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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. Let π :
Flat_vector_bundle
Defines a notion of parallel transport on a bundle
vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields
Connection_(vector_bundle)
values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential
Vector-valued differential form
Vector-valued_differential_form
Type of vector bundle
Simpson. A Higgs bundle can be thought of as a "simplified version" of a flat holomorphic connection on a holomorphic vector bundle, where the derivative
Higgs_bundle
Tangent spaces of a manifold
tangent bundle of a differentiable manifold M {\displaystyle M} is a manifold T M {\displaystyle TM} which assembles all the tangent vectors in M {\displaystyle
Tangent_bundle
vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may
Stable_vector_bundle
Correspondsnce between Higgs bundles and fundamental group representations
to a vector bundle with flat connection as follows. The universal cover X ^ {\displaystyle {\hat {X}}} of X {\displaystyle X} is a principal bundle over
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
their current result is a formula that counts the Euler number of a flat vector bundle in terms of vertices of transversal open coverings. Notoriously, the
Chern's conjecture (affine geometry)
Chern's_conjecture_(affine_geometry)
Connection on a vector bundle
a certain vector bundle over a base space S of a family of algebraic varieties V s {\displaystyle V_{s}} . The fibers of the vector bundle are the de
Gauss–Manin_connection
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
mathematics and theoretical physics, a Killing vector field or Killing field (named after Wilhelm Killing) is a vector field on a Riemannian manifold or pseudo-Riemannian
Killing_vector_field
Construct allowing differentiation of tangent vector fields of manifolds
the 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
Generalization of vector bundles
information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under
Coherent_sheaf
Concept in mathematics
any fiber bundle associated to P {\displaystyle P} via the associated bundle construction. In particular, on any associated vector bundle the principal
Connection_(principal_bundle)
Concept in algebraic geometry
canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle
Canonical_bundle
Partial differential equations whose solutions are instantons
system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of
Yang–Mills_equations
Topological invariant of manifolds that can distinguish homotopy-equivalent manifolds
and (Nicolaescu 2002, 2003). If M is a Riemannian manifold and E a vector bundle over M, then there is a Laplacian operator acting on the k-forms with
Analytic_torsion
Isomorphism between the tangent and cotangent bundles of a manifold
smooth vector bundles ♭ : T M → T ∗ M {\displaystyle \flat :\mathrm {T} M\to \mathrm {T} ^{*}M} . By non-degeneracy of the metric, ♭ {\displaystyle \flat }
Musical_isomorphism
Study of vector bundles, principal bundles, and fibre bundles
gauge 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)
Mathematic theorem about Riemann surfaces
says that a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it admits a flat unitary connection compatible with its
Narasimhan–Seshadri_theorem
Concept in differential geometry
holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of these cases, the
Holonomy
Vector bundles theorem
Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after Shoshichi
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Structure group sub-bundle on a tangent frame bundle
{\displaystyle GL(n)} -bundle, the frame bundle. In particular, every smooth manifold has a canonical vector bundle, the tangent bundle. For a Lie group G
G-structure_on_a_manifold
Function in mathematics
defines directional derivative for sections of a vector bundle more general than the tangent bundle. Connections also lead to convenient formulations
Connection_(mathematics)
Term in differential geometry
canonical vector-valued 1-form on the frame bundle, the torsion Θ {\displaystyle \Theta } of the connection form ω {\displaystyle \omega } is the vector-valued
Curvature_form
Classical field theories on fiber bundles
manifold M {\displaystyle M} is flat, there are simplifications which remove this subtlety. An associated vector bundle E → π M {\displaystyle E\xrightarrow
Covariant classical field theory
Covariant_classical_field_theory
Infinitesimal version of Lie groupoid
In mathematics, a Lie algebroid is a vector bundle A → M {\displaystyle A\rightarrow M} together with a Lie bracket on its space of sections Γ ( A ) {\displaystyle
Lie_algebroid
Characteristic classes of vector bundles
the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics
Chern_class
Type of derivative in differential geometry
bundles with a connection and vector-valued differential forms. A 'naïve' attempt to define the derivative of a tensor field with respect to a vector
Lie_derivative
Assignment of a tensor continuously varying across a region of space
the fiber is a vector space and the tensor bundle is a special kind of vector bundle. The vector bundle is a natural idea of "vector space depending
Tensor_field
Specification of a derivative along a tangent vector of a manifold
covariant differentiation in a vector bundle by means of what is known today as a Koszul connection or a connection on a vector bundle. Using ideas from Lie algebra
Covariant_derivative
Euclidean space without distance and angles
flat through the points. Any vector space may be viewed as an affine space; this amounts to "forgetting" the special role played by the zero vector.
