Curves or curved surfaces {manifold, topology}| can have Euclidean-space neighborhoods at all points, and neighborhoods connect continuously at overlapping open regions. Point or element sets are continuous functions specifiable by coordinates. Manifolds are generalized Riemann surfaces.
examples
Planes, spheres, and toruses are two-dimensional manifolds.
topological equivalence
Two two-dimensional manifolds can be topologically equivalent. For three-dimensional manifolds, the proof does not yet exist that any two manifolds can be topologically equivalent. For four-dimensional manifolds, no algorithm can prove that any two manifolds can be topologically equivalent.
metric
Riemannian manifolds can have a metric.
boundary
Manifolds with boundaries have neighborhood part on boundary.
Hausdorff space
All points can have open sets that do not intersect, so there is no branching {Hausdorff space}. Two distinct points have open sets that do not intersect other open set.
Volume depends on line element raised to power {Hausdorff dimension} and is not necessarily an integer.
fields: scalar
Manifolds can have smooth coordinate functions, making scalar fields. Manifolds are commutative scalar-field algebras.
fields: vector
Manifolds can have differentiation operator on scalar field, making vector field.
fields: covector field
Manifolds have symmetrical duals {1-form, topology} to vector fields {covector field}. Covector space is an n - 1 dimension hyperplane.
Vector spaces have tangents at points. Tangents have duals {covector space}.
p 1-form intersections make dimension n - p hyperplanes {p-form} {simple p-form}. p-forms are integrable {exterior calculus, manifold} to find density or gradient. Exterior derivative gradient is covector space.
Vector-field tensor differentiation depends on tangent vectors {covariant derivative operator} {connection, derivative}.
curvature
Curvature tensor measures vector change after parallel transport around loop.
torsion
If there is no torsion, curvature tensor is zero {first Bianchi identity, torsion} {Bianchi symmetry, torsion}. If there is no torsion, curvature-tensor derivative is zero {Bianchi identity, torsion} {second Bianchi identity, torsion}. Torsionless connections {Riemannian connection} {Christoffel connection} {Levi-Civita connection} can preserve vector length during parallel transport.
Non-metric spaces {2-manifold} can have two dimensions. Three geometrized 2-manifolds exist, depending on curvature. The simplest compact 2-manifold is a sphere with positive curvature. Torus is a 2-manifold with zero curvature. 2-manifolds can have negative curvature when they have two or more handles.
Non-metric spaces {3-manifold} can have three dimensions. 3-manifolds are possible three-dimensional topologies. The simplest compact 3-manifold is a 3-sphere. Other 3-manifolds have edges or have multiple paths from one region to another. People know all 3-manifold types. Manifolds can have orientation or not.
Eight geometries can make geometrized 3-manifolds {Thurston geometrization conjecture}: positive curvature, negative curvature, flat curvature, and five 2-sphere combinations with curvatures.
3-spheres are unique 3-manifolds {Poincaré conjecture} [1904]. If all loops in three-dimensional compact closed space can shrink to points, without breaking loop or space, space is 3-sphere, and no other such 3-space exists. 4-spheres are unique 4-manifolds, as proved by Michael Freedman [1983]. 5-spheres are unique 5-manifolds, and all higher dimensional spheres are unique manifolds, as proved by Stephen Smale [1960].
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225