Differential Geometry

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38 domain-specific abstractions whose origin domain is Differential Geometry.

• Abstract index notation — A basis-free tensor notation in which index letters name argument slots and variance rather than numerical components.
• Almost complex manifold — A smooth even-dimensional manifold equipped with a smoothly varying tangent-bundle endomorphism J whose square is minus the identity, providing pointwise complex linear structure without necessarily admitting complex coordinates.
• Bonnet Theorem — The fundamental theorem of surface theory reconstructs a surface immersion from compatible first and second fundamental forms, uniquely up to rigid motion.
• Chern–Weil homomorphism — The map sending invariant polynomials on a Lie algebra to de Rham cohomology classes represented by curvature forms of principal-bundle connections.
• Complex differential form — A differential form whose coefficients take complex values, often decomposed into holomorphic and antiholomorphic bidegrees.
• Complex hyperbolic space — Realize the simply connected complete Kähler manifold of complex dimension n and constant negative holomorphic sectional curvature, equivalently the rank-one Hermitian symmetric space acted on transitively by PU(n,1).
• Covariant derivative — A connection-defined derivative on vector or tensor fields that corrects ordinary differentiation so results transform consistently across changing bases on a manifold.
• Darboux vector — The instantaneous angular-velocity vector of a moving orthonormal frame along a space curve, combining curvature and torsion.
• Differentiable curve — A parametrized path in a manifold or Euclidean space whose coordinate representation has the declared degree of differentiability.
• Differentiable stack — Represent quotient-like smooth geometry as a stack on manifolds admitting a smooth atlas, equivalently through a Lie-groupoid presentation considered up to Morita equivalence.
• Differential form — An alternating covariant tensor field that can be integrated over oriented manifolds of matching dimension.
• Differential invariant — A function of variables and derivatives that remains unchanged under the prolonged action of a transformation group.
• Differential Structure — A maximal compatible atlas that determines which coordinate descriptions on a topological manifold count as differentiable.
• Double tangent bundle — The tangent bundle of the total space of a manifold’s tangent bundle, carrying two compatible vector-bundle projections and a canonical flip.
• Eguchi–Hanson space — A complete noncompact four-dimensional hyperkähler ALE manifold resolving the A1 quotient singularity and carrying a Ricci-flat self-dual metric.
• Einstein manifold — A Riemannian or pseudo-Riemannian manifold whose Ricci curvature tensor is everywhere a scalar multiple of its metric.
• Foliation — A decomposition of a manifold into connected immersed submanifolds of equal dimension that locally look like parallel coordinate slices.
• Frobenius manifold — A manifold whose tangent spaces carry smoothly varying commutative Frobenius-algebra products compatible with a flat metric and integrability conditions.
• Hedgehog (geometry) — A plane curve or higher-dimensional hypersurface represented as the envelope of oriented support lines or hyperplanes supplied by a differentiable support function, extending convex support geometry to self-crossing and projective cases.
• Intrinsic Equation of a Curve — A representation of curve shape by relations among arc length, tangent angle, curvature, and torsion that suppresses arbitrary position and coordinate frame.
• Kosmann lift — The canonical metric-dependent lift of a vector field on a Riemannian manifold to the orthonormal frame bundle, enabling a Lie derivative of spinors.
• Maurer–Cartan form — The canonical Lie-algebra-valued one-form that translates each tangent vector on a Lie group back to the identity.
• Mean curvature — The average of a hypersurface’s principal curvatures at a point, measuring its local extrinsic bending in an ambient manifold.
• Metric tensor — A smoothly varying nondegenerate bilinear form on tangent spaces that determines lengths, angles, volumes and causal or geodesic structure on a manifold.
• One-form — A smooth covector field assigning a linear functional on each tangent space of a differentiable manifold.
• Osculating plane — The plane through a space curve point spanned by its tangent and principal normal, giving second-order local contact when curvature is nonzero.
• Parabolic geometry (differential geometry) — A Cartan geometry modeled on a homogeneous quotient G/P of a semisimple Lie group by a parabolic subgroup, unifying conformal, projective and related structures.
• Parabolic line — The curve on a smooth surface where Gaussian curvature is zero and that generically separates elliptic from hyperbolic regions.
• Riemannian manifold — A smooth manifold equipped at every point with a smoothly varying positive-definite inner product on its tangent space.
• Spinc structure — A lift of an oriented manifold’s frame bundle to the group Spin-c, generalizing spin structure by coupling spinors to a complex line bundle.
• Symplectic spinor bundle — The infinite-rank Hilbert bundle associated to a metaplectic structure on a symplectic manifold through the metaplectic representation.
• Tangent bundle — The geometric bundle formed by assembling every tangent space of a smooth manifold into one smooth total space over that manifold.
• Tangent indicatrix — The curve traced on the unit sphere by the unit tangent vector of a regular space curve.
• Weakly symmetric space — A complete Riemannian homogeneous space in which an isometry can exchange any chosen pair of points.
• Weitzenböck identity — An identity expressing one Laplace-type operator as a rough Laplacian plus a curvature-dependent lower-order term.
• Yang–Mills flow — The negative gradient flow of the Yang–Mills energy on connections, evolving curvature toward Yang–Mills critical connections.
• Yau's conjecture — The statement that every closed Riemannian three-manifold contains infinitely many smooth closed immersed minimal surfaces, now a theorem.
• Yau's conjecture on the first eigenvalue — The conjecture that every closed embedded minimal hypersurface of the unit sphere S to the n plus one has first Laplace–Beltrami eigenvalue n.