  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.
