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V. M. Khatsymovsky - One of the best experts on this subject based on the ideXlab platform.
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affine Connection Form of regge calculus
Modern Physics Letters A, 2016Co-Authors: V. M. KhatsymovskyAbstract:Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general Connection as independent variables represents the Einstein–Hilbert action. The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of edge lengths and the world coordinates of the vertices. The conjugate variables are the general nondegenerate matrices on the three-simplices which play the role of a general discrete Connection. Our previous result on some representation of the Regge calculus action in terms of the local Euclidean (Minkowsky) frame vectors and orthogonal Connection matrices as independent variables is somewhat modified for the considered case of the general linear group GL(4, R) of the Connection matrices. As a result, we have some action invariant w.r.t. arbitrary change of coordinates of the vertices (and related GL(4, R) transFormations in the four-simplices). Excluding GL(4, R) Connection from this action via the equations of motion we have exactly the Regge action for the considered spacetime.
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affine Connection Form of regge calculus
arXiv: General Relativity and Quantum Cosmology, 2015Co-Authors: V. M. KhatsymovskyAbstract:Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general Connection as independent variables represents the Einstein-Hilbert action. The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of edge lengths and the world coordinates of the vertices. The conjugate variables are the general nondegenerate matrices on the 3-simplices which play a role of a general discrete Connection. Our previous result on some representation of the Regge calculus action in terms of the local Euclidean (Minkowsky) frame vectors and orthogonal Connection matrices as independent variables is somewhat modified for the considered case of the general linear group GL(4,R) of the Connection matrices. As a result, we have some action invariant w. r. t. arbitrary change of coordinates of the vertices (and related GL(4,R) transFormations in the 4-simplices). Excluding GL(4,R) Connection from this action via the equations of motion we have exactly the Regge action for the considered spacetime.
Villabon A. - One of the best experts on this subject based on the ideXlab platform.
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Flat Affine Manifolds And Their TransFormations
'Springer Science and Business Media LLC', 2020Co-Authors: Medina A., Saldarriaga O., Villabon A.Abstract:We give a characterization of flat affine Connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the Connection. From the infinitesimal point of view, this representation is determined by the 1-Connection Form and the fundamental Form of the bundle of linear frames of the manifold. We show that the group of affine transFormations of a real flat affine $n$-dimensional manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is greater than $n$. Moreover, when the dimension of the group of affine transFormations is $n$, this orbit has discrete isotropy. For any given Lie subgroup $H$ of affine transFormations of the manifold, we show the existence of an associative envelope of the Lie algebra of $H$, relative to the Connection. The case when $M$ is a Lie group and $H$ acts on $G$ by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transFormations admits a flat affine bi-invariant structure. The paper is illustrated with several examples.Comment: More references have been added. In particular, the reference to Jack Vey's thesis. We have corrected some typos and included some other changes. arXiv admin note: text overlap with arXiv:1707.0703
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Flat affine transFormations and their transFormations
Springer Verlag, 2020Co-Authors: Medina Alberto, Saldarriaga O., Villabon A.Abstract:More references have been added. In particular, the reference to Jack Vey's thesis. We have corrected some typos and included some other changes. arXiv admin note: text overlap with arXiv:1707.07030International audienceWe give a characterization of flat affine Connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the Connection. From the infinitesimal point of view, this representation is determined by the 1-Connection Form and the fundamental Form of the bundle of linear frames of the manifold. We show that the group of affine transFormations of a real flat affine n-dimensional manifold, acts on Rn leaving an open orbit when its dimension is greater than n. Moreover, when the dimension of the group of affine transFormations is n, this orbit has discrete isotropy. For any given Lie subgroup H of affine transFormations of the manifold, we show the existence of an associative envelope of the Lie algebra of H, relative to the Connection. The case when M is a Lie group and H acts on G by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transFormations admits a flat affine bi-invariant structure. The paper is illustrated with several examples
Hiroshi Yamashita - One of the best experts on this subject based on the ideXlab platform.
