Geometric structure of the Beltrami–Mitchell equations

K. N. Pestov^ab, M. A. Guzev^c, O. N. Lyubimova<sup>bd

a</sup> Vladivostok Branch of Russian Customs Academy
^b Khabarovsk Division of the Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences
^c Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, Vladivostok
^d Far Eastern Federal University, Vladivostok

Abstract: The paper demonstrates that the classical equations of stress in elasticity theory, known as the Beltrami–Mitchell equations, can be expressed as components of the Ricci tensor when considering linear deformations. This is provided that the conditions of equilibrium, Hooke's law, and the assumption of a Euclidean space for the material continuum are satisfied. It is proven that the divergence of the Ricci tensor is zero in this case. A relationship between the Ricci tensor and the strain tensor is derived, which is significant for describing the structural and deformational characteristics of the mechanical behavior of materials based on non-Euclidean geometries. It is demonstrated that in the elastic case, the Ricci tensor equals the Einstein tensor.

Keywords: Ricci tensor, Einstein tensor, Beltrami-Mitchell equations, compatibility conditions of Saint-Venant.

UDC: 531.36

Received: 01.02.2025
Revised: 17.06.2025
Accepted: 14.04.2025

DOI: 10.37972/chgpu.2025.63.1.009