The geometric notion meaning that at some point two curves (or a curve and a surface) hav
See also [[Tangent vector|Tangent vector]].
461 bytes (75 words) - 17:23, 7 February 2011
$#C+1 = 32 : ~/encyclopedia/old_files/data/V096/V.0906340 Vector,
''geometric''
4 KB (734 words) - 08:28, 6 June 2020
that associates with a differentiable vector field $ X $
and a differentiable geometric object $ Q $
9 KB (1,337 words) - 22:16, 5 June 2020
''of an operator (transformation) $A$ of a vector space $L$ over a field $k$''
An element $\lambda\in k$ such that there is a non-zero vector $x\in L$ satisfying the condition
2 KB (373 words) - 09:18, 12 December 2013
...is called Euclidean if each of its fibres has the structure of a Euclidean vector space with a scalar product $ \langle , \rangle $
is a smooth function on the base. A linear connection on a Euclidean vector bundle is called a Euclidean connection if for any parallel displacement of
2 KB (251 words) - 19:38, 5 June 2020
$#C+1 = 17 : ~/encyclopedia/old_files/data/D030/D.0300210 Darboux vector
The vector $ \pmb\delta $
2 KB (352 words) - 17:32, 5 June 2020
...ocity vector|Phase velocity vector]]) is defined, is non-zero and is not a vector tangent to the curve. The concept was introduced by H. Poincaré [[#Referen
...="top">[2]</TD> <TD valign="top"> S. Lefshetz, "Differential equations: geometric theory" , Interscience (1962)</TD></TR></table>
2 KB (281 words) - 18:48, 5 April 2020
...concentration of its probability distribution around a certain prescribed vector in terms of the second-order moments. Let $ X $
be a random vector assuming values $ x = ( x _ {1}, \dots, x _ {n} ) ^ {T} $
3 KB (375 words) - 16:43, 20 January 2022
Let a geometric object $ \Omega $(
cf. [[Geometric objects, theory of|Geometric objects, theory of]]) be given in a differentiable manifold $ X _ {n} $,
5 KB (725 words) - 19:35, 5 June 2020
...f $K$ along any non-zero integer vector. Here the "width" of $K$ along a vector $v$ in $\mathbf{R}^n$ is
The width of $K$ with respect to $\mathbf{Z}^n$ is greater or equal than the geometric width of $K$, which is the minimum width of $K$ along all unit-length vecto
1 KB (242 words) - 21:16, 8 April 2018
Adjoint norms (cf. [[Norm|Norm]]) in certain vector spaces dual to each other.
vector $ \alpha $,
4 KB (548 words) - 07:59, 6 June 2020
...cyclopediaofmath.org/legacyimages/c/c025/c025470/c02547012.png" /> for any
vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath
...02547060.png" /> (and this is equivalent). It is sometimes called the Reeb
vector field of <img align="absmiddle" border="0" src="https://www.encyclopediaofm
11 KB (1,538 words) - 17:29, 7 February 2011
...beginning of the 20th century. It differs from local geometry, in which a geometric object (field, mapping) is studied in sufficiently small domains only, as i
...ative results, as well as quantitative relations in the large, for regular geometric structures on multi-dimensional singularity-free manifolds.
5 KB (702 words) - 16:57, 15 April 2012
is a differential-geometric structure on $ M $
...ant differentiation|covariant differentiation]], which associates with two vector fields $ X , Y $
6 KB (870 words) - 22:17, 5 June 2020
...nifold similar to a [[Sasakian manifold]]. However, by its topological and geometric properties, such a manifold is closely related to a product manifold (unlik
...$(M,g)$ endowed with an endomorphism $\phi$ of the tangent bundle $TM$, a vector field $\xi$ and a $1$-form $\eta$ which satisfy the conditions
3 KB (562 words) - 15:54, 16 March 2024
...lidean space]] of signature $(1,3)$, suggested by H. Minkowski (1908) as a geometric model of space-time in the special theory of relativity (see ). Correspondi
...e is a time-like vector. The tangent vector to a light ray is an isotropic vector.
5 KB (692 words) - 19:18, 12 April 2014
A geometric object described in a coordinate system $ x = ( x ^ {1}, \dots, x ^ {n}
the tensor density is a tensor (cf. [[Tensor on a vector space|Tensor on a vector space]]). Concepts such as type, valency, covariance, contravariance, etc.
