Differentiating unit orthogonal vectors in a Riemannian space that admits and N-tuply orthogonal system of hypersurfaces.
In a normal coordinate system { y 𝑎 } , the identity 𝜕 2 𝑥 𝑝 /∂y 𝑛 ∂y 𝑚 = ∂ 2 𝑥 𝑝 /∂y 𝑚 ∂y 𝑛 is used to obtain expressions, involving the components of the metric tensor in this coordinate system, for the derivatives of the unit tangent vectors to the parametric curves. An example is given for a normal coordinate system in the De Sitter universe.