The free Python-based computer algebra system SageMath 9.6 has been released on 15 May 2022; it includes differential geometry and tensor calculus tools developed through the SageManifolds project (https://sagemanifolds.obspm.fr/); see the release tour https://trac.sagemath.org/wiki/ReleaseTours/sage-9.6 for an overview of novelties.
Regarding calculus on manifolds, SageMath 9.6 brings new features:
– symplectic structures have been introduced: symplectic forms, Poisson bivectors, Hamiltonian vector fields, Poisson brackets, Liouville volume form, symplectic Hodge duality
– projective spaces are now included in the manifold catalog
– some performance improvements have been implemented and a few bug have been fixed
See https://sagemanifolds.obspm.fr/changelog.html for details and examples.
The following people have contributed to these new features:
Tobias Diez, Marius Gerbershagen, Eric Gourgoulhon, Michael Jung, Trevor K. Karn, Matthias Koeppe and Travis Scrimshaw.
See https://sagemanifolds.obspm.fr/download.html for installation instructions.
See also https://www.youtube.com/playlist?list=PLnrOCYZpQUuJlnQbQ48zgGk-Ks1t145Yw for video tutorials by Christian Baer (Institute of Mathematics, University of Potsdam) about manifolds in SageMath.