2 KB (350 words) - 10:34, 5 March 2022
...ping|Transversal mapping]]); a concept in linear algebra, differential and geometric topology.
a) Two vector subspaces $ A, B $
5 KB (780 words) - 08:26, 6 June 2020
...stable vector bundles (of fixed rank and degree; cf. also [[Vector bundle|Vector bundle]]) over a given [[Riemann surface|Riemann surface]] $X$ of genus $g\
...#References|[a5]]]; and quantized Hitchin systems with applications to the geometric Langlands program [[#References|[a2]]].
5 KB (752 words) - 15:33, 4 October 2014
''of a set $M$ in an $n$-dimensional vector space''
In general vector spaces, where a hyperplane can be defined as a domain of constant value of
2 KB (331 words) - 04:59, 16 September 2014
A field of geometric quantities on a homogeneous space $ M = G / H $
is a [[Vector bundle|vector bundle]]. In this case, $ H $
8 KB (1,274 words) - 22:13, 5 June 2020
in a finite-dimensional vector space $ V $
and any non-zero vector $ v \in V $
6 KB (818 words) - 12:00, 16 December 2019
...[[Non-linear operator|Non-linear operator]]) between infinite-dimensional vector spaces and also certain classes of non-linear spaces and their mappings. Th
1) Differential calculus of non-linear mappings between Banach, topological vector and certain more general spaces, including theorems on the local inversion
4 KB (490 words) - 17:11, 7 February 2011
...ined as a certain set of objects, which are called its points; they can be geometric objects, functions, states of a physical system, etc. By considering such a
2) The "space of events", playing an important part in the geometric interpretation of relativity theory. Every event is characterized by its po
4 KB (580 words) - 14:27, 28 August 2014
...-geometric structures (cf. [[Differential-geometric structure|Differential-geometric structure]]) on a smooth [[Manifold|manifold]] $ M $
and for each tangent vector $ X _ {x} \in T _ {x} ( M) $
9 KB (1,365 words) - 18:18, 19 January 2021
...ction then one defines a vector field $X_f$ on $S$, called the hamiltonian vector field associated to $f$, by setting
...sely, if $\mathfrak{g}$ is a finite dimensional Lie algebra then its dual vector space $V:=\mathfrak{g}^*$ carries a linear Poisson bracket which is given
6 KB (1,109 words) - 13:50, 12 December 2013
$#C+1 = 46 : ~/encyclopedia/old_files/data/K055/K.0505420 Killing vector,
''more precisely, Killing vector field or infinitesimal motion''
5 KB (766 words) - 22:14, 5 June 2020
...^*$ of the problem of minimizing $f$, the convergence rate being that of a geometric progression with quotient $q<1$.
...this system is equivalent to finding a vector $x^*$ in an $n$-dimensional vector space minimizing the functional
3 KB (519 words) - 19:10, 6 April 2012
is constructed such that every vector $ b _ {i} $ ($ i = 1, \dots, k $)
are obtained from the condition of orthogonality of the vector $ b _ {i+ 1} $
4 KB (542 words) - 08:33, 13 May 2022
several metric concepts of vector algebra can be applied to trivectors. The measure of a trivector $ [ \mat
can be identified with a vector of $ A $,
4 KB (571 words) - 08:58, 10 April 2023
of a space (as a rule, a vector space) $ X $
into a vector space $ Y $
12 KB (1,902 words) - 08:03, 6 June 2020
A [[triple system]] is a [[Vector space|vector space]] $V$ over a field $K$ together with a $K$-[[trilinear mapping]] $V \
A vector space $V$ with triple product $[\,,\,,\,]$ is said to be a Lie triple syste
3 KB (559 words) - 16:43, 15 March 2023
of the real vector space $ \mathbf R ^ {n} $.
of the vector $ x ( p) \in \mathbf R ^ {n} $
7 KB (1,075 words) - 16:43, 4 June 2020
...ry constant. The integral curve is often identified with the solution. The geometric meaning of the integral curves of a scalar equation
the tangent of the angle of inclination of the vector with the $ x $-
3 KB (388 words) - 22:12, 5 June 2020
...Delta : \dim F = i\}\vert$, and call $(f_{-1},f_0,\ldots,f_{d-1})$ the $f$-vector of $\Delta$. Then
...dots,h_d)$, called the $h$-vector of $\Delta$, may be derived from the $f$-vector of $\Delta$ (and vice versa) by the equation
3 KB (544 words) - 20:16, 8 November 2023
The vector bundle consisting of tangent vectors to the ambient manifold that are norma
by the vector bundle $ T _ {Y} $.
