Riemannian & Symplectic Geometry

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Abstractions about curvature, symmetry, and structure in differential geometry, including manifolds defined by curvature conditions (Hadamard manifold, isoparametric manifold, complex hyperbolic space), minimal-surface constructions and conjectures (Björling problem, K-noid, Yau's conjecture), and symplectic or spinorial constructions (symplectization, symplectic spinor bundle, Kosmann lift).

16 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 Flat Manifold — A connected closed smooth manifold that admits metrics with absolute sectional-curvature bound times diameter squared below every positive threshold.
• Björling problem — The analytic minimal-surface problem of constructing a minimal surface through a prescribed real-analytic space curve with a prescribed compatible normal field.
• 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).
• 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.
• 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.
• Isoparametric manifold — Define a Euclidean submanifold with flat normal bundle whose shape operators have constant eigenvalues along every parallel normal field.
• K-noid — A genus-zero minimal surface with k catenoidal ends, topologically a sphere punctured at k points.
• 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.
• 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.
• 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.
• 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.
• 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.