On relation between rational and differential rational invariants of surfaces with respect to the motion groups (original) (raw)

Differential Invariants of Non-degenerate Surfaces

2019

This paper aims to prove that the set {gij(x), Lij(x), i, j = 1, 2} is a complete system of the SM(3)invariants of a nondegenerate surface in R3, where {gij(x)} and {Lij(x)}, i, j = 1, 2 are the sets of all coefficients of the first and second fundamental forms of a surface x in R3. A similar result was obtained for the group M(3).

Commutators and diffeomorphisms of surfaces

Ergodic Theory and Dynamical Systems, 2004

For any compact oriented surface we consider the group of diffeomorphisms of which preserve a given area form. In this paper we show that the vector space of homogeneous quasi-morphisms on this group has infinite dimension. This result is proved by constructing explicitly and for each surface an infinite family of independent homogeneous quasi-morphisms. These constructions use simple arguments related to linking properties of the orbits of the diffeomorphisms.

On differential rational invariants of patches with respect to motion groups

This paper can be considered as a research on Algebraic Differential Geometry. It is about differential rational invariants of subgroups of the Affine group over the constant fields of partial differential fields (characteristic zero). The obtained results can be formulated in terms of Differential Geometry as follows: 1. For any motion group represented by a subgroup H of the Affine group it is shown that systems of generators of a field of H-invariant (not differential) rational functions can be used to construct systems of generators for the differential field of H-invariant differential rational functions of parameterized surface (patch). 2. For some classic motion groups H the generating systems of the field of H-invariant differential functions are presented. 3. For motion groups, including all classical subgroups of the Affine group, separating systems of invariants, uniqueness and existence theorems are offered.

Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry

Turkish Journal of Mathematics

Denote the transformation group of the pseudo-Euclidean space of dimension n and index p by M(n,p). The authors prove that the system {(x (k) (t),x (k) (t))∣k=1,2,⋯,n} of M(n,p)-invariant differential rational functions of a curve with M(n,p)-invariant parametrization x is an algebraically independent generating system of the differential field of all M(n,p)-invariant differential rational functions of that curve. Augmenting this system by the determinant of x (1) (t),x (2) (t),⋯,x (n) (t) gives a generating system for the group of special pseudo-Euclidean transformations. Finally, the authors prove fundamental theorems for sets of relations between the elements of these systems.

A note on the relation between joint and differential invariants

Lobachevskii Journal of Mathematics, 2016

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allow to compute differential invariants from joint invariants.

Invariants and bonnet-type theorem for surfaces in $\r^4

2011

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map of Weingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion.

Invariants of Lines on Surfaces in $\R^4

2010

Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two tangents coincide with a given tangent, then we obtain the normal curvature of this tangent. Asymptotic tangents (curves) are characterized by zero normal curvature. Considering the invariant of the pair of a given tangent and its orthogonal one, we introduce the geodesic torsion of this tangent. We obtain that principal tangents (curves) are characterized by zero geodesic torsion.

Global differential invariants of nondegenerate hypersurfaces

Turkish Journal of Mathematics

Let {gij(x)} n i,j=1 and {Lij(x)} n i,j=1 be the sets of all coefficients of the first and second fundamental forms of a hypersurface x in R n+1. For a connected open subset U ⊂ R n and a C ∞-mapping x : U → R n+1 the hypersurface x is said to be d-nondegenerate, where d ∈ {1, 2,. .. n} , if L dd (x) ̸ = 0 for all u ∈ U. Let M (n) = {F : R n −→ R n | F x = gx + b, g ∈ O(n), b ∈ R n } , where O(n) is the group of all real orthogonal n × n-matrices, and SM (n) = {F ∈ M (n) | g ∈ SO(n)} , where SO(n) = {g ∈ O(n) | det(g) = 1}. In the present paper, it is proved that the set {gij(x), L dj (x), i, j = 1, 2,. .. , n} is a complete system of a SM (n + 1)-invariants of a d-non-degenerate hypersurface in R n+1. A similar result has obtained for the group M (n + 1) .