4 KB (554 words) - 06:02, 18 April 2023
is the general linear group of some finite-dimensional vector space $ U $ ,
In the modern geometric theory of invariants, by a comitant one often means any equivariant morphis
3 KB (511 words) - 13:16, 7 April 2023
...eferences|[5]]] consists in the fact that the transfer for the fibres of a vector bundle defined by a horizontal distribution ceases to have a linear charact
...g non-linear connections arose from the need to study various differential-geometric structures of higher orders (such as, for example, a [[Kawaguchi space|Kawa
7 KB (1,089 words) - 12:15, 18 February 2022
...regate of methods of finding various versions of [[Geometric approximation|geometric approximation]] to the solution of the corresponding problem.
If one adopts a geometric approximation for the wave equation and equates to zero the coefficients of
7 KB (1,036 words) - 05:39, 19 March 2022
2) the scalar product is distributive with respect to vector addition:
of a vector $ \mathbf a $
9 KB (1,270 words) - 08:54, 13 January 2024
...ure on a manifold, a geometric quantity, a geometric object, or a field of geometric objects, is a section of a bundle associated with the principal bundle of c
Intuitively, a geometric quantity can be considered as a quantity whose value depends not only on th
11 KB (1,705 words) - 14:55, 7 June 2020
...iteria. On the one hand, a non-Euclidean space may be a finite-dimensional vector space with a scalar product expressible in Cartesian coordinates as
...spaces may also be classified from the point of view of their differential-geometric properties as Riemannian spaces of constant curvature (this includes the ca
2 KB (256 words) - 08:02, 6 June 2020
...connection it is possible to proceed directly from a generalization of the geometric foundation of three-dimensional geometry, from certain axiom systems or fro
...more than three dimensions came gradually, primarily on the grounds of the geometric representation of powers: $ a ^ {2} $
12 KB (1,820 words) - 02:24, 21 January 2022
is the sheaf of holomorphic sections of the [[Negative vector bundle|negative vector bundle]] $ L $
for any [[Positive vector bundle|positive vector bundle]] of rank 1 (here $ K _ {X} $
5 KB (719 words) - 22:14, 5 June 2020
* A singular point of vector field of a special type (with all eigenvalues of the linear part being to o
==Node of a vector field==
5 KB (753 words) - 09:03, 12 December 2013
...y by points in the plane and operations on them correspond to the simplest geometric transformations of the plane. It is not possible to "organize" a number s
of the quaternion and its vector part
11 KB (1,563 words) - 14:54, 7 June 2020
is the cell obtained by shifting by a vector $ v $
and any vector $ v $,
5 KB (743 words) - 07:47, 13 May 2022
...) = A$, $x = p ( y )$. So, every vector field $X$ on $M$ is lifted into a vector field $\Gamma X$ on $Y$. The parallel transport on $Y$ along a curve $\gamm
2) As the connection form $T Y \rightarrow V Y$, transforming every vector of $T _ { y } Y$ into its first component with respect to the direct sum de
6 KB (926 words) - 16:46, 1 July 2020
or, in vector notation, $ \dot{w} = f ( w) $,
defined ( "generated" ) by a smooth vector field $ f ( w) $
3 KB (541 words) - 19:39, 5 June 2020
A [[Differential-geometric structure|differential-geometric structure]] on a smooth manifold $ M $,
is fixed, such that the vertex defined by the vector $ e _ {0} $
11 KB (1,486 words) - 17:46, 4 June 2020
be a basis of left-invariant vector fields on $ G $
also constitute a basis of left-invariant vector fields for the Lie subgroup $ H $.
15 KB (2,363 words) - 08:15, 18 August 2022
...he image $\{ p ( t ) : 0 \leq t \leq 1 \}$ of the interval $[0,1]$ under a vector-valued polynomial $p$. The coefficients $a _ { j }$ in that form readily pr
...td> <td valign="top"> G. Farin, "Curves and surfaces for computer aided geometric design" , Acad. Press (1993) (Edition: Third)</td></tr><tr><td valign="to
4 KB (598 words) - 16:55, 1 July 2020
...terior differentiation was developed by Ü. Lumiste [[#References|[a7]]]. A geometric interpretation of the Faddeev–Popov ghost $c$ and anti-ghost $\overline {
It should be noted that all geometric interpretations of the Faddeev–Popov ghosts mentioned above lay aside the
8 KB (1,181 words) - 20:50, 24 January 2021
...athcal{D} ^ { j }$, where $\mathcal{D} ^ { j }$ denotes the [[Vector space|vector space]] of compactly supported differential forms of degree $j$ on $V$ (cf.
