RiemPy

A symbolic toolkit for differential geometry and general relativity in Python.

RiemPy is a Python library for modeling Riemannian manifolds and computing fundamental geometric quantities in a symbolic way. It is designed for students, researchers, and enthusiasts of general relativity and differential geometry.

🚀 Installation

Clone the repository and install in dev mode:

git clone https://github.com/Joyboy0056/RiemPy.git
cd RiemPy
pip install -e .

or directly install the package from the public repo:

pip install git+https://github.com/Joyboy0056/RiemPy.git

Note

This is an early version of the library — useful, but still a draft.
Expect changes, improvements, and possibly breaking updates in future releases.

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Quick start

Please refer to the examples in this repository or see the code below and use it as a reference:

from riemmpy import Manifold, Submanifold, Sphere, Schwarzschild
from sympy import pprint, Matrix, exp, symbols

pprint(Sphere(2).metric)

x, y, z, t = symbols('x y z t')
R3 = Manifold(Matrix.eye(3), [x, y, z])

S2 = Submanifold(R3, Sphere(2).coords, Sphere(2).get_embedding())
S2.get_induced_metric()
pprint(Sphere(2).metric==S2.metric)

pprint(Schwarzschild(4).horizon.metric)

# Godel metric
g = Matrix([
    [1,      0,     exp(x),            0],
    [0,     -1,      0,            0],
    [exp(x),     0, -1/2 * exp(x)**2,      0],
    [0,      0,      0,           -1]
])

Godel = Manifold(g, [x, y, z, t])

Godel.get_scalar_curvature()
pprint(Godel.scalar_curvature)

📦 Module Documentation

Built on top of sympy, the library allows users to compute geometric quantities such as Christoffel symbols, Ricci tensor, curvature scalars and geodesics.

Class: Manifold

A base class representing a general (possibly pseudo) Riemannian manifold of dimension n.

Attributes:

• metric: The metric tensor expressed as a sympy.Matrix.
• coords: A list of symbolic coordinate variables.

Main methods:

• get_christoffel_symbols(): Computes the Christoffel symbols of the metric.
• get_riemann_tensor(): Computes the Riemann curvature tensor.
• get_ricci_tensor(): Computes the Ricci curvature tensor.
• get_ricci_scalar(): Computes the Ricci scalar.
• get_einstein_tensor(): Computes the Einstein tensor.
• is_flat(): Checks whether the Riemann tensor vanishes identically.
• is_einstein_mfd(): Checks whether the manifold is of Einstein type.
• print_sectional_curvatures(): Prints the sectional curvatures of the metric.
• get_kretschmann_scalar(): Computes the Kretschmann scalar.
• create_spacetime_metric(scale_factor): Constructs the Lorentzian metric from a pure Riemannian, with scale_factor as a sympy function.
• display_geodesic_equations(): Computes and print the geodesic equations symbolically.
• solve_geodesic(): Solves the geodesic equations numerically.
• get_geometrics(): Computes all the main geometric quantities of the manifold.

Minor methods: The Manifold class has been provided with some other minor methods, useful for computations of functions on manifolds, as covariant_derivative(X, T, ind), gradient(f), divergence(X), laplacian(f), hessian(f).

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Class: Submanifold

A subclass of Manifold representing a (n-k)-dimensional submanifold.

Attributes:

• ambient_manifold: A Manifold object.
• sub_coords: A list of symbolic coordinate variables.
• embedding: A list of symbolic functions of the coordinates.

Main methods:

• get_embedding_jacobian(): Computes the Jacobian matrix of the embedding map.
• get_induced_metric(): Computes the induced metric from the embedding.
• get_normal_field(): Computes the normal vector field.
• get_IInd_fundamental_form(): Computes the extrinsic curvature tensor.
• get_mean_curvatureII(): Computes the extrinsic curvature scalar.
• is_minimal(): Checks whether the mean curvature scalar vanishes identically.
• is_totally_geodesic(): Checks whether the mean curvature tensor vanishes identically.
• plot_geodesics_on_surface(domain, geodesics): For a 2-dimensional surface, plot the geodesics on the surface.
• get_geometrics_sub(): Computes all the main geometric quantities of the submanifold.

The subclass Submanifold inherits the methods of Manifold. Here below, you can find other classes for the main geometric spaces.

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Class: Sphere

A subclass for the spherical n-dimensional space S^n.

Attributes:

• dimension: An integer for the dimension of the manifold.

Main methods: Being a Manifold objects it inherits the base class methods. Moreover

• get_embedding(): Returns the embedding of S^n in R^n+1.

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Class: Hyp

A subclass for the hyperbolic n-dimensional space H^n.

Attributes:

• dimension: An integer for the dimension of the manifold.

Main methods: Being a Manifold objects it inherits the base class methods.

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Class: Eucl

A subclass for the Euclidean n-dimensional space R^n.

Attributes:

• dimension: An integer for the dimension of the manifold.

Main methods: Being a Manifold objects it inherits the base class methods.

Instances:

• polar: A Manifold object for the polar representation of Eucl(n).

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Class: Minkowski

A subclass for the Minkowskian (n+1)-dimensional space R^(1,n).

Attributes:

• neg: An integer for the negative part of the signature.
• pos: An integer for the positive part of the signature.

Main methods: Being a Manifold objects it inherits the base class methods.

Class: Schwarzschild

A subclass for the spacelike Schwarzschild n-dimensional space Sigma^n.

Attributes:

• dimension: An integer for the dimension of the manifold.

Main methods: Being a Manifold objects it inherits the base class methods.

Instances:

• spacetime: A Manifold object for the (n+1)-dimensional Schwarzschild spacetime.
• horizon: A Submanifold object for the (n-1)-dimensional event horizon.