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of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or
Bundle_metric
Construct in differenital geometry
In 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
Metric_connection
Isomorphism between the tangent and cotangent bundles of a manifold
be (non-canonically) endowed with a Riemannian metric, the musical isomorphisms show that a vector bundle on a smooth manifold is (non-canonically) isomorphic
Musical_isomorphism
Principal bundle associated to a vector bundle
In mathematics, a frame bundle is a principal fiber bundle F ( E ) {\displaystyle F(E)} associated with any vector bundle E {\displaystyle E} . The fiber
Frame_bundle
Mathematical parametrization of vector spaces by another space
Vector bundles are often given more structure. For instance, vector bundles may be equipped with a vector bundle metric. Usually this metric is required
Vector_bundle
Structure defining distance on a manifold
More generally, one may speak of a metric in a vector bundle. If E is a vector bundle over a manifold M, then a metric is a mapping g : E × M E → R {\displaystyle
Metric_tensor
Type of metric in Riemannian geometry
geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to
Kähler–Einstein_metric
correspondence asserts that a holomorphic vector bundle E {\displaystyle E} admits a Hermitian metric h {\displaystyle h} such that the associated Chern
Hermitian Yang–Mills connection
Hermitian_Yang–Mills_connection
Defines a notion of parallel transport on a bundle
equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles). Let M be a differentiable manifold, such
Connection_(vector_bundle)
Elliptic differential operators in geometry mathematics
operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e.
Laplace operators in differential geometry
Laplace_operators_in_differential_geometry
Unified field theory
fiber, one can construct a bundle metric defined on the entire bundle. Computing the scalar curvature of this bundle metric, one finds that it is constant
Kaluza–Klein_theory
bundle over a paracompact space admits a hermitian metric. The basic invariant of a complex vector bundle is a Chern class. A complex vector bundle is
Complex_vector_bundle
Continuous surjection satisfying a local triviality condition
associated bundle. A sphere bundle is a fiber bundle whose fiber is an n-sphere. Given a vector bundle E {\displaystyle E} with a metric (such as the
Fiber_bundle
Vector bundle of cotangent spaces at every point in a manifold
smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general
Cotangent_bundle
Math/physics concept
Riemannian metric. If one has a vector bundle E over M, then the metric can be extended to the entire vector bundle, as the bundle metric. One may then
Connection_form
bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M), UTM, or SM is the unit sphere bundle for the tangent bundle T(M). It is a fiber bundle
Unit_tangent_bundle
Complex vector bundle on a complex manifold
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
Holomorphic_vector_bundle
Metric on a determinant line bundle
and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel
Quillen_metric
manifold with metric g, then the Clifford bundle of M is the Clifford bundle generated by the tangent bundle TM. One can also build a Clifford bundle out of
Clifford_bundle
Quillen metric on the determinant line bundle, a Hermitian metric defined using the analytic torsion of a family of differential operators. Quillen metric Quillen
Quillen determinant line bundle
Quillen_determinant_line_bundle
2 {\displaystyle n+2} vector bundle T → M {\displaystyle {\mathcal {T}}\to M} equipped with the following data: a metric G : T ⊗ T → R {\displaystyle
Tractor_bundle
Riemannian manifold with SU(n) holonomy
non-trivial canonical bundle. For a compact complex n {\displaystyle n} -dimensional manifold M {\displaystyle M} that admits Kähler metrics, the following conditions
Calabi–Yau_manifold
Smooth manifold with an inner product on each tangent space
metric induces an isomorphism of bundles between the tangent bundle and the cotangent bundle. Namely, if g {\displaystyle g} is a Riemannian metric,
Riemannian_manifold
Concept in algebraic geometry
holomorphic line bundle L on X is said to be nef if for every ϵ > 0 {\displaystyle \epsilon >0} there is a smooth Hermitian metric h ϵ {\displaystyle
Nef_line_bundle
Manifold with Riemannian, complex and symplectic structure
canonical bundle has a Kähler–Einstein metric (with constant negative Ricci curvature), and every Calabi–Yau manifold has a Kähler–Einstein metric (with zero
Kähler_manifold
Fiber bundle whose fibers are group torsors
In the mathematical area of topology, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product
Principal_bundle
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
Riemann tensor (general relativity) Pseudo-Riemannian manifold Riemannian bundle metric Riemannian circle Riemannian cobordism Riemannian connection Riemannian
List of things named after Bernhard Riemann
List_of_things_named_after_Bernhard_Riemann
Array of numbers describing a metric connection
cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being
Christoffel_symbols
Assignment of a tensor continuously varying across a region of space
vector space and the tensor bundle is a special kind of vector bundle. (There are vector bundles that are not tensor bundles: the Möbius band for instance
Tensor_field
Tangent spaces of a manifold
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself.