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electric wire electric wire Connection method and wire harness
2004Co-Authors: Akinori Kurimoto, Kouji Fujita, Hiroshi YamashitaAbstract:A first marking and a second marking, indicating a Connection Form, are provided on a covering sheath at a Connection position of a first wire serving as a main wire. Therefore, an operation is carried out at a region between the first marking and the second marking of the first wire according to an operation Form indicated by the second marking. Therefore, the predetermined Form of operation can be carried out at the predetermined position of the first wire.
Joonsik Park - One of the best experts on this subject based on the ideXlab platform.
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torsion tensor Forms on induced bundles
Journal of the Chungcheong Mathematical Society, 2013Co-Authors: Hyun Woong Kim, Joonsik Park, Yongsoo PyoAbstract:Abstract. Let ` be a map of a manifold M into another manifold N , L ( N ) the bundle of all linear frames over N , and ` i 1 ( L ( N ))the bundle over M which is induced from ` and L ( N ). Then, weconstruct a structure equation for the torsion Form in ` i 1 ( L ( N ))which is induced from a torsion Form in L ( N ). 1. IntroductionLet ` : M ! N be a C 1 i map between smooth manifolds M and N , L ( N ) the bundle of all linear frames over N , and ` i 1 ( L ( N )) =: Q the bundle which is induced from ` and L ( N ). Then, the bundlehomomorphism `~ : Q ! L ( N ) between two principal flbre bundles ` i 1 ( TN ) =: Q and L ( N ) is deflned by `~ ( u;x ) = u (( u;x ) 2 Q;x 2M;… ( u ) = x;u 2 L ( N )) ([1]).Let i be an arbitrarily given Connection in L ( N ). Let ! and µ be theConnection Form and the canonical Form in L ( N ) which are deflned fromi ([1, 3]). Now, putting `~ ⁄ ! =: ~ ! and `~ ⁄ µ =: µ~ , we obtain the fact that !~ and µ~ are a Connection Form and the canonical Form for ~ ! in `
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yang mills Connection in the orthonormal frame bundle over a riemannian symmetric space
East Asian mathematical journal, 1998Co-Authors: Puyoung Kim, Joonsik ParkAbstract:The main result is that the Connection Form in the orthonormal frame bundle by the Levi-Civita Connection over a Riemannian symmetric space is a Yang-Mills Connection
Saldarriaga O. - One of the best experts on this subject based on the ideXlab platform.
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Flat Affine Manifolds And Their TransFormations
'Springer Science and Business Media LLC', 2020Co-Authors: Medina A., Saldarriaga O., Villabon A.Abstract:We give a characterization of flat affine Connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the Connection. From the infinitesimal point of view, this representation is determined by the 1-Connection Form and the fundamental Form of the bundle of linear frames of the manifold. We show that the group of affine transFormations of a real flat affine $n$-dimensional manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is greater than $n$. Moreover, when the dimension of the group of affine transFormations is $n$, this orbit has discrete isotropy. For any given Lie subgroup $H$ of affine transFormations of the manifold, we show the existence of an associative envelope of the Lie algebra of $H$, relative to the Connection. The case when $M$ is a Lie group and $H$ acts on $G$ by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transFormations admits a flat affine bi-invariant structure. The paper is illustrated with several examples.Comment: More references have been added. In particular, the reference to Jack Vey's thesis. We have corrected some typos and included some other changes. arXiv admin note: text overlap with arXiv:1707.0703
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Flat affine transFormations and their transFormations
Springer Verlag, 2020Co-Authors: Medina Alberto, Saldarriaga O., Villabon A.Abstract:More references have been added. In particular, the reference to Jack Vey's thesis. We have corrected some typos and included some other changes. arXiv admin note: text overlap with arXiv:1707.07030International audienceWe give a characterization of flat affine Connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the Connection. From the infinitesimal point of view, this representation is determined by the 1-Connection Form and the fundamental Form of the bundle of linear frames of the manifold. We show that the group of affine transFormations of a real flat affine n-dimensional manifold, acts on Rn leaving an open orbit when its dimension is greater than n. Moreover, when the dimension of the group of affine transFormations is n, this orbit has discrete isotropy. For any given Lie subgroup H of affine transFormations of the manifold, we show the existence of an associative envelope of the Lie algebra of H, relative to the Connection. The case when M is a Lie group and H acts on G by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transFormations admits a flat affine bi-invariant structure. The paper is illustrated with several examples