SageManifolds: examples

Example notebooks

The notebooks are in the Jupyter format (ipynb). They can be read directly in the browser by just clicking on their titles.

To download a notebook and run it on your computer, click on [ipynb] and type

sage -n jupyter example_file.ipynb

See also the tutorial for a basic introduction to SageManifolds (Japanese version is here).

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)
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 [pdf] [ipynb] (intrinsic and extrinsic geometry)