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 =…
Gauss-Bonnet: int_M K dA + oint k_g ds = 2 pi chi(M); topological-invariance of curvature
Gauss-Bonnet framework (Gauss 1827 Disquisitiones generales circa superficies curvas; Bonnet 1848 J Ec Polytech 19, 1; Chern 1944 Ann Math…
Atiyah-Singer index: ind(D) = int_M A-hat(M) ch(E); Fredholm-K-theory equality
Atiyah-Singer index theorem framework (Atiyah-Singer 1963 Bull AMS 69, 422; 1968 Ann Math 87, 484; Atiyah-Bott 1968 Ann Math 86, 374).…
Exterior-derivative nilpotency d circ d = 0; de Rham complex; cohomology
Exterior-derivative framework (Cartan 1899 Comptes Rendus 129, 522; de Rham 1931 Sur l'analysis situs des varietes a n dimensions; 1950…
Theorem: 4 pi - 2 pi * chi(S^2) = 0 (Gauss-Bonnet sphere instance, chi = 2)
Theorem (Gauss-Bonnet-sphere canonical): for the unit sphere S^2 with intrinsic Gaussian curvature K = 1 and total surface area 4 pi, the…
Theorem: ind(D_spin) = 1 - g = 0 at g = 1 (Dirac on torus T^2)
Theorem (Atiyah-Singer-genus-one canonical): for a Riemann surface of genus g, the Dolbeault operator D_partial-bar (Cauchy-Riemann…
Theorem: partial^2 f / partial-x partial-y - partial^2 f / partial-y partial-x = 0 (d^2 = 0 via Schwarz)
Theorem (d-squared-zero canonical): for a smooth function f(x, y), the equality of mixed partial derivatives partial^2 f / partial-x…
Riemannian metric tensor (physics)
Riemann 1854 + Levi-Civita 1917: g_{mu nu} symmetric non-degenerate (0,2)-tensor; defines lengths + angles + Levi-Civita connection; basis…
Connection / curvature (Yang-Mills)
Yang-Mills 1954: gauge connection A_mu in Lie-algebra-valued 1-form; F_{mu nu} = d A + A wedge A curvature 2-form; basis of Standard Model…
Vielbein / tetrad formalism
Cartan 1928 vierbein e^a_mu: orthonormal frame field; spin-connection omega^{ab}_mu; basis for fermion coupling to gravity (Dirac in curved…
Instanton / anti-self-dual connections
Belavin-Polyakov-Schwartz-Tyupkin 1975 ASD instanton: F = -*F minimizes Yang-Mills action; Atiyah-Singer index theorem links instantons to…
Supermanifold / superconnection
Berezin 1961 + Kostant 1977 + Quillen 1985 superconnection: Z2-graded fiber-bundles; basis of supersymmetric field theory + Mathai-Quillen…
Heat-kernel asymptotic (Mac Kean-Singer)
Mac Kean-Singer 1967: tr(e^{-t Lap}) ~ sum a_k(x) t^{(k-d)/2} as t -> 0+; Seeley-DeWitt coefficients a_k local invariants; basis of…
Riemann curvature (1854)
B Riemann 1854 sectional-curvature R^a_bcd; modern foundation of GR + Kähler + Calabi-Yau + holonomy classification.
Levi-Civita (1916)
T Levi-Civita 1916 metric-connection; modern Christoffel + parallel-transport + holonomy + spin-connection in physics.
Cartan formalism (1922)
É Cartan 1922 + 1928 'Sur les variétés à connexion affine'; modern frame-bundle + tetrad + spinor-formalism.
Hodge decomposition (1941)
W Hodge 1941 harmonic-form Δω=0 + d=δ=0; modern Hodge-de-Rham + Yang-Mills instanton classification.
Chern class (1946)
S-S Chern 1946 Chern-Weil theory characteristic-classes; modern foundation of topological-invariants in gauge-theory + cond-mat.
Instanton (Belavin 1975)
Belavin-Polyakov-Schwartz-Tyupkin 1975 + Atiyah-Singer 1968 ASD-instanton; modern Donaldson 1983 + Seiberg-Witten 1994 4-mfd.