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Suppose we are given a Riemannian Metric $g$ on $\mathbb{R}^n$. Let $v_1\in\mathbb{R}^n$ with $g(v_1,v_1)=1$. Is it possible to find a base $\{v_1,v_2,...,v_n\}$ of $\mathbb{R}^n$ in such a way that $g(v_i,v_j)=\delta_{ij}$, $\forall\ i,j=1,...,n$ ?

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Yes, the Gram–Schmidt procedure works in any Euclidean space. (From what you write, I assume that by Riemannian metric you mean just a constant inner product on the vector space $\mathbb{R}^n$, not an inner product that varies from point to point on the manifold $\mathbb{R}^n$.)

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