Tangent_bundle
Straight path on a curved surface or a Riemannian manifold
Hamiltonian flow on the cotangent bundle. The Hamiltonian is then given by the inverse of the (pseudo-)Riemannian metric, evaluated against the canonical
Geodesic
p} in these balls is convex. Cotangent bundle Covariant derivative Cubical complex Cut locus Diameter of a metric space is the supremum of distances between
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Structure group sub-bundle on a tangent frame bundle
G} , is a principal G {\displaystyle G} -subbundle of the tangent frame bundle F M {\displaystyle {\text{F}}M} (or GL ( M ) {\displaystyle \operatorname
G-structure_on_a_manifold
Tensor in differential geometry
trace of the Riemann curvature tensor of a Riemannian or pseudo-Riemannian metric. In Riemannian geometry, the Ricci curvature in a given tangent direction
Ricci_curvature
Generalization of an orientation of a vector space
those with positive determinant. If E is a real vector bundle of rank n, then a choice of metric on E amounts to a reduction of the structure group to
Orientation of a vector bundle
Orientation_of_a_vector_bundle
Type of Riemannian metric
The Sasaki metric is a natural choice of Riemannian metric on the tangent bundle of a Riemannian manifold. Introduced by Shigeo Sasaki in 1958. Let (
Sasaki_metric
Technique in photogrammetry and computer vision
points and many images, bundle adjustment is by definition tolerant to missing image projections, and if the distance metric is chosen reasonably (e.g
Bundle_adjustment
Branch of mathematics
vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be spacetime and the bundles and
Differential_geometry
Mathematical construct of fiber bundles
covariant metric tensor gives an isomorphism g : T M → T ∗ M {\displaystyle g\colon TM\to T^{*}M} from the tangent bundle to the cotangent bundle, which
Solder_form
Fiber bundle Principal bundle Frame bundle Hopf bundle Associated bundle Vector bundle Tangent bundle Cotangent bundle Line bundle Jet bundle Sheaf (mathematics)
List of differential geometry topics
List_of_differential_geometry_topics
Tensor that describes the 4D geometry of spacetime
relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Concept in differential geometry
integrable, then we have a Kähler structure. A Hermitian metric on a complex vector bundle E {\displaystyle E} over a smooth manifold M {\displaystyle
Hermitian_manifold
is provided with such a line bundle, along with its degenerate metric, to what extent is it possible to extend the metric off the null cone in a canonical
Ambient_construction
Concept in mathematics
\beta }\circ f).} Alternatively, in the bundle formalism, the Riemannian metrics on M and N induce a bundle metric on T *M ⊗ f *TN, and so one may define
Harmonic_map
Chinese mathematician (born 1958)
to Bergman metrics, Tian studied the following problem. Let L be a line bundle over a Kähler manifold M, and fix a hermitian bundle metric whose curvature
Tian_Gang
Metric on a complex projective space endowed with Hermitian form
\ldots ,\partial _{n}\}} of the holomorphic tangent bundle of CPn, in terms of which the Fubini–Study metric has Hermitian components g i j ¯ = h ( ∂ i , ∂
Fubini–Study_metric
Mathematics concept
vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle π : E → B
Vertical and horizontal bundles
Vertical_and_horizontal_bundles
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
Generalization of Riemannian manifolds
M together with a Finsler metric, which is a continuous nonnegative function F: TM → [0, +∞) defined on the tangent bundle so that for each point x of
Finsler_manifold
Topics referred to by the same term
surface Riemannian symmetric space Riemannian volume form Riemannian bundle metric List of topics named after Bernhard Riemann but may also refer to Hugo
Riemannian
Construct allowing differentiation of tangent vector fields of manifolds
infinitely many affine connections. If the manifold is further endowed with a metric tensor then there is a natural choice of affine connection, called the Levi-Civita
Affine_connection
