Abstract
The paper deals with the canonical deformed group of diffeomorphisms with a given length scale which describes the motion of single scales in a Riemannian space. This allows to measure the lengths of arbitrary curves implementing the length principle which was laid by B. Riemann at the foundation of geometry. We present a method of univocal extension of this group to a group which contains gauge rotations of vectors (the group of parallel translations) whose transformations leave unchanged the lengths of vectors and the corners between vectors. Thereby Klein’s Erlangen Program—the principle of equality—is implemented for Riemannian spaces.
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The results of the paper were presented at the International Conference “Geometry in Odessa-2013”.
Original Russian Text © S.E. Samokhvalov, E.B. Balakireva, 2015, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2015, No. 9, pp. 31–45.
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Samokhvalov, S.E., Balakireva, E.B. Group-theoretic matching of the length and the equality principles in geometry. Russ Math. 59, 26–37 (2015). https://doi.org/10.3103/S1066369X15090042
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DOI: https://doi.org/10.3103/S1066369X15090042