omnibias.geometry

Differential geometry on manifolds, built on the field substrate with torch + jax parity: metric tensor, Christoffel symbols, covariant derivative, Laplace–Beltrami, Riemann / Ricci / scalar curvature, geodesics, and exterior calculus.

import omnibias.geometry as og

Schemas (omnibias.geometry._core, re-exported)

Symbol	Role
MetricSpec	specification of a metric tensor field
ManifoldSpec	specification of a manifold
ChartSpec, ChartFn	a coordinate chart and its callable
MetricFn, AmbientMetricFn	metric callables (intrinsic / ambient)
DifferentialForm	a differential form whose components are fields

Backend ops (omnibias.geometry.torch / .jax)

The computational surface lives in the backend subpackages:

• metric: metric tensor, inverse, determinant.
• connection: Christoffel symbols, covariant derivative, covariant Hessian.
• operators: the Laplace–Beltrami operator.
• curvature: Riemann tensor, Ricci tensor, scalar/Gaussian curvature.
• geodesics: geodesic integration.
• exterior calculus: exterior derivative d, wedge ∧, Hodge star, codifferential.
• learned charts: the pullback metric g = JᵀhJ of a learned chart.

Honest exactness label

Field-function derivatives are closed-form exact; metric derivatives are autodiff-exact (forward-mode autodiff of the analytic metric), labelled as such — see exactness & scope.

See also

• Field substrate
• symbolic — geometric equation discovery builds on this.