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.

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