Variation of the Affine Connection in the kinematics of hypersurfaces

TL;DR

This paper investigates how the affine connection of hypersurfaces in Riemannian manifolds varies under infinitesimal deformations, linking geometric and mechanical quantities, and characterizing motions that preserve affine connections.

Contribution

It derives formulas for the variation of affine connections in hypersurfaces, connecting geometric and mechanical perspectives, and characterizes infinitesimal affine motions.

Findings

Formulas relating connection variation to metric and kinematic quantities

Conditions for motions to be infinitesimally affine

Examples of motions that are infinitesimally affine but not isometric

Abstract

Affine deformations serve as basic examples in the continuum mechanics of deformable 3-dimensional bodies (referred as homogeneous deformations). They preserve parallelism and are often used as an approximation to general deformations. However, when the deformable body is a membrane, a shell or an interface modeled by a surface, the parallelism is defined by the affine connection of this surface. In this work we study the infinitesimally affine time - dependent deformations (motions) after establishing formulas for the variation of the connection, but in the more general context of hypersurfaces of a Riemannian manifold. We prove certain equivalent formulas expressing the variation of the connection in terms of geometrical quantities related to the variation of the metric, as expected, or in terms of mechanical quantities related to the kinematics of the moving continuum. The latter is achieved using an adapted version of polar decomposition theorem, frequently used in continuum mechanics to analyze the motion. Also, we apply our results to special motions like tangential and normal motions. Further, we find necessary and sufficient conditions for this variation to be zero (infinitesimal affine motions), giving insight on the form of the motions and the kind of hypersurfaces that allow such motions. Finally, we give some specific examples of mechanical interest which demonstrate motions that are infinitesimally affine but not infinitesimally isometric.