  Extending and abstracting from Gauss’s treatment of curved surfaces  in Euclidian space, Riemann constructed an infinitesimal geometry of  \(n\)-dimensional manifolds. The coordinate assignments  \(x^{k}(p)\ [k \in \{1, \ldots ,n\}\)] of the points \(p\) in such an \(n\)-dimensional  Riemannian manifold are quite arbitrary, subject only to the  requirement of arbitrary differential coordinate   transformations.[33]  Riemann’s assumption that in an infinitesimal neighbourhood of a  point, Euclidean geometry and hence Pythagoras’s theorem holds, finds its formal expression in Riemann’s equation 