differential geometry. Riemannian manifolds, tangent spaces, geodesics, Laplace-Beltrami operator. the geometry of continuous curved space
| Op | Action |
|---|---|
chart(M, coords) |
Define coordinate patch on manifold |
metric(g_ij) |
Specify Riemannian metric tensor |
christoffel(g) |
Compute connection coefficients |
geodesic(p, v, t) |
Trace geodesic from point p with velocity v |
covariant_deriv(T, v) |
Parallel transport / covariant derivative |
curvature(g) |
Riemann curvature tensor |
laplacian(f, g) |
Laplace-Beltrami operator on manifold |
required for: latent space embeddings, tri-kernel diffusion formalized as heat flow on manifolds, physics simulation. programming model: coordinate charts, metric tensors, covariant derivatives — none of which exist in Ren. proof-hard over finite fields. research horizon
see cyb/languages for the complete language set. see cyb/multiproof for the proving architecture