If surface has more than two dimensions, closed surfaces {cycle, topology} exist.
orientable
Closed surfaces can have orientation.
simply connected
Closed surfaces can have no singularities or infinities.
independence
For closed surface, chain can meet itself, so boundary equals zero. Cycles are dependent if chain boundary is not zero. Number {Betti number} of manifold-dimension independent cycles is invariant. In orientable n-dimensional cycles, dimension-p Betti number equals dimension n - p Betti number {duality theorem}.
torsion
Number {torsion coefficient} of times cycle {torsion cycle} must multiply to bound is invariant. Simply connected closed n-dimensional manifolds are homeomorphic to spheres with same Betti number and torsion coefficients. Spheres are simply connected closed two-dimensional manifolds.
Meshes or nets can be on surfaces. Scale can vary from point to point {Gaussian coordinate} {curvilinear coordinate}, in which case magnitude is not important. Locally, surface manifolds can have curvature zero, be flat, and use Pythagorean theorem to find distances, vectors, and metric.
Topology can have geometry {geometrization}, which has constant curvature.
Like heat flows from hot to cold and makes uniform temperature, curvature can flow to make constant curvature {Ricci flow equation}. However, Ricci flow allows singularities, with different starting geometries. For example, dumbbell shape tends to make two spheres with point between, rather than one large sphere. Thin rod tends to have singular point at one end {cigar singularity}. If sphere replaces singularity, Ricci flow can continue. Ricci flow can find possible shapes in compact spaces (Richard Hamilton) [1982].
Grigory Perelman [2003] used Ricci flow to show that all Ricci-flow-procedure singularity types can morph into spheres or tubes in finite time, so procedures can remove them from space.
Two theorems {Riemann-Roch theorem} {residual theorem} are about regular and irregular surfaces.
Connectivity and closedness combine to make surface property {genus, surface}|. Closed Riemann surfaces with same genus are topologically equivalent. Genus is invariant under birational transformation.
sphere
Sphere has genus 0. Genus-0 closed surfaces can map onto spheres and connect simply. Sphere projective planes have genus zero, are closed, and look like circles with semi-circumference line at infinity.
handles
Sphere with n handles has genus n. The one-sided-surface Klein's bottle has genus one, because it has one handle.
holes
Sphere with n holes has genus n. Hole number equals function branch-point number divided by two, minus function-value number, plus one. For curves, genus equals 0.5 * (n - 1) * (n - 2) - d, where d equals double-point number and n equals function degree. Riemann surface corresponding to genus-p curve has connectivity 2*p + 1.
Number of possible closed curves {connectivity, topology}| can differ. Connectivity is number of closed surfaces that do not make disjoint regions. Spheres have one loop type. Toruses can have loops around and loops across. Closed curves can follow surface in different ways. Loops {loop out} completely bound closed surface. Connectivity is a global surface property.
Surface-topology indexes {homotopy, topology}| can measure how many ways a closed curve can be in a surface. First way is that loop can become point, as on sphere. For other topologies, loop cannot become point. Second way is that loop can become circle, as across torus. Loops and mathematical groups are homotopic. Functions or structures have symmetries.
For all genuses, using parameters can make multiple-valued functions into single-valued functions {uniform function} {uniformization problem} {uniformization theorem}. For genus zero {unicursal curve, genus}, parameterized function f(w,z) can equal zero. For genus equal one {bicursal curve}, parameterized function f(w,z) can equal zero.
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].
Topology can define direction {orientation, topology}. To turn bounded surfaces inside out, project at right angles through plane defined by surface boundary line, to make projected-figure points same distance from plane as original bounded surface {inside out}. Turning figures with surface arrows inside out reverses arrow direction.
After one loop, vector direction is either same or opposite {orientability}|. Orientability is a global property.
On triangulated surfaces {orientable surface}, triangles can orient so sides common to two triangles orient oppositely on two triangles. Spheres are orientable surfaces. Projective planes are not orientable surfaces. If and only if two orientable closed surfaces have same genus, surfaces are homeomorphic.
If arrow goes around Klein bottle or Möbius strip once, arrow reverses direction {version, topology}|, the same as turning figure inside out.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225