On construction of the second basic shape of the surface of almost limited curvature of positive curvature
- DOI
- 10.5922/0321-4796-2026-57-1-4
- Pages
- 52-63
Abstract
This article proposes a method for constructing the second fundamental form of a surface of positive curvature. It is well known that constructing the second fundamental form of a surface requires at least a twice continuously differentiable parametrization. This paper uses a probabilistic approach to this problem, introducing the concept of the second Itô derivative along the direction of Brownian motion. In more detail, we define the second derivative “without the second derivative”. For this, we use the Itô formula, restricted to special random processes. Using the proposed construction, we can write Weingarten's equations for surfaces of almost bounded curvature (the paper has been accepted for publication in the Ufa Mathematical Journal) and determine the mean curvature almost surely over the entire surface pointwise. To some extent, this paper can be viewed as a generalization of I. Ya. Bakelman’s work “Differential Geometry of Smooth Irregular Surfaces” (see [5]). In the author’s opinion, further reduction of surface smoothness is difficult to achieve using analytical methods (the technique developed in [5]), and new research methods are needed. This paper proposes one such method: a stochastic framework for studying low-smoothness surfaces. Previously, a slightly different approach was used, based on transition functions and transition densities of random processes; see, for example, [6] (which also contains references to earlier works).
Reference
1. Watanabe, S., Ikeda, N.: Stochastic differential equations and diffusion processes. Moscow (1986).
2. Anulova, S. V., Veretennikov, A.Yu., Krylov, N. V., Liptser, R. Sh., Shiryaev, A. N.: Stochastic calculus. Itogi Nauki i Tekhn. Series Sovrem. probl. mat. Fundam. directions, 45, 5—253 (1989).
3. Dynkin, E. B.: Markov processes. Moscow (1963).
4. Rashevsky, P. K.: Course of differential geometry. Moscow, Leningrad (1939).
5. Bakelman, I. Y.: Differential geometry of smooth irregular surfaces. Russian Mathematical Surveys, 11:2(68), 67—124 (1956).
6. Klimentov, D. S.: Stochastic analogue of fundamental theorem of surface theory for surfaces of bounded distortion and positive curvature. Ufa Math. J., 11:4, 40—48 (2019).
7. Aleksandrov, A. D., Zalgaller, V. A.: Two-dimensional manifolds of bounded curvature (Fundamentals of intrinsic geometry of surfaces). Proceedings of the Steklov Mathematical Institute of the USSR, 63, Moscow, Leningrad, 3—262 (1962).