Affine_space
Mathematical concept that extends the intuitive idea of gluing in topology
descent implies a vector bundle on Y (so, a bundle given on each Xi), and our concern is to 'glue' those bundles Vi, to make a single bundle V on X. What we
Descent_(mathematics)
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space)
Hopf_fibration
Mathematical measure of how much a curve or surface deviates from flatness
Curvature form for the appropriate notion of curvature for vector bundles and principal bundles with connection Curvature of a measure for a notion of curvature
Curvature
Direct summand of a free module (mathematics)
compact manifold is the space of smooth sections of a smooth vector bundle). Vector bundles are locally free. If there is some notion of "localization"
Projective_module
Differential geometry construct on fiber bundles
on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear
Ehresmann_connection
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
Analogs of homology groups for algebraic varieties
because they can all be constructed as the vanishing loci of the same vector bundle. For example, given two homogeneous polynomials of degree d {\displaystyle
Chow_group
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
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
Circulation density in a vector field
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional
Curl_(mathematics)
Line or vector perpendicular to a curve or a surface
reflected ray. Dual space – In mathematics, vector space of linear forms Ellipsoid normal vector Normal bundle – Concept in mathematics Pseudovector – Physical
Normal_(geometry)
Generalization of the Dirac equation
spacetime. Mathematically, this is a section of a vector bundle associated to the spin-frame bundle by the representation ( 1 / 2 , 0 ) ⊕ ( 0 , 1 / 2
Dirac equation in curved spacetime
Dirac_equation_in_curved_spacetime
Concept in mathematics
of flat connections on algebraic vector bundles on X with regular singularities to the category of local systems of finite-dimensional complex vector spaces
Riemann–Hilbert correspondence
Riemann–Hilbert_correspondence
Non-tensorial representation of the spin group
symplectic manifold) has a Spinc structure. Likewise, every complex vector bundle on a manifold carries a Spinc structure. A number of Clebsch–Gordan
Spinor
Object in differential geometry
frame bundle FM of the manifold M. This principal bundle is equipped with a connection form ω, a gl(n)-valued one-form which maps vertical vectors to the
Torsion_tensor
Riemannian manifold with SU(n) holonomy
surfaces give examples of complex manifolds that have Ricci-flat metrics, but their canonical bundles are not trivial, so they are Calabi–Yau manifolds according
Calabi–Yau_manifold
Scheme in algebraic geometry
normal bundle, the vector bundle on X corresponding to the dual of the sheaf I/I2. If X is a point, then the normal cone and the normal bundle to it are
Normal cone (algebraic geometry)
Normal_cone_(algebraic_geometry)
Expression that may be integrated over a region
tangent space of M, and therefore it factors through the trivial bundle M × R. The vector bundle morphism ⋀ k T M → M × R {\textstyle {\textstyle \bigwedge
Differential_form
This can be interpreted in contact geometry as well using Reeb vector flow. Reeb vector flow applies more generally to the geodesic flow on Finsler manifolds
Geodesics as Hamiltonian flows
Geodesics_as_Hamiltonian_flows
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
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
smooth manifold. An affine flat structure on M is a sheaf Tf of vector spaces that pointwisely span TM the tangent bundle and the tangent bracket of pairs
Frobenius_manifold
Intrinsic geometric structures in mathematics
frame bundle. In the case of an embedded surface, this lift is very simply described in terms of orthogonal projection. Indeed, the vector bundles associated
Riemannian connection on a surface
Riemannian_connection_on_a_surface
Algebraic geometry analog of a principal bundle in algebraic topology
In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski
Torsor_(algebraic_geometry)
Singularity theorem in Yang–Mills theory
B^{n}\setminus \{0\}:=\{x\in \mathbb {R} ^{n}|\|x\|\leq 1\}} and a vector bundle η ↠ B 4 ∖ { 0 } {\displaystyle \eta \twoheadrightarrow B^{4}\setminus
Uhlenbeck's singularity theorem
Uhlenbeck's_singularity_theorem
Smooth manifold
J^{2}=-1} when regarded as a vector bundle isomorphism J : T M → T M {\displaystyle J\colon TM\to TM} on the tangent bundle. A manifold equipped with an
Almost_complex_manifold
Generalization of Riemannian manifolds
→ [0, +∞) defined on the tangent bundle so that for each point x of M, F(v + w) ≤ F(v) + F(w) for every two vectors v,w tangent to M at x (subadditivity)
