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