  In his famous inaugural lecture at Göttingen, entitled  Über die Hypothesen, welche der Geometrie zu Grunde liegen (About the hypotheses which lie at the  basis of geometry), Riemann (1854) examined how metric relations can be determined on a continuous manifold; that is, what specific  form \(f : M \times M \rightarrow \mathbb{R}\) should  have. Consider the coordinates \(x^{i}(p)\)  and \(x^{i} (p) + dx^{i} (p)\) of two neighboring  points \(p, q \in M\). The measure of the  distance \(ds = f(p,q)\) must  be some function \(F_{p}\) at \(p\) of the  differential increments \(dx^{i}(p)\); that is, 