See also [[Geometric measure theory|Geometric measure theory]].
5 KB (733 words) - 17:00, 1 July 2020
...lopediaofmath.org/legacyimages/b/b015/b015280/b0152801.png" />-dimensional
vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath
...kes it possible to consider the barycentric coordinates of the points of a
geometric complex with respect to all of its vertices. Barycentric coordinates are us
6 KB (811 words) - 17:11, 7 February 2011
...relations between them, on which the definitions of figures are based and geometric propositions are proved, was recognized in the works of even the ancient ge
...the clearness of the drawing and the physical feasibility of the necessary geometric constructions, and were not strictly logically deduced from the axioms and
25 KB (3,631 words) - 19:39, 5 June 2020
$#C+1 = 125 : ~/encyclopedia/old_files/data/V096/V.0906490 Vector measure
or, more generally, a topological vector space). A vector measure $ F $
10 KB (1,436 words) - 09:13, 6 January 2024
...), and linear, bilinear and quadratic functions (functionals and forms) on vector spaces.
...inear transformation, and a linear, bilinear and multilinear function on a vector space.
9 KB (1,394 words) - 08:15, 9 January 2024
One of the basic geometric concepts. A straight line is usually implicitly defined by the axioms of ge
determine the coordinates of the normal vector of this straight line.
2 KB (335 words) - 08:23, 6 June 2020
...of measures, subject to some regularity conditions. Such translation by a vector space of "arrows" is called an affine structure, and it is taken to be th
in terms of its displacement vector $ l ( x, \theta ) $
10 KB (1,420 words) - 08:23, 6 June 2020
...control system defined on a smooth manifold by a pair of smooth admissible vector fields has the strong accessibility property if this pair belongs to some o
...polysystems and control theory" D.Q. Mayne (ed.) R.W. Brockett (ed.) , ''Geometric Methods in System Theory. Proc. NATO Advanced Study Institute, London, Augu
2 KB (329 words) - 15:33, 20 November 2014
...heory]] of the energy function on $\Omega G$ [[#References|[a6]]]. Certain geometric aspects of Birkhoff stratification may be described in terms of non-commuta
...top"> B. Bojarski, "On the stability of Hilbert problem for holomorphic vector" ''Bull. Acad. Sci. Georgian SSR'' , '''21''' (1958) pp. 391–398</td><
4 KB (630 words) - 17:01, 1 July 2020
A generalization of [[Vector analysis|vector analysis]], a part of [[Tensor calculus|tensor calculus]] studying differen
...re general geometric objects than tensor fields, such as tensor densities, vector-valued differential forms, etc. is considered as a part of tensor analysis.
11 KB (1,607 words) - 08:25, 6 June 2020
...respect to area, volume, etc.). The concept of a derivative is extended to vector-valued point functions in an abstract space (see [[Differentiation of a map
For a geometric and mechanical interpretation of the derivative, the simplest rules of diff
4 KB (596 words) - 11:47, 5 July 2016
A topological space associated with a vector (or sphere) bundle or spherical fibration.
be a vector bundle over a CW-complex $ X $.
6 KB (876 words) - 08:25, 6 June 2020
...the [[Parallel displacement(2)|parallel displacement]] of the plane by the vector $ ( a , b ) $,
and parallel displacement by a vector $ ( a , b ) $.
7 KB (1,100 words) - 03:33, 21 March 2022
are the Fourier coefficients of the vector $ f $
The geometric meaning of Bessel's inequality is that the orthogonal projection of an elem
4 KB (620 words) - 04:37, 9 May 2022
is less than one. If in some norm, compatible with the norm of a vector $ x $,
then the simple-iteration method converges at the rate of a geometric series and the estimate
4 KB (568 words) - 20:21, 10 January 2024
...oncept of a CR-manifold (CR-structure) has been defined having in mind the geometric structure induced on a real hypersurface of $ \mathbf C ^ {n} $,
is involutive, i.e., for any complex vector fields $ U $
6 KB (924 words) - 11:07, 26 March 2023
...e _ { k } D _ { k }$ is $\mu$-ample (cf. also [[Ample vector bundle|Ample vector bundle]]).
