Area Formula for Surfaces in Non-Holonomic Structures

Abstract: Nonholonomic structures can be considered as a natural generalization of the structures of Riemannian geometry. One of their main features is a specific metric, relative to which one can traverse distances of different orders along different directions ($t$, $t^2$, $t^3$, etc.) in time $t$. Therefore, mappings that are Lipschitz in the classical sense are generally not such in the nonholonomic sense, and vice versa. Nevertheless, in the second half of the 20th century, the theory of sub-Riemannian differentiability was created, which allows one to approximate "complicated" mappings by regular ones. Carnot groups are one of the well-known examples of nonholonomic structures. The talk will discuss the sub-Riemannian analogue of the area formula for surfaces obtained under intrinsically Lipschitz mappings of open sets of Carnot groups. Such groups and their generalizations, Carnot manifolds, arise naturally in both theoretical and applied fields, such as neurobiology, robotics, and astrodynamics.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods