LeviPy

A pure-Python, SymPy-backed library for symbolic differential geometry and general relativity computations — the Levi-Civita connection, curvature tensors, geodesics, and more, all computed exactly as closed-form symbolic expressions.

What sets LeviPy apart is its Lean 4 bridge: it doesn't just compute geometric quantities, it can export them as machine-checkable proof obligations in Lean 4 against Mathlib — turning a symbolic curvature calculation into a formal theorem a proof assistant can verify. See Lean 4 formal verification.

Features

• Metric tensor g_{ij} with automatic inverse g^{ij}
• Christoffel symbols Γ^k_{ij} of the Levi-Civita connection
• Riemann curvature tensor R^l_{ijk}
• Ricci tensor R_{ij} and Ricci scalar R
• Einstein tensor G_{ij}
• Geodesic equations for an affinely parametrised curve
• Parallel transport of a vector field along a curve
• Covariant derivatives of vectors, covectors, and (1,1)-tensors
• Lie bracket [X, Y] of two vector fields
• Lean 4 bridge — translate SymPy expressions to Lean 4 terms, emit theorem stubs, and link to relevant Mathlib lemmas for formal verification

All tensors derived from the metric are computed lazily and cached.

Requirements

• Python ≥ 3.12
• SymPy ≥ 1.14

This project uses uv for environment and dependency management.

Installation

# Clone, then sync the environment
uv sync

# Build a wheel / sdist into dist/
uv build

Quick start

import sympy as sp
from levipy import Manifold

# Coordinates for the 2-sphere
theta, phi = sp.symbols('theta phi', real=True)
r = sp.Symbol('r', positive=True)

# Build a manifold and attach a metric g_{ij}
M = Manifold('S2', coords=[theta, phi])
g = M.metric([[r**2, 0],
              [0, r**2 * sp.sin(theta)**2]])

# Derived geometric quantities
g.christoffel()      # Christoffel symbols  Γ^k_ij
g.riemann()          # Riemann tensor       R^l_ijk
g.ricci_tensor()     # Ricci tensor         R_ij
g.ricci_scalar()     # Ricci scalar         R
g.einstein_tensor()  # Einstein tensor      G_ij

Most objects provide a .display() method for pretty-printed / LaTeX output.

Geodesics

geo = g.geodesic_equations()
geo.display()

Parallel transport

from levipy import ParallelTransport

pt = ParallelTransport(g)
pt.equations(curve=[sp.cos(pt.lam), sp.sin(pt.lam)])

Lie bracket

from levipy import lie_bracket

X = [theta, 0]
Y = [0, phi]
lie_bracket(X, Y, coords=[theta, phi])

Lean 4 formal verification bridge

Symbolic computation tells you what the curvature of a space is. It does not tell you the result is correct — a typo in a metric or a SymPy simplification quirk can silently produce a wrong tensor. LeviPy's Lean 4 bridge closes that gap by translating your computed geometry into formal statements that the Lean 4 proof assistant can check against Mathlib.

This is a proof-obligation layer, not an automated prover. Given a fully computed metric (with its Christoffel symbols, Riemann/Ricci tensors, etc.), TheoremBuilder emits a .lean source file containing:

1. The necessary Mathlib imports
2. Coordinate symbols declared as (x : ℝ)
3. The metric as a noncomputable def
4. One theorem per nonzero Christoffel symbol
5. Riemann flatness / curvature theorems
6. The Ricci scalar theorem
7. Geodesic equation statements

Each theorem comes with a sorry-based proof skeleton and tactic hints that point at real Mathlib lemmas — so a human (or Lean's automation) can fill in the proofs. The output is a set of checkable obligations: if the geometry is wrong, the Lean statements won't close.

Three components make up the bridge:

Component	Role
SymPyToLean4	Translates a SymPy expression into a Lean 4 term string
TheoremBuilder	Emits a full .lean file of proof obligations from a metric
MathlibLinker	Suggests relevant Mathlib 4 lemmas/tactics for an expression

import sympy as sp
from levipy import Manifold, TheoremBuilder

# Name the symbols in Greek so the generated Lean uses θ, φ identifiers
theta, phi = sp.symbols('θ φ', real=True)
M = Manifold('S2', coords=[theta, phi])
g = M.metric([[1, 0], [0, sp.sin(theta)**2]])

builder = TheoremBuilder(
    g,
    manifold_name='Sphere2',
    extra_hypotheses=['(hth : 0 < θ)', '(hth2 : θ < π)'],  # must match the symbol names
)
builder.save('Sphere2.lean')   # writes a Lean 4 source file

Note for pip install users: the PyPI package ships the Python bridge (which generates .lean files), but not the companion Lean project itself. To build and check the generated proofs, clone the GitHub repository — the Lean Lake project lives in lean/ (pinned to a Mathlib toolchain).

Lower-level pieces are also available:

from levipy import SymPyToLean4, MathlibLinker

SymPyToLean4().translate(sp.sin(theta)**2)   # SymPy expr → Lean 4 term string
MathlibLinker()                              # map expressions to Mathlib lemmas

Package layout

levipy/
├── geometry/      Manifold, covariant derivative, Lie bracket
├── tensors/       Metric, Christoffel, Riemann, Ricci, Einstein
├── gr/            General-relativity tools (geodesics, parallel transport)
└── lean4/         Lean 4 bridge (translator, theorem builder, Mathlib linker)

lean/              Companion Lean 4 / Lake project (Mathlib-backed proofs)

Development

# Run the test suite (pytest is a dev dependency)
uv run pytest

# Verbose
uv run pytest -v

License

Released under the MIT License. © 2026 Miraj Samarakkody.