<s>[INST] The curvature tensor \(R\) for the surface \(\Sigma\) in \(\mathbb{R}^{3}\) is described by the equation:
\[R(X,Y)Z= \left(g(Y,Z)A(X)-g(X,Z)A(Y)\right)N\]
where \(X\), \(Y\), and \(Z\) are vector fields in \(T\Sigma\); \(g\) is the unit normal vector field to \(\Sigma\); \(A\) is a certain second order symmetric tensor field; \(N\) is the unit normal vector field to \(\Sigma\); \(g\) is the first fundamental form on \(\Sigma\).
Question 1. Is the curvature tensor skew symmetry in the first two indices, \(X\) and \(Y\)?[/INST]