Riemannian Calculus of Variations using Strongly Typed Tensor Calculus

TL;DR

This paper introduces a strongly typed tensor calculus framework for Riemannian calculus of variations, enabling error detection and natural formulations in coordinate-free tensor computations.

Contribution

It develops a novel strongly typed tensor formalism for Riemannian calculus of variations, enhancing error detection and providing a uniform coordinate-free computational method.

Findings

Derived Euler-Lagrange equations using the formalism

Formulated second variation of energy functionals

Demonstrated the framework's effectiveness in tensor calculus

Abstract

In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to "telescope" to accomodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first order Lagrangians are the focus of the second half of this paper, in which in the first variation, the Euler-Lagrange equations, and the second variation of such functionals will be derived.