In the book Mathematical Masterpieces, chapter 3, section 1, the authors have talked about the curvature and the ideas around it.
They wrote
If the curvature is given in $\dfrac{1}{2}n(n-1)$ surface directions at every point, then the metric relations of the manifold may be determined.
Could you please explain for me what did they mean? I can't understand what is $\dfrac{1}{2}n(n-1)$ surface directions at every point
Thanks.