Tangent / cotangent bundles; Lie derivative; principal fiber bundles; Cartan connection; Gauss-Bonnet; Ricci flow; second fundamental form; exterior derivative; Hodge star; sheaves; vector-bundle classification. Complementary to existing…
Differential Geometry
Tangent bundle TM
Disjoint union of tangent spaces TM = sqcup_p T_pM forms 2n-dim smooth manifold; vector field is section of TM; foundation for Lie…
Cotangent bundle T*M
T*M carries canonical symplectic form omega = -d theta where theta is Liouville 1-form sum p_i dq^i; phase-space of mechanics; basis of…
Lie derivative L_X
Cartan magic formula L_X = i_X d + d i_X on differential forms; measures change along flow of vector field X; commutes with d and…
Principal fiber bundle
(P, M, G, pi) with G-action free + transitive on fibers; sections trivialize bundle; gauge theories (Yang-Mills) live as connections on…
Cartan connection form
Connection 1-form omega: TP -> g (Lie algebra) defines horizontal subspaces; gauge field A is local pullback; curvature F = d omega + 0.5…
Gauss-Bonnet formula (2D)
For closed orientable surface S: int_S K dA = 2 pi chi(S) where K is Gauss curvature and chi Euler characteristic; topological invariant…
Ricci flow
Hamilton 1982 equation partial_t g = -2 Ric(g) evolves Riemannian metric to homogenize curvature; Perelman 2003 used…
Second fundamental form
II(X, Y) = nabla_X N (Y) where N is unit normal of submanifold; Gauss equation relates II to intrinsic + ambient curvature; mean-curvature…
Exterior derivative d
Anti-derivation on differential forms with d^2 = 0; satisfies Leibniz d(omega ^ eta) = d omega ^ eta + (-1)^|omega| omega ^ d eta; defines…
Hodge star operator
On oriented Riemannian n-manifold, * : Lambda^k -> Lambda^(n-k) defined via inner-product and volume form; Laplacian on forms Delta = d d*…
Sheaf on manifold
Assignment of abelian-group / ring / module to every open set, compatible with restriction and gluing; basis for sheaf cohomology; coherent…
Vector bundle classification
Real / complex rank-k vector bundles over CW-complex X classified by homotopy classes [X, BO(k)] / [X, BU(k)]; Chern classes c_i in…
Einstein tensor
G_mu_nu = R_mu_nu - 0.5 g_mu_nu R; symmetric divergence-free tensor encoding spacetime curvature in Einstein field equations G + Lambda g =…
Ricci tensor
Trace of Riemann curvature tensor: R_mu_nu = R^alpha_mu_alpha_nu; symmetric 2-tensor; vanishing characterizes Ricci-flat manifolds (e.g.…
Dirac operator
First-order elliptic differential operator on spinor bundle: D = gamma^mu nabla_mu; squares to Laplacian (Lichnerowicz formula);…
Chern-Gauss-Bonnet theorem
Generalizes Gauss-Bonnet to even-dim closed Riemannian manifolds: int_M Pf(Omega/2pi) = chi(M); Pfaffian of curvature 2-form integrates to…
Parallel transport (connection)
Levi-Civita 1917 parallel-transport on Riemannian manifold preserves vector along curves; defined by connection nabla_X Y = X^mu…
Curvature 2-form Omega
Cartan structure equations: Omega = d omega + omega ^ omega (so(n)-valued); Omega = R^a_b_c_d e^c ^ e^d / 2 in orthonormal-frame basis;…
Killing vector field (isometry)
X is Killing iff Lie-derivative L_X g = 0; equivalently nabla_(mu X_nu) = 0; generates infinitesimal isometries of Riemannian manifold;…
Mean curvature flow
Hamilton-Huisken evolution: surface evolves at velocity H along inward-normal; minimizes area; Huisken 1984 proved convex surfaces shrink…
Poincare duality (cohomology)
Poincare 1893: closed orientable n-manifold M has H^k(M) iso H_(n-k)(M) iso H^(n-k)(M); intersection-pairing nondegenerate on H^k x…
Spin structure (Spin -> SO double cover)
Lift of orientable Riemannian frame-bundle SO(n) -> Spin(n) (the double cover); exists iff w_2(M) = 0 (second Stiefel-Whitney class);…
Cartan connection (1923)
E Cartan 1923 Cartan connection; modern Ehresmann-connection + principal-bundle-curvature; foundational gauge theory.
Chern classes (1946)
S S Chern 1946 (Wolf 1984) characteristic classes c_k(E); modern Chern-Weil + index-theorem applications.
Ricci flow (Hamilton 1982)
R Hamilton 1982 dg/dt = -2 Ric; Perelman 2002-2003 (Fields 2006 declined); Poincare + geometrization.
Calabi-Yau (Calabi 1957 / Yau 1978)
Calabi 1957 conjecture + Yau 1978 proof (Fields 1982); modern string-theory compactification + mirror-symmetry.
Donaldson (1983)
S Donaldson 1983 (Fields 1986) Donaldson-invariants 4-manifold; modern Seiberg-Witten 1994 simpler reformulation.
APS eta (1975)
Atiyah-Patodi-Singer 1975 eta-invariant + index-theorem manifolds with boundary; modern parity-anomaly applications.