Methods-and-theory sub-discipline: differential-geometric machinery as used by physicists — connections, curvatures, principal / associated bundles, gauge-theoretic fibrations, Lie-algebra-valued forms. Core objects: Levi-Civita…
differential-geometry-physics
Levi-Civita Γ^k_{ij} = ½ g^{kl}(∂_i g_{jl}+∂_j g_{il}−∂_l g_{ij})
Christoffel symbols of the second kind Γ^k_{ij} are the connection coefficients of the unique torsion-free metric-compatible…
Riemann R^ρ_{σμν}=∂_μΓ^ρ_{νσ}−∂_νΓ^ρ_{μσ}+ΓΓ−ΓΓ; (4,0) symmetries
Riemann curvature tensor R^ρ_{σμν} of a Levi-Civita connection measures the non-commutativity of second covariant derivatives: [∇_μ, ∇_ν]…
Bianchi I (algebraic): R^ρ_{[σμν]}=0; II (differential): ∇_{[λ}R_{|σ|μν]}=0
Two Bianchi identities of the Levi-Civita connection. First ('algebraic') Bianchi identity: the cyclic sum R^ρ_{σμν} + R^ρ_{μνσ} +…
Minkowski g=diag(−1,1,1,1): all 64 Γ^λ_{μν} = 0; residual 0
Sympy-exact symbolic witness of the vanishing of all Christoffel symbols in 4D Minkowski spacetime under Cartesian coordinates (t, x, y,…
S² round: R^θ_{φθφ} = sin²θ; Ricci scalar R = 2; antisym residual 0
Sympy-exact symbolic witness of the round-2-sphere curvature. Setup: unit sphere S² of radius 1 with metric g = dθ² + sin²θ dφ²,…
First Bianchi R^ρ_{[σμν]}=0 on S²: cyclic-sum scan ≡ 0
Sympy-exact symbolic witness of the first Bianchi identity on the round 2-sphere. Setup: same metric as the sphere-Riemann witness (g =…