Minkowski parallel-translation and their coordinatization

1977 ◽  
Vol 9 (1-2) ◽  
pp. 19-27 ◽  
Author(s):  
R. Artzy
Keyword(s):  
Parallel Translation

Parallel Translation in Vector Bundles with Abelian Structure Group and the Gauss-Bonnet Formula

1973 ◽  
Vol 25 (4) ◽  
pp. 765-771
Author(s):  
Hansklaus Rummler
Keyword(s):  
Riemannian Manifold ◽  
Simple Proof ◽  
General Result ◽  
Tangent Bundle ◽  
Vector Bundles ◽  
Structure Group ◽  
Parallel Translation ◽  
Result Theorem

Most proofs for the classical Gauss-Bonnet formula use special coordinates, or other non-trivial preparations. Here, a simple proof is given, based on the fact that the structure group SO(2) of the tangent bundle of an oriented 2-dimensional Riemannian manifold is abelian. Since only this hypothesis is used, we prove a slightly more general result (Theorem 1).

scholarly journals Some New Characterizations Of Parallel Translation Surface According To Bishop Frame With Timelike M1 In Minkowski 3-Space

2018 ◽  
Vol 22 ◽  
pp. 01030
Author(s):  
Gülden Altay Suroǧlu
Keyword(s):  
Translation Surface ◽  
Parallel Translation ◽  
Translation Surfaces ◽  
Bishop Frame

In this paper we consider parallel translation surfaces, which are generated by spacelike curves, according to Bishop frame with timelike M1 in Minkowski 3- space. Then, we obtain some characterizations of these surface.

REPRESENTATION OF ITERATIVE PATTERNS

1989 ◽  
Vol 03 (03n04) ◽  
pp. 479-495
Author(s):  
HIROMOTO ASO ◽  
NAMIO HONDA
Keyword(s):  
Compact Representation ◽  
Parallel Translation ◽  
Translation Group ◽  
Colored Tile

Iterative patterns can be described by the component pattern of iteration and their way of connection for iteration. It is shown that the shape of the component pattern can be a rectangle for any iterative pattern, which is called a tile. The tile is related to the basis of the parallel translation group of the iterative pattern. The component pattern itself is identified by a "tessera" which is defined by a non-degenerate colored tile. The tessera is a compact representation of iterative patterns. Some natures of tesseras and tiles are also discussed. The number of the kinds of distinct iterative patterns for each way of iteration is evaluated.

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