Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 1063-1072

COVERINGS BY n-CUBES AND THE GAUSS–BONNET THEOREM

Emilija Celakoska and Kostadin Trencevski

SS Cyril and Methodius University, Faculty of Mechanical Engineering, Department of Mathematics and Informatics
P.O. Box 464, 1000 Skopje, Republic of Macedonia; emilija.celakoska 'at' mf.edu.mk

SS Cyril and Methodius University, Faculty of Natural Sciences and Mathematics, Institute of Mathematics
P.O. Box 162, 1000 Skopje, Republic of Macedonia; kostadin.trencevski 'at' gmail.com

Abstract. Instead of standard n-simplexes we deal with n-dimensional cubes with coordinates on real manifolds. The transition matrices for any two cubes having (n – 1)-dimensional common side form a group Hn of orthogonal matrices composed of zeros and exactly one non-zero value 1 or –1 in each row (column). Considering the coverings, a theorem of Gauss–Bonnet type which holds also for odd-dimensional or non-orientable manifolds is proved. We conjecture that a real manifold admits a restriction of the transition matrices to a Lie subgroup G of GL(n,R) of dimension ≥ 1, or the unit element in GL(n,R) if and only if the manifold can be covered by n-cubes such that the transition matrices take values in the intersection of Hn and G or the unit matrix. The complex case uses GL(2n,R) and transition matrices of even dimension. The conjecture is supported with 5 examples. We give methods for calculation of the smallest admissible subgroup of Hn and finally, some conclusions and open questions are presented.

2010 Mathematics Subject Classification: Primary 53C23; Secondary 52B05, 51M20, 57M10.

Key words: Gauss–Bonnet theorem, transition matrices, coverings.

Reference to this article: E. Celakoska and K. Trencevski: Coverings by n-cubes and the Gauss–Bonnet theorem. Ann. Acad. Sci. Fenn. Math. 43 (2018), 1063-1072.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4365

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