Finsler_manifold
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
Geometric space whose points represent algebro-geometric objects of some fixed kind
physics, the number of moduli of vector bundles and the closely related problem of the number of moduli of principal G-bundles has been found to be significant
Moduli_space
on the moduli stack of vector bundles on a curve" (PDF), in Schmitt, Alexander (ed.), Affine flag manifolds and principal bundles, Trends in Mathematics
Moduli stack of principal bundles
Moduli_stack_of_principal_bundles
Symmetries in a gravitational theory
morphism, unless Γ {\displaystyle \Gamma } is flat. However, there is a category of above mentioned natural bundles T → X {\displaystyle T\to X} which admit
General covariant transformations
General_covariant_transformations
Operation on differential forms
of the cotangent bundle, that gives a local linear approximation to f {\displaystyle f} in the cotangent space at each point. A vector field V = ( v 1
Exterior_derivative
Type of manifold in differential geometry
Y)=\omega ^{\flat }(X)(Y)} for any two vector fields X , Y {\displaystyle X,Y} , and ω ♯ ∘ ω ♭ = Id {\displaystyle \omega ^{\sharp }\circ \omega ^{\flat }=\operatorname
Symplectic_manifold
Generalizations of codimension-1 subvarieties of algebraic varieties
complex vector space of meromorphic functions on X with poles at most given by D, called H0(X, O(D)) or the space of sections of the line bundle associated
Divisor_(algebraic_geometry)
Tensor in differential geometry
{z}}^{\beta }}}\right).} If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the
Ricci_curvature
Mathematical concept
of the tangent bundle to itself: ( L g ) ∗ : T h G → T g h G . {\displaystyle (L_{g})_{*}:T_{h}G\to T_{gh}G.} A left-invariant vector field is a section
Maurer–Cartan_form
Doughnut-shaped surface of revolution
(R+P\sin v)\sin u,P\cos v).} If R and P in the above flat torus parametrization form a unit vector (R, P) = (cos(η), sin(η)) then u, v, and 0 < η < π/2
Torus
Affine connection on the tangent bundle of a manifold
pseudo-Riemannian manifold. TM is the tangent bundle of M. g is the pseudo-Riemannian metric of M. X, Y, Z are smooth vector fields on M, i. e. smooth sections of
Levi-Civita_connection
Superconductivity theory
Ginzburg–Landau functional can be formulated in the general setting of a complex vector bundle over a compact Riemannian manifold. This is the same functional as given
Ginzburg–Landau_theory
Manifold
that is, the tangent bundle is equipped with a linear complex structure. Concretely, this is an endomorphism of the tangent bundle whose square is −I;
Complex_manifold
Study of angle-preserving transformations of a geometric space
embedding. Thus the 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
Conformal_geometry
characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class
Segre_class
Type of metric in Riemannian geometry
on any holomorphic vector bundle over X {\displaystyle X} (note that the Levi-Civita connection on the holomorphic tangent bundle is precisely the Chern
Kähler–Einstein_metric
holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space of holomorphic
Quadratic_differential
Application of Lagrangian mechanics to field theories
on a fiber bundle, and the derivatives of the field are understood to be sections of the jet bundle. The above can be generalized for vector fields, tensor
Lagrangian_(field_theory)
Type of mathematical space
of a highest weight vector in a projectivized representation of G. The complex projective homogeneous varieties are the compact flat model spaces for Cartan
Generalized_flag_variety
Formulation of physics
in a flat multidimensional Euclidean space, which is called the configuration space of this system. Its points are marked by the radius-vector r {\displaystyle
Newtonian_dynamics
Tensor index notation for tensor-based calculations
of a vector. In flat spacetime with linear coordinatization, a tuple of differences in coordinates, Δxμ, can be treated as a contravariant vector. With
Ricci_calculus
Generalization of affine connections
a connection on the frame bundle (principal bundle) of M (or equivalently, a connection on the tangent bundle (vector bundle) of M). A key aspect of the
Cartan_connection
Operation that pairs a left and a right R-module into an abelian group
construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative
Tensor_product_of_modules
Generalization of vector spaces from fields to rings
to the module of sections of some vector bundle; the category of C∞(X)-modules and the category of vector bundles over X are equivalent. If R is any
Module_(mathematics)
Every vector bundle E → X {\displaystyle E\to X} over a scheme is a smooth morphism. For example, it can be shown that the associated vector bundle of O
