Geometrical Tools for Pdes on a Surface

This is an excerpt from my paper with A. Mahalov [1]. PDE theories with Riemannian geometry are long studied subject — c.f. for example the texts cited in the bibliography. Here, I only present a very brief and elementary explanation, mainly about differential operators on a 2D surface. Generalization to higher dimension needs more caution. Let M denote a 2-dimensional, compact, Riemannian manifold without boundary, typically the unit sphere M = S3−1, endowed with metric g inherited from the embedding Euclidean space R3. Let p ∈M denote a point with local coordinates (p1, p2). Any vector field u in the tangent bundle TM is identified with a field of directional differentials, which is written in local coordinates as