Manifold Space

A Manifold Space is a topological space that is locally similar to Euclidean space and where each space point has a neighborhood that is homeomorphic to an open subset of an n-dimensional Euclidean space.

• Context:
  • It can (typically) be covered by a collection of coordinate charts.
  • It can (often) be classified by its dimension, which determines the local Euclidean space it resembles.
  • It can (often) be visualized with Manifold Shapes.
  • ...
  • It can range from being a Differentiable Manifold to a Topological Manifold.
  • ...
  • It can serve as the foundation for advanced studies in areas such as differential geometry, topology, and theoretical physics.
  • ...
• Example(s):
  • Differentiable Manifolds, such as:
    • Smooth surfaces in [math]\displaystyle{ \mathbb{R}^3 }[/math] (e.g., a Sphere or Torus)
    • The real number line [math]\displaystyle{ \mathbb{R} }[/math]
    • Tangent Bundles of smooth manifolds
    • Open subsets of [math]\displaystyle{ \mathbb{R}^n }[/math]
  • Topological Manifolds, such as:
    • The Möbius Strip
    • The Klein Bottle
    • Arbitrary-dimensional spheres [math]\displaystyle{ S^n }[/math]
    • Real projective spaces [math]\displaystyle{ \mathbb{RP}^n }[/math]
    • Topological Groups (e.g., the circle group [math]\displaystyle{ S^1 }[/math])
  • Complex Manifolds, such as:
    • Complex projective spaces [math]\displaystyle{ \mathbb{CP}^n }[/math]
    • Riemann Surfaces (e.g., complex tori, hyperelliptic curves)
    • Stein Manifolds (e.g., [math]\displaystyle{ \mathbb{C}^n }[/math], domains of holomorphy)
    • Complex Lie Groups (e.g., [math]\displaystyle{ \text{GL}(n,\mathbb{C}) }[/math], [math]\displaystyle{ \text{SL}(n,\mathbb{C}) }[/math])
    • Kähler Manifolds (e.g., compact Riemann surfaces, projective algebraic varieties)
  • Lie Groups (e.g., the group of rotations [math]\displaystyle{ \text{SO}(3) }[/math])
  • an Euclidean Space, which showcases the simplest example of a manifold.
  • a Projective Plane, which is a manifold that can be visualized as the set of all lines through the origin in three-dimensional space.
  • ...
• Counter-Example(s):
  • Discrete Spaces, which lack the local Euclidean neighborhood structure.
  • Singular Spaces, which have points where the local neighborhood is not homeomorphic to Euclidean space.
• See: Differentiable Manifold, Topological Manifold, Coordinate Chart, Differential Geometry, Topology, Manifold Learning, Neighbourhood (Mathematics), Homeomorphic, Line (Geometry), Circle, Lemniscate, Surface, Plane (Geometry), Dimensionality Compression Algorithm, Torus.

---

---

References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Manifold Retrieved:2015-2-6.
  • In mathematics, a 'manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

    Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane (in the context of manifolds they are called charts). When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map.

    The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds.

    This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.

---