Symbolic tensor calculus on manifolds: a SageMath implementation

TL;DR

This paper introduces a flexible symbolic tensor calculus method implemented in SageMath that operates on fully specified manifolds without coordinate or frame restrictions, supporting non-parallelizable manifolds and multiple backends.

Contribution

It presents a novel, general approach for symbolic tensor calculus on manifolds within SageMath, overcoming limitations of previous methods.

Findings

Supports tensor calculus on arbitrary manifolds

Operates without coordinate or frame limitations

Implemented in open-source SageMath software

Abstract

These lecture notes present a method for symbolic tensor calculus that (i) runs on fully specified smooth manifolds (described by an atlas), (ii) is not limited to a single coordinate chart or vector frame, (iii) runs even on non-parallelizable manifolds and (iv) is independent of the symbolic backend used to perform calculus at the level of coordinate expressions. In addition to the main ideas, we discuss some details of the implementation in the open-source mathematics software system SageMath, which has been performed via the SageManifolds project.