SageManifolds

SageManifolds
Name	SageManifolds
Developer	SageMath developers
Released	2013
Latest release	2024
Programming language	Python, Cython
Operating system	Linux, macOS, Windows
License	GPLv3

SageManifolds is a component of the SageMath ecosystem for symbolic and numerical computations on differentiable manifolds. It provides tools for tensor calculus, differential forms, Riemannian geometry and connections, enabling work relevant to researchers using Albert Einstein, Bernhard Riemann, Emanuel Lasker-level formalism and applications in projects associated with Institut Fourier, University of Washington, CNRS and Princeton University. Its design interoperates with libraries and projects such as NumPy, SciPy, SymPy, Singular, Pari/GP and Maxima to serve needs across academic groups including those at Massachusetts Institute of Technology, University of Cambridge, ETH Zurich and University of Toronto.

Overview

SageManifolds offers a high-level interface to model smooth manifolds, coordinate charts, tensor fields, vector fields, differential forms and connections compatible with users from Fields Medal-level research groups and institutional users like NASA, CERN-affiliated teams and faculty at Harvard University and Yale University. It emphasizes symbolic manipulation, numeric evaluation and visualizations that integrate with tools from Matplotlib, Jupyter Notebook, LaTeX workflows and computational formats used by researchers at Stanford University and Caltech.

History and Development

Development began as part of the broader SageMath initiative coordinated by contributors from University of Washington, University of Paris-Sud, École Normale Supérieure and volunteers connected to Python Software Foundation communities. Early milestones drew inspiration from classical work by Élie Cartan, Bernhard Riemann and modern computational geometry efforts at institutions like Max Planck Society and IBM Research. Major releases aligned with community events such as workshops at PyCon, Sage Days and conferences held at Institut Henri Poincaré, with contributions from researchers associated with University of Oxford, Columbia University and University of California, Berkeley.

Features and Capabilities

The core capabilities include symbolic tensor algebra, coordinate chart management, Levi-Civita and arbitrary connections, curvature computations, geodesic integration and manipulation of exterior calculus primitives familiar to users from General Relativity research groups at Perimeter Institute and Kavli Institute for Theoretical Physics. It supports exact computations via backends like SymPy, polynomial algebra via Singular, number theory routines from Pari/GP and numeric linear algebra via LAPACK and BLAS. Visualization of manifolds and curvature quantities integrates with Matplotlib, 3D rendering engines used by teams at Blender Foundation and plotting frameworks deployed in research at Imperial College London.

Architecture and Implementation

Architecturally, the project is implemented in Python with performance-critical components in Cython and interfaces to low-level libraries like GMP and MPFR. It follows modular patterns promoted by the SageMath governance and packaging practices used in large collaborations such as those at Debian and Anaconda, Inc.; testing and continuous integration leverage infrastructure from GitHub and Travis CI-style services adopted in projects at Mozilla Foundation and Google. Data structures reflect tensor algebra conventions found in literature from Roger Penrose, Stephen Hawking and computational geometry projects at University of Illinois Urbana-Champaign.

Usage and Examples

Typical workflows are demonstrated in Jupyter notebooks prepared by academics from University of Cambridge, University of Oxford and University of Chicago for courses modeled after curricula at MIT and Princeton University. Examples include computing the Riemann tensor for models inspired by metrics studied by Karl Schwarzschild, solving geodesic equations relevant to simulations by European Space Agency teams, and exploring curvature invariants used in research at Max Planck Institute for Gravitational Physics. Integration examples show coupling with NumPy arrays, symbolic simplifications via SymPy and solving PDEs with solvers used in projects at Los Alamos National Laboratory.

Community and Contributions

The contributor base spans academic researchers, postdoctoral fellows and volunteers affiliated with institutions like CNRS, University of Warwick, University of Michigan and organizations such as the Python Software Foundation. Development, issue tracking and code review occur on platforms used by major open-source projects like GitHub and in community meetings similar to Sage Days and workshops hosted at Institut Henri Poincaré. Funding and collaborative work have intersected with initiatives supported by grants from agencies such as NSF, ERC and collaborations with entities like European Research Council and university research offices at University of California campuses.

Category:Mathematical software