Smooth_morphism
Smooth manifold with an inner product on each tangent space
\cdot )} is a isomorphism of smooth vector bundles from the tangent bundle T M {\displaystyle TM} to the cotangent bundle T ∗ M {\displaystyle T^{*}M} . An
Riemannian_manifold
at the point p {\displaystyle p} . Sections of the tangent bundles, also known as vector fields, are typically denoted as X , Y , Z ∈ Γ ( T M ) {\displaystyle
Exterior_calculus_identities
then, by an observation of Shoshichi Kobayashi, the circle bundle S in its canonical line bundle admits a Sasaki–Einstein metric, in a manner that makes
Sasakian_manifold
Property of a differential manifold that includes complex structures
tangent bundle of M, which will be denoted T, is the vector bundle over M whose fibers consist of all tangent vectors to M. A section of T is a vector field
Generalized_complex_structure
Tensor field in Riemannian geometry
manifold, and X ( M ) {\displaystyle {\mathfrak {X}}(M)} be the space of all vector fields on M {\displaystyle M} . We define the Riemann curvature tensor as
Riemann_curvature_tensor
known as the GJMS operators. A related construction is the tractor bundle. The model flat geometry for the ambient construction is the future null cone in
Ambient_construction
In mathematics, a partition of a manifold into submanifolds
dimensional subbundle of the tangent bundle of a manifold) to be tangent to the leaves of a foliation, is that the set of vector fields tangent to the distribution
Foliation
Topological space that locally resembles Euclidean space
what a tangent vector might be, and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle. The n-sphere
Manifold
Combination of higher category theory with Chern–Weil theory
ways to describe the k {\displaystyle k} -th Chern class of complex vector bundles of rank n {\displaystyle n} , which is as a: (1-categorical) natural
∞-Chern–Weil_theory
Mathematics of smooth surfaces
frame bundle so that its tangent vectors lie in a special subspace of codimension one in the three-dimensional tangent space of the frame bundle. The projection
Differential geometry of surfaces
Differential_geometry_of_surfaces
Module over a sheaf of differential operators
D-module structure is nothing else than equipping the vector bundle associated to M with a flat (or integrable) connection. As the ring DX is noncommutative
D-module
Mathematical object
admits nonvanishing vector fields (sections of its tangent bundle). One can even find three linearly independent and nonvanishing vector fields. These may
3-sphere
Construct in mathematics
objects, but the stacky version remembers automorphisms of vector bundles. For any stable vector bundle E {\displaystyle E} the automorphism group A u t ( E
Gerbe
the geodesics. Geodesic flow is a flow on a tangent bundle TM of a manifold M, generated by a vector field whose trajectories are of the form ( γ ( t )
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Electromagnetic quantum-mechanical effect in regions of zero magnetic and electric field
describe the quantum electron by a section in a complex line bundle with an "external" flat connection ∇ {\displaystyle \nabla } with monodromy α = {\displaystyle
Aharonov–Bohm_effect
Approach to general relativity
tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent vector fields
Tetrad_formalism
Property of a mathematical space
dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion
Dimension
FLAT VECTOR-BUNDLE
FLAT VECTOR-BUNDLE
Male
Greek
(á¼ÎºÏ„ωÏ) Variant spelling of Greek Hektor, EKTOR means "defend; hold fast."
Male
English
Roman Latin name VICTOR means "conqueror."Â
Boy/Male
English American
Doctor; teacher.
Boy/Male
American, British, Christian, Danish, Dutch, English, Finnish, French, German, Greek, Hindu, Indian, Irish, Jamaican, Latin, Romanian, Slovenia, Spanish, Swedish, Swiss, Tamil, Ukrainian
Victorious; Conqueror; Winner; Champion; One who Conquers; Victory
Male
Scandinavian
 Scandinavian form of Roman Latin Victor, VIKTOR means "conqueror." Compare with another form of Viktor.
Boy/Male
Australian, Basque, Czech, Czechoslovakian, Danish, Finnish, French, German, Hungarian, Latin, Polish, Slovenia, Swedish, Swiss, Ukrainian
The Conqueror; Victory; Victorious; Conquer
Boy/Male
Tamil
Vijayketu | விஜயகேதà¯
Flag of victory
Vijayketu | விஜயகேதà¯
Male
Russian
(Cyrillic Виктор): Slavic form of Roman Latin Victor, VIKTOR means "conqueror." In use by the Bulgarians, Russians and Serbians. Compare with another form of Viktor.
Surname or Lastname
English (East Anglia) and Jewish (Ashkenazic)
English (East Anglia) and Jewish (Ashkenazic) : metonymic occupational name for someone who grew, sold, or treated flax for weaving into linen cloth, from (respectively) Middle English flax, German Flachs.
