Some Curvature Functions and their Arithmetic Properties

We define curvatures of a non-singular parallelepiped in a real vector space equipped with a non-degenerate quadratic form of type (p, q), and relate them to curvature functions of parametrized curves in a pseudo-Riemannian manifold. These functions are patterned over the classical Frenet-Serret curvature functions as extended by Blaschke, [1]. We consider arithmetic properties of these and other curvature functions. For example, we prove that if M is a non-singular real algebraic variety defined over a subfield 𝔽 of ℝ, equipped with a pseudo-Riemannian metric also defined over 𝔽, then the Riemannian sectional curvature at a point and the section defined over 𝔽 lies in 𝔽.