Differential geometry of surfaces plays a key role in the development of the Laplace-Beltrami elliptic system for mesh enhancement used in the last two chapters of this book. The exposition here is elementary, essentially in the style of [Struik, 1961]. The book by [Gray, 1998] is a modern text using symbolic manipulation software as an aid to learning. Finally, a book by [Frankel, 1997] provides a formulation targeted at physicists. Note that Appendix A presents a background for the basic mathematical concepts used here.

The emphasis of this chapter is on the inner or intrinsic geometric properties of surfaces. These properties are independent of the way the surface is immersed in a three-dimensional space. The metric or first form, for example, is introduced as an induced metric form of a three-dimensional Euclidean space and characterizes the intrinsic properties of the surface. The same can be said for the second fundamental form, which may be thought of as describing the bending of a surface in Euclidean space. This approach also provides intrinsic information about the surface, and is the subject of the Gauss theorema egregium.

The metric tensor and associated first fundamental form are basic tools to allow one to formulate an elliptic system of equations for enhancement of structured and unstructured grids. With the aid of the Christoffel symbols given in terms of the derivatives of the metric tensor, one may define covariant derivatives of vectors and tensors leading to the invariant expression for the Laplace operator. The Riemann...