Boy/Male
Spanish
Victor.
Male
Russian
(Филат) Pet form of Russian Feofilakt, FILAT means "God-guard."
Boy/Male
Hindu, Indian, Marathi
Flag of Victory
Male
Arthurian
, sir Hector de Maris; (defender).
Male
English
 Anglicized form of Scottish Gaelic Eachann, HECTOR means "brown horse." Compare with another form of Hector.
Surname or Lastname
Scottish
Scottish : Anglicized form of the Gaelic personal name Eachann (earlier Eachdonn, already confused with Norse Haakon), composed of the elements each ‘horse’ + donn ‘brown’.English : found in Yorkshire and Scotland, where it may derive directly from the medieval personal name. According to medieval legend, Britain derived its name from being founded by Brutus, a Trojan exile, and Hector was occasionally chosen as a personal name, as it was the name of the Trojan king’s eldest son. The classical Greek name, HektÅr, is probably an agent derivative of Greek ekhein ‘to hold back’, ‘hold in check’, hence ‘protector of the city’.German, French, and Dutch : from the personal name (see 2 above). In medieval Germany, this was a fairly popular personal name among the nobility, derived from classical literature. It is a comparatively rare surname in France.
Boy/Male
French
From the flat land.
Surname or Lastname
English (chiefly East Anglia)
English (chiefly East Anglia) : topographic name for someone who lived on a flat, a patch of level or low-lying ground (Old Norse flat, flǫt).South German : variant of Flath 2.
Male
Portuguese
Galician-Portuguese form of Roman Latin Victor, VITOR means "conqueror."
Male
Portuguese
Portuguese form of Latin Hector, HEITOR means "defend; hold fast."
Boy/Male
Hindu
Flag of victory
FLAT VECTOR-BUNDLE
FLAT VECTOR-BUNDLE
Boy/Male
Hindu
Boy/Male
Hindu
Girl/Female
Assamese, Bengali, Celebrity, Gujarati, Hebrew, Hindu, Indian, Kannada, Latin, Malayalam, Marathi, Oriya, Sanskrit, Sindhi, Tamil, Telugu
Queen
Girl/Female
Hindu
A bond between friendship and Love
Boy/Male
Tamil
Prasiddhi | பà¯à®°à®¸à®¿à®¤à¯à®¤à®¿Â
Accomplishment, Fame
Girl/Female
Indian, Marathi
Soft
Male
Greek
(Γάδ) Greek form of Hebrew Gad, GAD means "troop." In the bible, this is the name of a tribe descended from Gad, mentioned in the New Testament in Rev vii. 5. Compare with other forms of Gad.
Girl/Female
Hindu
Name of a Goddess, Beautiful eyed
Boy/Male
Muslim
Kind and friendly
Girl/Female
Indian
Reflection, Image, Radiance
FLAT VECTOR-BUNDLE
FLAT VECTOR-BUNDLE
FLAT VECTOR-BUNDLE
FLAT VECTOR-BUNDLE
FLAT VECTOR-BUNDLE
superl.
Lacking liveliness of commercial exchange and dealings; depressed; dull; as, the market is flat.
n.
A flat-bottomed boat, without keel, and of small draught.
n.
Same as Radius vector.
n.
The flat or broad side of a sword.
n.
A directed quantity, as a straight line, a force, or a velocity. Vectors are said to be equal when their directions are the same their magnitudes equal. Cf. Scalar.
n.
Plain; flat; level.
superl.
Below the true pitch; hence, as applied to intervals, minor, or lower by a half step; as, a flat seventh; A flat.
n.
A flat stone used for paving.
v. i.
A float board. See Float board (below).
adv.
In a flat manner; directly; flatly.
superl.
Not sharp or shrill; not acute; as, a flat sound.
n.
Something broad and flat in form
superl.
Tasteless; stale; vapid; insipid; dead; as, fruit or drink flat to the taste.
superl.
Unanimated; dull; uninteresting; without point or spirit; monotonous; as, a flat speech or composition.
adv.
Level with the ground; flat.
v. t.
To make flat; to flatten; to level.
a.
Pertaining to a rector or a rectory; rectoral.
n.
The ratio of one vector to another in length, no regard being had to the direction of the two vectors; -- so called because considered as a stretching factor in changing one vector into another. See Versor.
v. i.
To become flat, or flattened; to sink or fall to an even surface.
v. t.
To lay with flags of flat stones.