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
*H*_{n} 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
*H*_{n} and *G* or the unit matrix.
The complex case uses *GL*(2*n*,**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 *H*_{n} 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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