Category

Manifolds

page 1

thumb|upright=0.65|The Klein bottle immersed in three-dimensional space

type of non-orientable surface

group that is also a smooth manifold with group operations that are smooth

topological manifold

topological space (which may also be a separated space) which locally resembles real n-dimensional space

metrizable space

topological space that is homeomorphic to a metric space

collection of charts on a mathematical manifold

Poincaré duality

duality that relates homology and cohomology groups for oriented closed manifolds

in knot theory, a set of knots that do not intersect but are intertwined

3-manifold that is a quotient of S³ by ℤ/p actions: (z,w) ↦ (exp(2πi/p)z, exp(2πiq/p)w)

tubular neighborhood

neighborhood of a submanifold homeomorphic to that submanifold’s normal bundle

closed manifold

mathematical concept of a compact manifold without boundary

uniformization theorem

theorem that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere

geodesic curvature

mathematical measure in Riemannian geometry

thumb|160px|Immersed manifold straight line with self-intersections In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.

Whitehead manifold

open 3-manifold that is contractible, but not homeomorphic to R³

parallelizable manifold

a differentiable manifold whose (co)tangent bundle is topologically trivial

Hilbert manifold

manifold modeled on Hilbert spaces; separable Hausdorff space in which each point has a neighborhood homeomorphic to an infinite dimensional Hilbert space

global analysis

study of the global and topological properties of differential equations on manifolds and vector space bundles

prime decomposition of a 3-manifold

Decomposes compact, orientable 3-manifolds uniquely into finitely many prime 3-manifolds

Stiefel manifold

the manifold of all orthonormal k-frames in n-dimensional Euclidean space

configuration space

moduli space of n points on a space M; if M is a manifold, in general forms an orbifold

internal groupoid in the category of smooth manifolds

section of the trivial line bundle associated to the determinant representation of the frame bundle

stable manifold

portion of phase space of a dynamical system that exhibits stable motion