The SageManifolds project aims at extending the modern computer algebra system SageMath towards differential geometry and tensor calculus. All SageManifolds code is included in SageMath 7.5 (and higher versions), i.e. it does not require any separate installation.
SageManifolds deals with differentiable manifolds of arbitrary dimension. Various coordinate charts and vector frames can be introduced on the manifold, which does not need to be parallelizable. A given tensor field is then described by its representations (sets of components) in each vector frame.
Generic pseudo-Riemannian manifolds can be considered, among which Riemannian manifolds and Lorentzian manifolds, with applications to General Relativity. In particular, the computation of the Riemann curvature tensor and associated objects (Ricci, Weyl, Schouten and Cotton tensors) is implemented. SageManifolds can also deal with generic affine connections, not necessarily Levi-Civita ones.
At present (version 1.0.1), SageManifolds 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)
Open and free software
The list of current authors is here.