The geometric notion meaning that at some point two curves (or a curve and a surface) hav
See also [[Tangent vector|Tangent vector]].
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$#C+1 = 32 : ~/encyclopedia/old_files/data/V096/V.0906340 Vector,
''geometric''
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that associates with a differentiable vector field $ X $
and a differentiable geometric object $ Q $
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''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
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...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
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$#C+1 = 17 : ~/encyclopedia/old_files/data/D030/D.0300210 Darboux vector
The vector $ \pmb\delta $
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...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>
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...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} $
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Let a geometric object $ \Omega $(
cf. [[Geometric objects, theory of|Geometric objects, theory of]]) be given in a differentiable manifold $ X _ {n} $,
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...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
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Adjoint norms (cf. [[Norm|Norm]]) in certain vector spaces dual to each other.
vector $ \alpha $,
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...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
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...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.
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is a differential-geometric structure on $ M $
...ant differentiation|covariant differentiation]], which associates with two vector fields $ X , Y $
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...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
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...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.
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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.
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...ping|Transversal mapping]]); a concept in linear algebra, differential and geometric topology.
a) Two vector subspaces $ A, B $
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...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]]].
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''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
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A field of geometric quantities on a homogeneous space $ M = G / H $
is a [[Vector bundle|vector bundle]]. In this case, $ H $
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in a finite-dimensional vector space $ V $
and any non-zero vector $ v \in V $
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...[[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
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...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
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...-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) $
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...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
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$#C+1 = 46 : ~/encyclopedia/old_files/data/K055/K.0505420 Killing vector,
''more precisely, Killing vector field or infinitesimal motion''
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...^*$ 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
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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} $
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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 $,
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of a space (as a rule, a vector space) $ X $
into a vector space $ Y $
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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
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of the real vector space $ \mathbf R ^ {n} $.
of the vector $ x ( p) \in \mathbf R ^ {n} $
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...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 $-
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...Delta : \dim F = i\}\vert$, 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
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The vector bundle consisting of tangent vectors to the ambient manifold that are norma
by the vector bundle $ T _ {Y} $.
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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
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...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
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...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
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2) the scalar product is distributive with respect to vector addition:
of a vector $ \mathbf a $
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...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
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...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} $
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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} $
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* 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==
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...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
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is the cell obtained by shifting by a vector $ v $
and any vector $ v $,
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...) = A$, $x = p ( y )$. 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
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or, in vector notation, $ \dot{w} = f ( w) $,
defined ( "generated" ) by a smooth vector field $ f ( w) $
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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