Abstract
In this paper, we trace the development of concepts of differential geometry such as a first order differential calculus, an algebra of differential forms and a connection from Euclidean geometry to noncommutative geometry. We begin with basic structures of differential geometry in n-dimensional Euclidean space such as vector field, differential form, connection and then, having explained a general idea of non-commutative geometry, we show how these notions can be developed in the assumption that the algebra of smooth functions on Euclidean space is replaced by its non-commutative analog and the differential graded algebra of differential forms is replaced by a q-differential graded algebra, where q is a primitive Nth root of unity and a differential d of this algebra satisfies the equation d N = 0.
To the 70th birthday of professor Wolfgang Sprössig
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Acknowledgements
The authors gratefully acknowledge that this paper is written on the basis of lectures delivered to the students of School of Education, Culture and Communication (UKK) of University of Mälardalen and financially supported within the framework of the project NPHE-2017/10119 (NordPlus Higher Education 2017). The authors express their sincere appreciation to the colleagues from the Division of Applied Mathematics of University of Mälardalen for their care and hospitality. The authors also wish to thank Estonian Ministry of Education for financial support within the framework of the institutional funding IUT20-57.
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Abramov, V., Liivapuu, O. (2019). Connections in Euclidean and Non-commutative Geometry. In: Bernstein, S. (eds) Topics in Clifford Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23854-4_14
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DOI: https://doi.org/10.1007/978-3-030-23854-4_14
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