affine connection in a sentence
1) A more mathematically motivated account of affine connections .
affine collocations
2) An affine connection defines a notion of development of curves.
3) On any manifold of positive dimension there are infinitely many affine connections .
affine connection example sentences
4) Affine connections can be defined within Cartan's general framework.5) Affine connections from the point of view of Riemannian geometry.
6) Cartan connections generalize affine connections in two ways.
7) Let "M" be a manifold with an affine connection ∇.
8) Curvature and torsion are the main invariants of an affine connection .
9) The main invariants of an affine connection are its torsion and its curvature.
10) Cartan's treatment of affine connections as motivated by the study of relativity theory.
11) The second motivation for affine connections comes from the notion of a covariant derivative of vector fields.
12) However, this approach does not explain the geometry behind affine connections nor how they acquired their name.
13) The notion of an affine connection was introduced to remedy this problem by "connecting" nearby tangent spaces.
14) The affine connection may be used for defining curvature and torsion, like is usual in Riemannian geometry.
15) The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection .
example sentences with affine connection
16) These last two points are quite hard to make precise, so affine connections are more often defined infinitesimally.17) Two articles by Lumiste, giving precise conditions on parallel transport maps in order that they define affine connections .
18) In the modern approach, this is closely related to the definition of affine connections on the frame bundle.
19) His discovery prompted Weyl and Cartan to introduce various notions of connection, including in particular that of affine connection .
20) For the mentioned __FORMULA__-families the affine connection is called the __FORMULA__-connection and can also be expressed in more ways.
21) Because an affine connection is not unique, it is an additional piece of data that must be specified on the manifold.
22) This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle.
23) In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection .
24) Finally, Berger's paper lists possible holonomy groups of manifolds with only a torsion-free affine connection ; this is discussed below.
25) Given any affine connection ∇, there is a unique torsion-free connection ∇' with the same family of affinely parametrized geodesics.
26) The first three properties state that __FORMULA__ is an affine connection compatible with the metric, sometimes also called a "hermitian" or metric connection.
27) Given a system of parametrized geodesics, one can specify a class of affine connections having those geodesics, but differing by their torsions.
28) An affine connection , which is not uniquely defined, but generalizes in a more complete manner the features of ordinary directional differentiation.
How to use affine connection in a sentence
29) This tensor measures curvature by use of an affine connection by considering the effect of parallel transporting a vector between two points along two curves.30) In General Relativity, one usually refers to "the" covariant derivative, which is the one associated with Levi-Civita affine connection .
31) By definition, an affine connection is a bilinear map __FORMULA__, where __FORMULA__ is a space of all vector fields on the spacetime.
32) An affine connection is a rule which describes how to legitimately move a vector along a curve on the manifold without changing its direction.
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