Hermitian vector bundle E {\displaystyle E} over a smooth manifold M {\displaystyle M} which is compatible with the Hermitian metric ⟨ ⋅ , ⋅ ⟩ {\displaystyle
Hermitian_connection
the tangent bundle of M admits a splitting into a Whitney sum of two Df-invariant subbundles, called the stable bundle and the unstable bundle and denoted
Hyperbolic_set
Correspondsnce between Higgs bundles and fundamental group representations
pair ( E , Φ ) {\displaystyle (E,\Phi )} . A Hermitian metric h {\displaystyle h} on a Higgs bundle ( E , Φ ) {\displaystyle (E,\Phi )} gives rise to a Chern
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
Canonical connection on a pseudo-Riemannian manifold
the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free. The fundamental theorem
Levi-Civita_connection
Mathematical formulation of gravity theory with a general affine connection
(that is, a general linear connection on the tangent bundle), in addition to a pseudo-Riemannian metric. This is in contrast to standard formulations of General
Metric-affine gravitation theory
Metric-affine_gravitation_theory
Concept in algebraic geometry
bundle form an important class; for example, over the complex numbers, these are the curves with a metric of negative curvature. The canonical bundle
Ample_line_bundle
Type of vector bundle
In mathematics, a Higgs bundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle
Higgs_bundle
Covariant derivative of the metric tensor
covariant derivative of the metric tensor. It can be interpreted as the failure of a connection to parallelly transport the metric. Physically, this corresponds
Nonmetricity_tensor
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
Study of angle-preserving transformations of a geometric space
line bundle N+ → S is identified with the bundle of conformal scales on S: to give a section of this bundle is tantamount to specifying a metric in the
Conformal_geometry
Vector bundles theorem
correspondence for Higgs bundles, as well as the Yau–Tian–Donaldson conjecture about the existence of Kähler–Einstein metrics on Fano varieties, and the
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Mathematical operation on vector spaces
in general relativity, the gravitational field is described through the metric tensor, which is a tensor field with one tensor at each point of the space-time
Tensor_product
Property of a differential manifold that includes complex structures
structures. Consider an N-manifold M. The tangent bundle of M, which will be denoted T, is the vector bundle over M whose fibers consist of all tangent vectors
Generalized_complex_structure
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
Method for specifying point positions
Mixed tensor Antisymmetric tensor Symmetric tensor Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative
Coordinate_system
Bundle of linear subspaces of the tangent bundle
geometry, a contact bundle is a particular type of fiber bundle constructed from a smooth manifold. Like how the tangent bundle is the manifold that
Contact_bundle
Algebraic object with geometric applications
known as a metric) g, the term contraction is used for removing two contravariant or two covariant indices by forming a trace with the metric tensor or
Tensor
Branch of topology
topological space. Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on
General_topology
Expression that may be integrated over a region
generally a pseudo-Riemannian manifold, the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles. This makes it possible to convert vector
Differential_form
Mathematical concept
fibration. Complex projective space carries a (Kähler) metric, called the Fubini–Study metric, in terms of which it is a Hermitian symmetric space of
Complex_projective_space
Type of derivative in differential geometry
the spinor bundle. Such is not the case: the quantities on the right-hand side of Kosmann's local expression combine so as to make all metric and connection
Lie_derivative
Shorthand notation for tensor operations
→ V ∗ {\displaystyle V\to V^{*}} , for instance a Riemannian metric or Minkowski metric), one can raise and lower indices. A basis gives such a form (via
Einstein_notation
Application of topology
