On differential invariants of an equivalence group and their geometric meaning

Abstract

Previously, the properties of the Lie group G, which is an equivalence group of the eikonal equation, wave equation, and other differential equations (DEs), have been studied by the author in the two-dimensional case; various applications to mathematical physics and differential geometry have been obtained. This paper presents a study of the three-dimensional analogue of the G group, the ten-parameter G 10 group, which is a subgroup of the main equivalence group of the three-dimensional eikonal equation, acoustics equation, and other DEs. Its differential invariants (DIs) up to the third order and invariant differentiation operators (IDOs) were calculated. The geometric meaning of some DIs of the group G 10 (the scalar curvature R of Riemannian space with the metric dl 2 = n 2(x, y, z)(dx 2 + dy 2 + dz 2), its first and second Beltrami differential parameters Δ1 u and Δ2 u, and other quantities) and IDOs was found. An expression for R was derived in terms of other DIs of the group G 10. To obtain this expression, and DIs and IDOs of the group G 10, we use the geometric analogy with the two-dimensional case and differential and Riemannian geometry.