The SageManifolds project aims at extending the modern computer algebra system SageMath (http://www.sagemath.org/) towards differential geometry and tensor calculus. All SageManifolds 1.0 code is included in SageMath 7.5, so that it does not require any separate installation. Key features of SageMath are being open-source, using the Python language and running in the powerful Jupyter Notebook (http://jupyter.org/).
SageManifolds is devoted to explicit tensor calculus (as opposed to “abstract tensor calculus”): the dimension of the manifold must be specified and some atlas must be provided. SageManifolds 1.0 functionalities include
– topological manifolds: charts, open subsets, maps between manifolds, scalar fields
– differentiable manifolds: tangent spaces, vector frames, tensor fields, curves, pullback and pushforward operators
– standard tensor calculus (tensor product, contraction, symmetrization, etc.), even on non-parallelizable manifolds
– taking into account any monoterm tensor symmetry
– exterior calculus (wedge product, exterior derivative, Hodge duality)
– Lie derivatives of tensor fields
– affine connections (curvature, torsion)
– pseudo-Riemannian metrics
– some plotting capabilities (charts, points, curves, vector fields)
Example of use, in particular in the context of general relativity, are posted at
http://sagemanifolds.obspm.fr/examples.html
Visit http://sagemanifolds.obspm.fr/ for free download and run.
Eric Gourgoulhon (on behalf of the SageManifolds team: http://sagemanifolds.obspm.fr/authors.html )