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 R2,3, 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
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.