X-Nico

Curvature of Riemannian manifolds

The curvature of n-dimensional Riemannian manifold is given by an antisymmetric n×n matrix $\backslash Omega^\{\}\; \{\}=\backslash Omega^i\; \{\backslash \; j\}$ of 2-forms (or equivalently a 2-form with values in $so(n)$, the Lie algebra of the orthogonal group $O(n)$, which is the structure group of the tangent bundle of a Riemannian manifold).

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For a more elementary discussion see the article on curvature which discusses the curvature of curves and surfaces in 2 and 3 dimensions, as well as Differential geometry of surfaces.

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