geometry
mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any
Metric_connection
Supergeometric generalization of a manifold
equipped with an Grassmann-odd symplectic structure. All natural geometric objects on a supermanifold are graded. In particular, the bundle of two-forms is
Supermanifold
British mathematician and philosopher (1845–1879)
British mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra. This is a special
William_Kingdon_Clifford
Vector behavior under coordinate changes
indices, because it has parts that live in the tangent bundle as well as the cotangent bundle. A contravariant vector is one which transforms like d x
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Differential form of degree one or section of a cotangent bundle
cotangent bundle. Equivalently, a one-form on a manifold M {\displaystyle M} is a smooth mapping of the total space of the tangent bundle of M {\displaystyle
One-form
Non-tensorial representation of the spin group
forms a spinor bundle associated to the principal spin bundle and a chosen spin representation; spinor fields are sections of this bundle. In flat spacetime
Spinor
Second-rank tensor in quantum chromodynamics
the spacetime with values in the adjoint bundle of the chromodynamical SU(3) gauge group (see vector bundle for necessary definitions). Throughout this
Gluon_field_strength_tensor
Algebraic object with geometric applications
tensors, and the Riemann curvature tensor. The exterior algebra of Hermann Grassmann, from the middle of the nineteenth century, is itself a tensor theory
Tensor
Object in differential geometry
characterization of torsion, applies to the frame bundle FM of the manifold M. This principal bundle is equipped with a connection form ω, a gl(n)-valued
Torsion_tensor
French mathematician (1905–1979)
classical Lie groups, such as Grassmann manifolds and other homogeneous spaces. He developed the concept of fiber bundle, and the related notions of Ehresmann
Charles_Ehresmann
Function that is invariant under all permutations of its variables
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Symmetric_function
Tensor invariant under permutations of vectors it acts on
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Symmetric_tensor
Differential form
{\displaystyle n} -form. It is an element of the space of sections of the line bundle ⋀ n ( T ∗ M ) {\displaystyle \textstyle {\bigwedge }^{n}(T^{*}M)} , denoted
Volume_form
Tensor describing energy momentum density in spacetime
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Stress–energy_tensor
Type of physical quantity
pseudo-volume form, due to the additional sign twist (tensoring with the sign bundle). The volume element is a pseudotensor density according to the first definition
Pseudotensor
Tensor operator generalizes the notion of operators which are scalars and vectors
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Tensor_operator
Mathematical object that describes the electromagnetic field in spacetime
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Electromagnetic_tensor
Coordinate-free definition of a tensor
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Tensor_(intrinsic_definition)
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Tensors in curvilinear coordinates
Tensors_in_curvilinear_coordinates
Four-dimensional analog of the icosahedron
due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility
600-cell
Polish military leader (1746–1817)
Julian Ursyn (1965). Budka, Mechie J. (ed.). Under Your Vine and Fig Tree. Grassmann Pub. Co., 398 pages. ISBN 9780686818083. Niemcewicz, Julian Ursyn (1844)
Tadeusz_Kościuszko
Decomposition in multilinear algebra
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Tensor_rank_decomposition
In mathematics, invariant of square matrices
n} -dimensional vectors of anti-commuting Grassmann numbers (aka "supernumbers"), taken from the Grassmann algebra. The exp {\displaystyle \exp } here
Determinant
Abbreviation in the fields of special and general relativity
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Four-tensor
Element of an exterior algebra
have properties similar to the homogeneous coordinates of points, called Grassmann coordinates. Points in a real projective space Pn are defined to be lines
Multivector
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Moment_of_inertia
algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic
Standard_monomial_theory
Physics concept
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Covariant_transformation
Tensor index notation for tensor-based calculations
}B^{\gamma }{}_{\delta \cdots ;\epsilon }\,.} A Koszul connection on the tangent bundle of a differentiable manifold is called an affine connection. A connection
Ricci_calculus
Branch of physics which studies the behavior of materials modeled as continuous media
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Continuum_mechanics
Tensor used in general relativity
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Einstein_tensor
Generalization of tensor fields
also be regarded as a section of the tensor product of a tensor bundle with a density bundle. In physics and related fields, it is often useful to work with
Tensor_density
Mapping from p forms to p-1 forms
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Interior_product
Antisymmetric permutation object acting on tensors
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Levi-Civita_symbol
Electromagnetism in general relativity
F(\nabla )} of a U(1)-connection ∇ {\displaystyle \nabla } on a principal U(1)-bundle whose sections represent charged fields. The connection is much like the
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Tensor equal to the negative of any of its transpositions
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Antisymmetric_tensor
Mathematical Concept
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Voigt_notation
Measure of the curvature of a pseudo-Riemannian manifold
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Weyl_tensor
Property of a mathematical space
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Dimension
Array of numbers
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Matrix_(mathematics)
Theory of gravitation as curved spacetime
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
General_relativity
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Symmetrization
Algebra based on a vector space with a quadratic form
references to Schönberg's papers of 1956 and 1957 as described in section "The Grassmann–Schönberg algebra Gn" of Bolivar 2001 See for ex. Oziewicz & Sitarczyk
Clifford_algebra
Relativistic quantum mechanical wave equation
the spinor field ψ ( x ) {\displaystyle \psi (x)} is an anti-commuting Grassmann-valued field that only acts as an integration variable. The Dirac equation
Dirac_equation
Representation of a tensor in Euclidean space
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Cartesian_tensor
Mathematical notation for tensors and spinors
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Abstract_index_notation
Tool to track locally defined data attached to the open sets of a topological space
to easily compute the cohomology groups of all line bundles on projective space and grassmann manifolds. In many cases there is a duality theory for
Sheaf_(mathematics)
Covariant derivative of the metric tensor
and let ∇ {\displaystyle \nabla } be an affine connection on the tangent bundle T M {\displaystyle TM} . The nonmetricity tensor is defined (some authors
Nonmetricity_tensor
Tensor having both covariant and contravariant indices
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Mixed_tensor
Method of calculating chiral anomalies
where { a i , b ¯ i } {\displaystyle \{a^{i},{\overline {b}}^{i}\}} are Grassmann valued coefficients, and { ψ i } {\displaystyle \{\psi _{i}\}} are eigenvectors
Fujikawa_method
Tensor that describes the 4D geometry of spacetime
Christoffel Albert Einstein Leonhard Euler Carl Friedrich Gauss Hermann Grassmann Tullio Levi-Civita Gregorio Ricci-Curbastro Bernhard Riemann Jan Arnoldus
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Notation used for Weyl spinors
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Van_der_Waerden_notation
GRASSMANN BUNDLE
GRASSMANN BUNDLE
GRASSMANN BUNDLE
GRASSMANN BUNDLE
GRASSMANN BUNDLE
GRASSMANN BUNDLE
GRASSMANN BUNDLE
GRASSMANN BUNDLE
GRASSMANN BUNDLE