Another Quadratic Form on the Tangent Space at a Point on a Smooth Surface Embedded in ℝ��

Abstract

The second fundamental form of an embedded surface evaluated at a tangent vector is the component of the shape operator of that vector in the same direction. However, the orthogonal component of the shape operator seems neglected, and this study will be an insight into this idea. This component of the shape operator gives rise to a new quadratic form on the tangent space which will be called as the fourth fundamental form, and it measures the geodesic torsion of the surface. Unlike the first two fundamental forms which would together describe all the geometric properties of a considered surface, the fourth fundamental form will be victimized to certain disadvantages. But once restricted to constant mean curvature or constant Gaussian curvature surfaces, it successfully reconstructs all geometric properties. Here, we will discuss the special case of minimal surfaces in detail.