 Mean curvature

The mean curvature vector of an $n$-dimensional embedded submanifold is the trace $\mathbf{H}={\sum }_{i}A(e_i,e_i)$. For a surface in ${\mathbb{R}}^3$, the scalar mean curvature is half the trace of the shape operator:

Mean curvature is a geometric concept that arises in differential geometry, particularly in the study of surfaces. It measures the average curvature of a surface at a given point and is an important characteristic in the study of minimal surfaces and the geometry of manifolds. For a surface defined in three-dimensional space, the mean curvature \( H \) at a point is given by the average of the principal curvatures \( k_1 \) and \( k_2 \) at that point.