Differential Geometry & Manifolds¶

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Abstractions about smooth and geometric structure on manifolds, including tangent and tensor fields, differential forms, connections, curvature, geodesics, foliations, and symplectic constructions. They describe how local coordinates and derivatives assemble into global geometry.

53 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

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.
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.
Collapsing manifold — A Riemannian manifold or sequence whose metric geometry approaches a lower-dimensional limit while specified curvature or diameter controls are retained.
Complex differential form — A differential form whose coefficients take complex values, often decomposed into holomorphic and antiholomorphic bidegrees.
Courant bracket — An antisymmetric differential-geometric bracket on sections of a tangent-plus-cotangent bundle that extends the Lie bracket and has an exact-form Jacobiator.
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.
Curvilinear coordinates — A locally invertible coordinate system on Euclidean space or a manifold whose coordinate curves or surfaces may be curved, with geometry represented through basis variation, metric coefficients and a Jacobian.
Darboux vector — The instantaneous angular-velocity vector of a moving orthonormal frame along a space curve, combining curvature and torsion.
Ddbar lemma — A complex-geometric result stating, under its standard hypotheses, that a differential form closed under both ∂ and ∂̄ and exact for d is also ∂∂̄-exact.
Dehn surgery — A 3-manifold construction that removes tubular neighborhoods of link components and glues solid tori back along specified boundary slopes.
Differentiable curve — A parametrized path in a manifold or Euclidean space whose coordinate representation has the declared degree of differentiability.
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.
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.
Equivariant differential form — A group-equivariant polynomial map from a Lie algebra to differential forms on a manifold, representing a cochain in the Cartan model of equivariant cohomology.
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.
Geodesic — A geometry-relative straight path whose tangent transports parallel to itself, yielding locally length-minimizing curves for a Riemannian metric while permitting broader affine and spacetime forms.
Geodesic convexity — The extension of convex sets and functions to curved spaces by replacing straight line segments with minimizing geodesics.
Geometric quantization — A construction that seeks a quantum Hilbert space and observables from a classical symplectic phase space while preserving its geometric structures.
Hadamard manifold — A complete, simply connected Riemannian manifold with everywhere nonpositive sectional curvature.
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.
Holomorphic tangent bundle — The complex vector bundle of type-(1,0) tangent directions on a complex manifold, with holomorphic transition functions.
Horocycle — A curve in the hyperbolic plane orthogonal to geodesics converging to one ideal boundary point, equivalently a limiting circle tangent to the boundary at that point.
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.
Large deformation diffeomorphic metric mapping — A computational-anatomy framework that registers shapes or dense images through smooth invertible flows generated by a metric on a diffeomorphism group.
Mathai–Quillen formalism — Represent a vector bundle's Thom class by a canonical Gaussian differential form built with a connection and curvature, linking cohomological localization, superconnections and topological quantum field theory.
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.
Nilmanifold — A homogeneous manifold represented as a quotient of a nilpotent Lie group by a closed subgroup, with compact nilmanifolds commonly arising from cocompact discrete lattices.
One-form — A smooth covector field assigning a linear functional on each tangent space of a differentiable manifold.
Orthogonal coordinates — A curvilinear coordinate system whose coordinate curves or hypersurfaces meet mutually at right angles, making the metric tensor diagonal in the coordinate basis.
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.
Quadratic differential — A section of the square of a Riemann surface’s holomorphic cotangent bundle, locally written as a coefficient times the square of a coordinate differential.
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.
Stable manifold — The invariant manifold consisting locally or globally of states whose forward trajectories converge to a hyperbolic fixed point or invariant set, tangent to its stable eigenspace.
Submersion (mathematics) — A smooth map between manifolds whose differential is surjective at every point, making each target direction locally attainable.
Symplectic spinor bundle — The infinite-rank Hilbert bundle associated to a metaplectic structure on a symplectic manifold through the metaplectic representation.
Symplectization — The canonical construction that associates a symplectic manifold to a contact manifold by adjoining a nonzero scale coordinate to its contact covectors.
Tacnode — A plane-curve double singularity where two smooth local branches share the same tangent, canonically modeled by y²=x⁴ and carrying higher contact than an ordinary node.
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.
Tensor field — A smoothly or otherwise regularly varying assignment of a tensor of fixed type to every point of a manifold or region.
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 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.