...B )$ is a Cartier divisor, and let $d$ be the maximum of the dimensions of geometric fibres of $f$. Express $\lambda / r = p / q$ for relatively prime positive
6 KB (873 words) - 09:30, 3 February 2024
...n+1}$. When $\sigma = 1$, $f$ is called a strict contact transformation. A vector field $X$ on a contact manifold is called a contact (or strict contact) inf
<TR><TD valign="top">[1]</TD> <TD valign="top"> P.K. Rashevskii, "Geometric theory of partial differential equations", Moscow-Leningrad (1947) (In Ru
3 KB (384 words) - 07:34, 13 February 2024
dimensional vector spaces $ T ^ {n} $
...of general type present a [[Differential-geometric structure|differential-geometric structure]] of higher order.
6 KB (908 words) - 19:48, 12 January 2024
be a class of objects in algebraic geometry (varieties, schemes, vector bundles, etc.) on which an equivalence relation $ R $
...continuously depend on parameters; 2) the assignment and study of algebro-geometric structures on the parameter sets. The second part forms the matter of modul
16 KB (2,402 words) - 11:49, 16 December 2019
...ration is to the construction of divergence-free vector fields and related geometric problems.
2 KB (330 words) - 17:00, 1 July 2020
...n the appropriate setting, where non-smooth competitors are allowed (cf. [[Geometric measure theory]]); the setting of Bombieri, De Giorgi and Giusti is that of
2 KB (319 words) - 09:01, 4 November 2013
dimensional vector-columns with these numbers as coordinates is found in the columns of this s
see [[Orthogonal Latin squares|Orthogonal Latin squares]]. In geometric terms, an $ \mathop{\rm OA} ( \lambda n ^ {2} , k, n, 2, \lambda ) $
3 KB (395 words) - 08:04, 6 June 2020
can be regarded as symplectic vector fields on $ M $,
are the values of the vector fields $ X, Y \in \mathfrak g $
7 KB (1,006 words) - 08:24, 6 June 2020
A [[Differential-geometric structure|differential-geometric structure]] on a smooth manifold $ M $,
smooth vector fields, linearly independent at each point $ x $
15 KB (2,221 words) - 10:58, 17 March 2023
...41 : ~/encyclopedia/old_files/data/M063/M.0603250 Measure in a topological vector space
...eral problem encountered in the construction of a measure in a topological vector space is that of extending a [[Pre-measure|pre-measure]] to a measure. Let
7 KB (1,017 words) - 08:00, 6 June 2020
...er curves and Bernstein polynomials is that Bézier curves are a parametric vector-valued representation based on control points, whereas Bernstein polynomial
The usefulness of Bézier curves resides in their many geometric and analytical properties. There are elegant and efficient algorithms for e
5 KB (759 words) - 09:49, 27 March 2023
The position vector $ \mathbf r $
...of a [[Homogeneous space|homogeneous space]]. In an arbitrary differential-geometric homogeneous space $ G / H $
4 KB (613 words) - 22:17, 5 June 2020
...visors, imbeddings of varieties in a projective space, differential forms, vector fields and automorphisms, deformations of varieties and subvarieties, in on
...<TD valign="top">[4]</TD> <TD valign="top"> K. Kodaira, "On a differential-geometric method in the theory of stacks" ''Proc. Nat. Acad. Sci. USA'' , '''39''' (1
4 KB (573 words) - 18:20, 17 April 2017
...of all straight lines $l$ in $\mathbf{R} ^ { 2 }$ as follows: let the unit vector $\theta$ represent the direction of $l$ and let $p$ be its signed distance
...="top">[a3]</td> <td valign="top"> L.A. Santaló, "Integral geometry and geometric probability" , ''Encycl. Math. Appl.'' , Addison-Wesley (1976)</td></tr><t
4 KB (682 words) - 06:33, 10 May 2022
...ath.org/legacyimages/m/m062/m062160/m06216079.png" />. The arithmetic-mean
geometric-mean inequality corresponds to the special case <img align="absmiddle" bord
...2/m062160/m06216098.png" />, extended with zeros if necessary to make up a
vector of length <img align="absmiddle" border="0" src="https://www.encyclopediaof
34 KB (4,631 words) - 18:28, 30 November 2016
of a finite-dimensional [[Vector space|vector space]] $ V $,
...001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E. Artin, "Geometric algebra" , Interscience (1957) {{MR|1529733}} {{MR|0082463}} {{ZBL|0077.021
4 KB (558 words) - 16:18, 6 June 2020
