Riemannian and non-Riemannian geometries of filament rods and elastic walls

Abstract

The Riemannian geometry of elastica in one and two dimensions is considered. An example is given of the deflexion or Frenet curvature of the elastic filament rod where the Riemannian curvature vanishes, since the curve is one dimensional. However the Frenet curvature scalar appears on the Levi-Civita-Christoffel symbol of the Riemannian geometry. A second example is the bending of a planar two-dimensional wall where only the horizontal lines of the planar wall are bent, or a plastic deformation without cracks or fractures. In this case since the vertical lines are approximately not bent, and remain vertical while the horizontal lines are slightly bent in the limit of small deformations. This implies that the Gaussian curvature vanishes. However the Riemann curvature does not vanish and again may be expressed in terms of the elastic properties of the planar wall. Non-Riemannian geometry in its own is applied to rods with nonhomogeneous cross-sections and computation of Cartan torsion in terms of the twist and the Riemann tensor are computed from the twist of the rod. In the homogeneous case the Riemann tensor maybe also obtained from Kirchhoff equations by comparing them to the non-geodesic equations and computing the affine connection. The external force acting on the rod represents the term responsible for the geodesic deviation in the space of the total Frenet curvature and twist. The Riemann tensor appears in terms of the mechanical torsional moment.