There are three types of so-called fundamental forms. The most important are the first and second (since the third can be expressed in terms of these). The fundamental forms are extremely important and useful in determining the metric properties of a surface, such as line element, area element, normal curvature, Gaussian curvature, and mean curvature. Let M be a regular surface with v_p, w_p points in the tangent space M_p of M. Then the first fundamental form is the inner product of tangent vectors, I(v_p, w_p) = v_p·w_p.

arc length | area element | first fundamental form | Gaussian curvature | geodesic | Kähler manifold | line element | line of curvature | mean curvature | normal curvature | Riemannian metric | scale factor | second fundamental form | surface area | third fundamental form | Weingarten equations