This article defines a scalar function on a manifold, viz a function from the manifold to real numbers. The scalar function may be intrinsic or defined in terms of some other structure/functions
In terms of the shape operator
The mean curvature of a Riemannian manifold at a point on the Riemannian manifold is defined as the trace of the shape operator at that point. In other words, the mean curvature is a function that associates to every point the trace of the shape operator at the point.
The mean curvature function is denoted as .