Orthogonality on Riemannian Manifolds

Given a Riemannian manifold \((M,g)\). For a given point \(p\in M\) we are able to compare vectors on \(T_pM\). We call them orthogonal if

\begin{equation*} g_p(v,w)=0. \end{equation*}

A system of vector on \(T_pM\) is called orthogonal if they are pairwise orthogonal. It is orthonormal if for every vector \(v\) in the system holds

\begin{equation*} g_p(v,v)=1. \end{equation*}