Visual Images in Some Other Fields of Geometry and in Its Applications

Abstract

Suppose in ℝ³ there are several adjoining but not mixing physical media, for instance, in a large vessel there are several immiscible fluids. Suppose, the whole system is in equilibrium. Since the media are immiscible, the boundaries (interfaces) between them are determined. These interfaces J can be tought of (in the first approximation) as two-dimensional piece-wise smooth surfaces separating the adjoining media. We consider for simplicity the case of two media which we denote by A₁ and A₂. Let the pressures in the media be respectively equal to p₁ and p₂. The equilibrium condition for the media proves to impose a strong restriction upon the geometry of their interface. To formulate this restriction, we require an important concept of local differential geometry, namely, the concept of mean curvature of the surface.