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.
Highlights
The invariant is one of the main concepts in mathematics
When we consider the concept of the invariant, we understand a number or an algebraic expression associated with the figures, which keeps their values when they perform a certain transformation on that figure
It is known that if all points of a surface in Euclidean space are composed by elliptic points, this surface is said to be convex surface
Summary
The invariant is one of the main concepts in mathematics. When we consider the concept of the invariant, we understand a number or an algebraic expression associated with the figures, which keeps their values when they perform a certain transformation on that figure. We refer to the invariants of motion groups. They help to consider the important problems. In addition to these invariants, the invariants are involved when determining parameters of a surface, a line, that’s getting two parameters instead of one parameter. We determine constants in the following linear transformation in Euclidean space: x′= x + a y′. Let be given a surface F in Euclidean space R3 and Mathematics and Statistics 7(4): 106-115, 2019. Our objective is to learn a curvature of the surface when a linear transformation holds for the indicatrix equation of the surface (2)
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