The Riemannian geometry of elastica in one and two dimensions is considered. Two examples from engineering science are given. The first is 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 LeviCivita-Christoffel symbol of the Riemannian geometry. A second e...

We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to this theorem.