Metric Geometry & Curvature

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Abstractions about distance structures and curved-space geometry, spanning metric-space foundations (complete field, formal ball, injective metric space), curvature and convexity notions (geodesic convexity, Hadamard space, nonpositive-curvature CAT(0) spaces), and geometric measure objects (varifold, collapsing manifold, relative convex hull).

14 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.

• Collapsing manifold — A Riemannian manifold or sequence whose metric geometry approaches a lower-dimensional limit while specified curvature or diameter controls are retained.
• Complete field — Equip a field with a compatible absolute value or metric and require every Cauchy sequence to converge within the field, making limits available without leaving its algebraic carrier.
• Conjugate index — For a Banach space, the largest exponent for which its dual is guaranteed to have the corresponding finite cotype, expressed through Hölder-conjugate type and cotype behavior under the governing convention.
• Effective Polish space — A complete separable metric space supplied with a computable dense presentation that makes basic distance comparisons effectively decidable or enumerable.
• Formal ball — An ordered pair of a metric-space point and a nonnegative radius, ordered so one pair computationally approximates another; generalized formal balls allow negative radii.
• Geodesic convexity — The extension of convex sets and functions to curved spaces by replacing straight line segments with minimizing geodesics.
• Hadamard space — A complete geodesic metric space of nonpositive curvature in the CAT(0) sense, equivalently satisfying a strong midpoint convexity inequality.
• Hilbert metric — A projectively invariant metric on the interior of a bounded convex domain, defined by a logarithmic cross ratio of the two boundary intersections on the line through a point pair.
• Injective metric space — A metric space into which every nonexpansive map from a subspace extends nonexpansively over the containing space, equivalently a hyperconvex space.
• Korn's inequality — A rigidity inequality bounding the full gradient of a vector field, modulo rigid motions, by its symmetric gradient under specified domain and boundary conditions.
• Motion (geometry) — Transform a metric space by a surjective distance-preserving map, with Euclidean motions further classified by orientation, fixed-point structure, and decomposition into translations, rotations, reflections, and glide operations.
• Polyhedral space — A metric space assembled by gluing constant-curvature simplices isometrically along compatible faces.
• Relative convex hull — The smallest geodesically convex set containing given points while constrained to remain inside a surrounding polygon or simple closed region.
• Varifold — A Radon measure on position–tangent-plane space representing a generalized surface with mass and orientation-free tangent information.