Manifolds, Tensors & Curvature

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Abstractions about the differential-geometric machinery for describing manifolds and objects on them, including tensor and index notation, tangent bundles, connections and covariant derivatives, curvature and space curves, and coordinate systems such as curvilinear and orthogonal frames.

48 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.

• Abstract index notation — A basis-free tensor notation in which index letters name argument slots and variance rather than numerical components.
• 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.
• Covariant transformation — A component-transformation rule in which lower-index tensor components change with the inverse-coordinate or dual-basis matrix so the underlying geometric object and contractions remain invariant.
• Curl (mathematics) — The vector differential operator measuring the local infinitesimal circulation and rotation axis of a three-dimensional vector field.
• 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.
• Del — The vector differential operator whose combinations with scalar or vector fields denote gradient, divergence and curl.
• 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.
• Differential operator — An operator built from derivatives and coefficient functions that maps functions or sections to new functions or sections according to a declared finite order.
• Differentiation of trigonometric functions — The calculus rule family that maps trigonometric functions and their compositions to derivatives through their periodic identities, limit behavior and the chain rule.
• 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.
• Einstein manifold — A Riemannian or pseudo-Riemannian manifold whose Ricci curvature tensor is everywhere a scalar multiple of its metric.
• Elliptic operator — A differential operator whose principal symbol is invertible away from the zero covector, excluding real characteristic directions and supporting strong regularity for its solutions.
• 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.
• 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.
• Inflection point — A point on a sufficiently smooth curve where signed curvature changes sign, or on a function graph where local concavity changes from one side to the other.
• Integral curve — A parametrized curve whose tangent at every point equals a specified vector field, representing a solution trajectory of an ordinary differential equation.
• Inverse problem for Lagrangian mechanics — The problem of determining whether a given system of differential equations is equivalent to Euler–Lagrange equations for some Lagrangian and, if so, constructing one.
• Inversion transformation — A conformal coordinate transformation that maps a nonzero point to a reciprocal radial position and, with translations and rotations, extends Poincaré symmetry toward the conformal group.
• 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.
• 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.
• Nijenhuis–Richardson bracket — A graded Lie bracket on alternating vector-valued multilinear forms, defined by antisymmetrized insertion and used to encode Lie algebra structures and their deformations.
• 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.
• Painlevé transcendents — New special functions defined by the six canonical nonlinear second-order Painlevé equations, whose movable singularities are poles rather than movable branch points.
• Penrose graphical notation — A diagrammatic tensor notation in which shapes, lines and contractions visually encode multilinear maps, indices and composition.
• 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.
• Scalar field — A function assigning one scalar quantity to every point of a space or spacetime region, invariant under coordinate changes appropriate to a scalar.
• Sobolev spaces for planar domains — A Hilbert-space framework that encodes weak derivatives and boundary traces to formulate elliptic boundary-value and eigenvalue problems on bounded planar domains.
• Special conformal transformation — A conformal map obtained by composing inversion, translation, and inversion, represented in Euclidean or Minkowski coordinates by a characteristic fractional transformation.
• 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.
• Superquadrics — A parameterized family of three-dimensional shapes that generalizes quadrics by replacing squared coordinate terms with adjustable powers, producing rounded, boxlike or pinched forms.
• Surface integral — An integral of a scalar or vector field over a parameterized surface, combining local field values with induced area or oriented flux elements.
• 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.
• Torus action — An algebraic or smooth group action of a torus on a variety or manifold, organizing points into orbits and exposing weights, fixed points, quotients, and combinatorial structure.
• Two-point tensor — A tensor-like linear map whose indices transform in two different vector spaces, commonly connecting a material reference configuration with a current spatial configuration.
• Weakly symmetric space — A complete Riemannian homogeneous space in which an isometry can exchange any chosen pair of points.