Group-theoretic matching of the length principle and equality principle in geometry

The paper deals with canonical deformed group of diffeomorphisms with a given length scale which describes the motion of the single scales in the Riemannian space. This allows to measure lengths of arbitrary curves, implementing length principle which is laid by B. Riemann in the basis of the geometry. We present the way of univocal extension of the given group to a group, which contains gauge rotations of vectors (parallel transports group) whose transformations leave unchanged the lengths of the vectors and corners between them. Thereby Klein's Erlanger Program – the principle of equality – is implemented for Riemannian spaces.