SageManifolds Differential geometry and tensor calculus with SageMath

SageManifolds: examples: "Example worksheets The worksheets are in the Jupyter Notebook format (ipynb). The command to start the Jupyter notebook (with a SageMath kernel) is sage -n jupyter worksheet_file.ipynb All the examples below can also be read directly in the browser by just clicking on their title. Standard 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 simplified version on the SageMathCloud is here Mercator projection and connection with torsion on S2 [ipynb] (a nice example of an affine connection with non-zero torsion) 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 Euclidean space Vector calculus in Cartesian and spherical coordinates [pdf] [ipynb] (Cartesian and spherical frames in the Euclidean 3-space; changes of frames) Tensor algebra Tensors on free modules of finite rank [pdf] [ipynb] (pure algebraic part of SageManifolds) Black hole spacetimes Schwarzschild spacetime [pdf] [ipynb] (Einstein equation, Bianchi identity, change of chart, change of vector frame) 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) 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) 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) de Sitter spacetime [pdf] [ipynb] (mapping between manifolds, induced metric, curvature, maximally symmetric space) Anti-de Sitter spacetime [pdf] [ipynb] (mapping between manifolds, induced metric, curvature, maximally symmetric space) 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) 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 See also the tutorial for a basic introduction to SageManifolds"

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