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**Laurent** This is just the decomposition of $\displaystyle \textbf{X}_{ij}$ in a basis. Indeed, $\displaystyle (\textbf{X}_1,\textbf{X}_2)$ is a basis of the tangent plane, while $\displaystyle \textbf{U}$ is orthogonal to that plane, so that $\displaystyle (\textbf{X}_1,\textbf{X}_2,\textbf{U})$ is a basis of $\displaystyle \mathbb{R}^3$ (which depends on $\displaystyle (u_1,u_2)$). Then for instance $\displaystyle \Gamma_{ij}^1$ is the $\displaystyle \textbf{X}_1$-component of $\displaystyle \textbf{X}_{i j}$.