pseudo-Riemannian metric and an associated space-time structure is a space-time. Gravitation theory is formulated as classical field theory on natural bundles over
World_manifold
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
Exterior algebraic map taking tensors from p forms to n-p forms
the global sections of the bundle ⋀ k T ∗ M → M {\textstyle \bigwedge ^{k}\mathrm {T} ^{*}\!M\to M} . The Riemannian metric induces a scalar product on
Hodge_star_operator
Concept in Riemannian geometry
structure of a differentiable manifold equipped with a metric. If M is a manifold equipped with a metric g, then an orthonormal frame at a point P of M is
Orthonormal_frame
Principal bundle formulation of the Higgs field
bundle P / H → X {\displaystyle P/H\to X} . This section is treated as a classical Higgs field. A key point is that there exists a composite bundle P
Higgs_field_(classical)
Operation in mathematics
tensors. Over a Riemannian manifold, a metric (field of inner products) is available, and both metric and non-metric contractions are crucial to the theory
Tensor_contraction
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
Manifold
tangent bundle, which is Hermitian with respect to the complex structure on the tangent space at each point. As in the Riemannian case, such metrics always
Complex_manifold
Concept in geometry
{\displaystyle H} naturally admits a bundle metric coming from the quaternionic algebra structure, and, with this metric, H {\displaystyle H} splits into
Quaternionic_manifold
(anti)-canonical line bundle (i.e. in the case of Fano or Calabi–Yau manifolds) the notions of K-stability and K-polystability coincide, cscK metrics are precisely
Constant scalar curvature Kähler metric
Constant_scalar_curvature_Kähler_metric
2009 studio album by Metric
deluxe bundle packages. There was a Limited Edition Package available at first that was limited to 500 copies, which has now sold out. Metric opted to
Fantasies_(album)
Measure of the curvature of a pseudo-Riemannian manifold
Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the
Weyl_tensor
Study of vector bundles, principal bundles, and fibre bundles
theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused
Gauge_theory_(mathematics)
Tensor index notation for tensor-based calculations
bundle of a differentiable manifold is called an affine connection. A connection is a metric connection when the covariant derivative of the metric tensor
Ricci_calculus
Fiber bundle
theory of fiber bundles with a structure group G {\displaystyle G} (a topological group) allows an operation of creating an associated bundle, in which the
Associated_bundle
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
Algebra associated to any vector space
{\displaystyle F_{[ij,k]}=F_{[ij;k]}=0.} None of this requires a metric. Adding the Lorentz metric and an orientation provides the Hodge star operator ⋆ {\displaystyle
Exterior_algebra
observation of Shoshichi Kobayashi, the circle bundle S in its canonical line bundle admits a Sasaki–Einstein metric, in a manner that makes the projection from
Sasakian_manifold
with eigenvalue 1 generates the Lie algebra. The subbundle of the tangent bundle associated to this eigenspace is called horizontal. On a Carnot group, any
Carnot_group
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
Branch of mathematics
Euclidean spaces and more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations
Topology
Section of a certain line bundle
a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume
Density_on_a_manifold
Approach to general relativity
general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis
Tetrad_formalism
Type of Riemannian manifold
flat Euclidean metric is a hyperkähler manifold. The first non-trivial example discovered is the Eguchi–Hanson metric on the cotangent bundle T ∗ S 2 {\displaystyle
Hyperkähler_manifold
Distance-preserving mathematical transformation
congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from
Isometry
Riemannian manifold which satisfies vacuum Einstein equations
proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the
Einstein_manifold
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