### Journals Information

**
Mathematics and Statistics Vol. 7(4), pp. 106 - 115 DOI: 10.13189/ms.2019.070403 Reprint (PDF) (420Kb) **

## Invariants of Surface Indicatrix in a Special Linear Transformation

**A. Artykbaev ^{1}^{,*}, B. M. Sultanov ^{2}**

^{1}Railway Engineering Institute, Uzbekistan

^{2}Department of Geometry and Topology, National University of Uzbekistan, Uzbekistan

**ABSTRACT**

The linear transformation of the plane is considered, whose matrix belongs to the Heisenberg group. The transformation matrix is neither symmetric nor orthogonal. But the determinant is one. The class of the second-order curves is studied, which is obtained from each other by the transformation under consideration. The invariant values of curves of this class are proved. In particular, the conservation of the product of semi-axes of curves in this class is proved, as well as the equality of the areas for the ellipses of the class under consideration. The obtained invariants of the second order curves are applied to curves of the second order, which is the indicatrix of the surface. Conclusion: a theorem is obtained which proves the invariance of the total curvature of a surface in a Euclidean space of the class under consideration is a transformation, which is a deformation.

**KEYWORDS**

Second Order Curve, Heisenberg Group, Curve Indicatrix, Surface, Gauss Curvature, Total Curvature, Principal Curvature

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] A. Artykbaev , B. M. Sultanov , "Invariants of Surface Indicatrix in a Special Linear Transformation," Mathematics and Statistics, Vol. 7, No. 4, pp. 106 - 115, 2019. DOI: 10.13189/ms.2019.070403.

(b). APA Format:

A. Artykbaev , B. M. Sultanov (2019). Invariants of Surface Indicatrix in a Special Linear Transformation. Mathematics and Statistics, 7(4), 106 - 115. DOI: 10.13189/ms.2019.070403.