A programming interface to the Riemannian manifold in a finite element environment

In this paper we present a user-programmable interface to the Riemannian manifold. At the interface, starting from the preprocessor coordinate system (chart), one can define other charts for the manifold and determine the manifold metric properties independently of the preprocessor coordinate chart representation of the manifold. Further, the interface allows one to manipulate vector fields and differential forms as abstract mathematical entities, rather than as their coefficient representations. In contrast to interfaces that use classical vector analysis, the metric is fully isolated at the interface. As an example of finite element modeling, the interface is applied to set up a boundary value problem on a Riemannian manifold.