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Graph Surfaces in Non-Holonomic Geometry

Name
Maria
Surname
Karmanova
Scientific organization
Sobolev Institute of Mathematics
Doctor of Sciences
Position
Scientific discipline
Mathematics & Mechanics
Topic
Graph Surfaces in Non-Holonomic Geometry
Abstract
We investigate graph mappings on non-holonomic structres, deduce their differential properties and prove area formulas
Keywords
sub-Lorentzian structure, sub-Riemannian structure, polynomial sub-Riemannian differentiability, area formula
Summary

Five-dimensional sub-Riemannian and sub-Lorentzian structures are our main topic of research. We consider classes of graph mappings and corresponding graph surfaces on them. It is easy to see that these mappings are not Lipschitz and sub-Riemannian sense and therefore the non-holonomic differentiability theory founded by S. Vodopyanov is not applicable. Nevertheless, our new approach enables us to approximate graph mappings by some polynomials; in other words, they possess the property of polynomial sub-Riemannian differentiability, where the "differential" depends polynomially on coordinates of one point with respect to another. The first result is

Theorem 1. Graph mappings constructed by classes of intrinsically Lipschitz mappings are polynomially sub-Riemannian differentiable.

The next result is the area formula for corresponding graph surfaces. Emphasize that it depends on values of sub-Riemannian differential of the initial mapping (but not the polynomial differential).

Theorem 2. If $$\varphi$$ is a Lipschitz mapping of two one-dimensional Heisenberg groups $$\mathbb H$$ and $$\widetilde{\mathbb H}$$, $$\varphi_{\Gamma}: x\mapsto \exp\Bigl(\sum\limits_{j=1}^3\varphi_i(x)X_i\Bigr)(x)$$ is a graph mapping, then the intrinsic sub-Riemanian measure of $$\varphi_{\Gamma}(\mathbb H)$$ equals

$$\int\limits_{\Omega}\sqrt{1+(X_1\varphi_4)^2(x)+(X_2\varphi_4)^2(x)}\sqrt{1+(X_3\varphi_5)^2(x)}\,d\mathcal H^4(x).$$

The intrinsic sub-Lorentzian measure equals

$$\int\limits_{\Omega}\sqrt{1-(X_1\varphi_4)^2(v)}\sqrt{1-(X_2\varphi_4)^2(v)}\sqrt{1-(X_3\varphi_5)^2(v)}\,d\mathcal H^{4}(v).$$

Here the sub-Lorentzian measure in the image is constructed with respect to "distance"

$$(\widetilde{d}^{SL_2}_{\infty})^2(v,y)= \max\bigl\{\max\{y_1^2, y_2^2\}-y_4^2, {sgn}(y_3^2-y_5^2)\sqrt{|y_3^2-y_5^2|}\bigr\}.$$

Recently, it also turned out that results similar to Theorems 1 and 2 hold for large classes of graph mappings of Carnot groups, and general area formulas are valid for the corresponding graph surfaces.

Main part of these results are pblished in [1-3].

[1] Karmanova M. B. Area Formula for Graph Surfaces on Five-Dimensional Sub-Lorentzian Structures // Doklady Mathematics, 2016. V. 93, No. 2.

[2] Karmanova M. B. Variations of Non-Holonomic-Valued Mappings and Their Applications to Maximal Surface Theory // Doklady Mathematics, 2016. V. 93, No. 3.

[3] Karmanova M. B. Codimension Two Graph Surfaces over Three-Dimensional Carnot-Caratheodory Spaces // Dokl. AN, 2016. V. 468, No. 6.