A [[Differential-geometric structure|differential-geometric structure]] on a smooth manifold $ M $;
is fixed, with the vertex determined by the vector $ e _ {0} $
10 KB (1,455 words) - 06:59, 6 March 2024
possesses a global time-like vector field, then $ {} ^ {1} V _ {n} $
...a world-line having non-space-like tangents only. The coupling between the geometric model and the physical data is given by Einstein's field equations (cf. [[E
6 KB (888 words) - 08:08, 6 June 2020
A space with a [[differential-geometric structure]] whose points can be provided with coordinates from some algebra
...a module over the algebra, whose definition can be obtained from that of a vector space over a skew-field by replacing the skew-field by an associative algeb
8 KB (1,172 words) - 16:55, 7 June 2020
The [[Geometric genus|geometric genus]] $ p _ {g} $
...{n} \simeq P ( {\mathcal L} _ {n} ) $ (projectivization of two-dimensional vector bundles over the projective line $ \mathbf P ^ {1} $),
5 KB (707 words) - 16:26, 2 March 2022
...and taking values in the extended real line (or, more generally, a normed vector space), with respect to which general sets in the domain of definition $\ma
...efinitions apply to [[Signed measure|signed measures]] or [[Vector measure|vector measures]] $\mu$: in such cases the assumptions above are required to hold
5 KB (751 words) - 09:47, 16 August 2013
these define a vector in $ \mathbf R ^ {2m } $
and can thus be associated with a column vector $ \mathbf u \in \mathbf R ^ {2m \times 1 } $.
11 KB (1,540 words) - 22:11, 5 June 2020
...invariant way the notions of a derivative and a differential for fields of geometric objects on manifolds, such as vectors, tensors, forms, etc. The basic conce
be a smooth vector field, $ X _ {p} \neq 0 $,
15 KB (2,257 words) - 17:31, 5 June 2020
''of a vector space''
be a [[Vector space|vector space]] over a field $ k $
8 KB (1,152 words) - 08:29, 6 June 2020
...gent vector bundles (cf. also [[Lie algebra|Lie algebra]]; [[Vector bundle|Vector bundle]]; [[Tangent bundle|Tangent bundle]]). For a comprehensive treatment
...r space $\Gamma ( A )$ of smooth global sections of $A$, and a morphism of vector bundles $q _ { A } : A \rightarrow T M$, where $T M$ is the tangent bundle
12 KB (1,916 words) - 16:55, 1 July 2020
...al manifolds in non-commutative algebras. The most remarkable examples are geometric quantization (B. Kostant and J.M. Souriau [[#References|[a11]]], [[#Referen
...non-Lagrangian partial differential equations did fail, until some recent geometric studies on the quantization of partial differential equations [[#References
16 KB (2,356 words) - 19:53, 13 January 2018
...$V \mapsto \operatorname { Hom } _ { k } ( V , k )$ for finite-dimensional vector spaces over a field $k$.
...- 1 - k } ( S ^ { n } \backslash X )$, and this forms the basic intuitive geometric idea behind Spanier–Whitehead duality.
3 KB (432 words) - 16:46, 1 July 2020
...the [[Buffon problem|Buffon problem]] (see also [[Geometric probabilities|Geometric probabilities]]; [[Stochastic geometry|Stochastic geometry]]). The uniform
defined as a vector from the origin to a random point on the surface of the unit sphere, is uni
6 KB (911 words) - 12:00, 22 December 2019
...ngent bundle]]) of $X$. A line bundle $L$ on $X$ (cf. also [[Vector bundle|Vector bundle]]) is said to be nef if the degree of the restriction of $L$ to any
...rem. In the late 19th century, the numbers $h ^ { i } ( L )$ intervened in geometric arguments in much the same way as they intervene today, e.g., [[#References
8 KB (1,253 words) - 15:30, 1 July 2020
...eaf of automorphisms of the trivial vector bundle of rank $ n $. Algebraic vector bundles of rank $ 1 $ are said to be '''line bundles'''; they correspond to
...one can associate the projective bundle $ \mathbf{P}(E) $, just like to a vector space one can associate a [[Projective scheme|projective space]].
14 KB (2,169 words) - 08:12, 14 December 2016
...r bundles, endomorphisms, etc.), with the study of their various algebraic-geometric properties and with compactification techniques for moduli spaces (see [[Mo
...2}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Geometric invariant theory" , Springer (1965) {{MR|0214602}} {{ZBL|0147.39304}} </TD>
5 KB (777 words) - 22:00, 7 July 2014