SageManifolds

Example notebooks

The notebooks are in the Jupyter format (ipynb). They can be read directly in the browser by just clicking on their titles. The notebooks are opened in read-only mode, but you can access to an interactive version by clicking on Execute on Binder in the top right menu.

To download a notebook and run it on your computer, click on [ipynb] (or on the download button in the notebook top right menu) and type

sage -n jupyter example_file.ipynb

See also the manifold tutorial for a basic introduction (Japanese version is here), the plot tutorial (plots of coordinate charts, manifold points, vector fields and curves) and the tutorial on pseudo-Riemannian manifolds (metric, Levi-Civita connection, curvature, geodesics) [video].

2-dimensional manifolds

• Sphere S² [pdf] [ipynb] (multiple domains and charts, transition maps, scalar and vector fields, tangent spaces, curves, plot of charts and vector fields, embedding, pullback, Riemannian metric); a version using SymPy as symbolic backend, instead of SageMath default is here; a simplified version on CoCalc is here
• Mercator projection and connection with torsion on S² [ipynb] (a nice example of an affine connection with non-zero torsion)
• Euclidean plane E² [ipynb] [CoCalc] (Cartesian and polar coordinates, vector calculus)
• Hyperbolic plane H² [pdf] [ipynb] (many charts associated with various models of H², embedding, pullback, curvature, changes of chart, graphics)
• Real projective plane RP² [pdf] [ipynb] (minimal atlas with 3 charts, transition maps, differential mappings, Roman surface, Boy surface)

3-dimensional manifolds

• Euclidean space E³ (Cartesian, spherical and cylindrical coordinates, vector calculus)
• Sphere S³: charts, quaternions and Hopf fibration [ipynb] (various charts, embedding in R⁴, maps, curves, 3D graphics)
• Sphere S³: vector fields and left-invariant parallelization [ipynb] (right translations, global vector fields and global frame, link with the Hopf fibration)
• Sphere S³: round metric [ipynb] (round metric as the pullback of Euclidean metric on R⁴, Riemann tensor, volume 3-form)

Tensor algebra

• Tensors on free modules of finite rank [pdf] [ipynb] (pure algebraic part of SageManifolds)

Maximally symmetric spacetimes

• Minkowski spacetime [ipynb] (null coordinates, induced metric, conformal completion, Penrose diagram, embedding in the Einstein cylinder)
• Anti-de Sitter spacetime [ipynb] (immersion in R^2,3, induced metric, curvature, geodesics, Einstein cylinder, Poincaré patch)
• de Sitter spacetime [pdf] [ipynb] (map between manifolds, induced metric, curvature, maximally symmetric space)

Black hole spacetimes

• Schwarzschild spacetime (basics) [ipynb] (Christoffel symbols, Riemann tensor, Kretschmann scalar)
• Schwarzschild spacetime [pdf] [ipynb] (Einstein equation, Bianchi identity, change of chart, change of vector frame, Kruskal-Szekeres coordinates)
• Introducing pseudo-Riemannian manifolds on the Schwarzchild spacetime example [ipynb] (manifold, charts, points, vector fields, metric, curvature, geodesics)
• Computing a geodesic in Schwarzschild spacetime [ipynb] (numerical integration of the geodesic equation)
• More geodesics in Schwarzschild spacetime [ipynb] (numerical integration of the geodesic equation)
• Image of an accretion disk around a Schwarzschild black hole [ipynb] (numerical integration of many geodesics)
• Carter-Penrose diagram of Schwarzschild spacetime [pdf] [ipynb] (change of chart, change of vector frame)
• Kerr spacetime [pdf] [ipynb] (Killing equation, Einstein equation, Bianchi identity, Kretschmann scalar)
• Kerr-Newman spacetime [pdf] [ipynb] (Maxwell equations, Killing equation, Bianchi identity, Einstein equation, Kretschmann scalar)
• Principal null directions in Kerr spacetime [ipynb] (Weyl tensor, principal null vectors, index notations)
• Walker-Penrose Killing tensor in Kerr spacetime [ipynb] (Killing equation, principal null vectors, symmetrization, Killing tensor)
• 3+1 slicing of Kerr spacetime [pdf] [ipynb] (3-metric, lapse, shift, extrinsic curvature, 3+1 Einstein equations, electric and magnetic parts of the Weyl tensor)
• Simon-Mars tensor and Kerr spacetime [pdf] [ipynb] (Weyl tensor, self-dual Killing form, Simon-Mars tensor)
• 3+1 Simon-Mars tensor in Kerr spacetime [pdf] [ipynb] (3+1 Einstein equations, 3+1 decomposition of the Simon-Mars tensor)
• Near-horizon geometry of the extremal Kerr black hole [ipynb] (coordinate changes, tensor series expansion, Killing form)
• 5-dimensional Kerr-AdS spacetime [ipynb] (5-dimensional Einstein equation)

Other examples regarding black hole spacetimes are posted here.

Examples regarding black branes in 5-dimensional spacetimes: black branes in Lifshitz-like spacetimes and Vaidya-Lifshitz solution

Cosmological spacetimes

• Friedmann equations [pdf] [ipynb] (FLRW metric, Einstein equation)

Other examples in General Relativity

• Tolman-Oppenheimer-Volkoff equations [pdf] [ipynb] (Derivation of the TOV equations from the Einstein equation, numerical resolution to get models of relativistic stars)
• Lemaître-Tolman solutions [ipynb] (Solving the Einstein equation for spherically symmetric pressureless fluids)
• Curzon-Chazy spacetime: Simon-Mars tensor [pdf] [ipynb] (Weyl tensor, self-dual Killing form, Simon-Mars tensor)
• Tomimatsu-Sato spacetime: Einstein equations [pdf] [ipynb] (3+1 Einstein equations)
• Tomimatsu-Sato spacetime: Simon-Mars tensor [pdf] [ipynb] (3+1 decomposition of the Simon-Mars tensor, 2D and 3D graphics)

Examples in solid state physics and electromagnetism

• Elasticity theory in Euclidean space (Cartesian coordinates) [pdf] [ipynb] (strain and stress tensors, Hooke's law)
• Elasticity theory in Euclidean space (spherical coordinates) [pdf] [ipynb] (strain and stress tensors, Hooke's law)
• Electromagnetism in Minkowski spacetime [ipynb] (Electromagnetic field 2-form from E and B, Maxwell equations, conserved current, Lorentz force, Poynting vector)

Analysis on manifolds

• Limiting Carleman weights on 4-dimensional Riemannian manifolds: example 1, example 2, example 3

Submanifolds

• Foliation of Kerr spacetime by spacelike hypersurfaces [ipynb] (intrinsic and extrinsic geometry)
• Manifolds and submanifolds equipped with a degenerate metric [ipynb] (degenerate metric, rigging, screen distribution, Weingarten map, shape operator)
• Event horizon of Schwarzschild black hole as a degenerate submanifold [ipynb] (degenerate metric, embedding, screen distribution, Weingarten map)

Vector bundles

• Simple vector bundles [ipynb] (vector bundle, tensor bundle, section)
• Mixed differential forms and characteristic classes [ipynb] (graded algebra of mixed differential forms, characteristic class, Chern class, Euler class)
• Characteristic classes in SageMath (a general introduction)

See also the manifold tutorial for a basic introduction (Japanese version is here) and the plot tutorial for plots of coordinate charts, manifold points, vector fields and curves.