Manifolds (n-Dimensional Manifold (n-Manifold)

is a topological space that locally resembles Euclidean space near each point
More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has an open neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space

Manifolds - Introduction

An n-dimensional topological manifold 𝑀 is a topological Hausdorff space with a countable base that is locally homeomorphic to ℝ^𝑛. This means that for every point 𝑝 in 𝑀, there is an open neighborhood 𝑈 of 𝑝 and a homeomorphism 𝜑: 𝑈 → 𝑉 which maps the set 𝑈 onto an open set 𝑉⊂ℝ^𝑛. Additionally:

The mapping 𝜑: 𝑈→𝑉 is called a chart or coordinate system
The set 𝑈 is the domain or local coordinate neighborhood of the chart
The image of the point 𝑝∈𝑈, denoted by 𝜑(𝑝)∈ℝ^𝑛, is called the coordinates or local coordinates of 𝑝 in the chart
A set of charts, {𝜑_𝛼|𝛼∈ℕ}, with domains 𝑈_𝛼, is called the atlas of 𝑀 if ⋃_𝛼∈ℕ𝑈_𝛼= 𝑀

For example

Two intersecting patches (green and purple with cyan/teal as the intersection) on a manifold with different charts (continuous 1-1 mappings) to ℝ^𝑛 Euclidean space. Notice that the intersection of the patches has a smooth 1-1 mapping in ℝ^𝑛 Euclidean space, making it a differential manifold

The mapping between the intersecting parts of 𝑈_𝛼 and 𝑈_𝛽 in their respective chart coordinates called a transition map, given by 𝜏_𝛼𝛽= 𝜑_𝛽∘𝜑_𝛼⁻¹ and 𝜏_𝛽𝛼 = 𝜑_𝛼∘𝜑_𝛽⁻¹ (their domain is restricted to either 𝜑_𝛼(𝑈_𝛼∩𝑈_𝛽) or 𝜑_𝛽(𝑈_𝛼∩𝑈_𝛽), respectively), ∘ denotes function composition.

These transition functions are important because depending on their differentiability, they define a new class of differentiable manifolds (denoted by 𝐶^𝑘 if they are 𝑘-times continuously differentiable). The most important one for our conversation is transition maps that are infinitely differentiable, which we call smooth manifolds.

Once we have smooth manifolds, we can do things like calculus. Performing analysis on a manifold embedded in a high-dimensional space could be a major pain in the butt, but analysis in a lower-dimensional Euclidean space is easy (relatively)!

Manifolds - Examples

Manifolds - Categorizations

Generic Families	Specific Dimensions	Special Types	Other Categories
Click here to expand... Euclidean space (ℝ^𝑛) n-sphere (𝑆^𝑛) n-torus (𝑇^𝑛) Real projective space (𝑅𝑃^𝑛) Complex projective space (𝐶𝑃^𝑛) Quaternionic projective space (𝐻𝑃^𝑛) Flag manifold Grassmann manifold Stiefel manifold	Click here to expand... 1-Manifolds Circle (𝑆¹ = 𝑇¹) Long line Real line (ℝ¹) Real projective line (𝑅𝑃¹≅ 𝑆¹) 2-Manifolds Cylinder (𝑆¹ × ℝ) Klein bottle (𝑅𝑃² # 𝑅𝑃²) Klein quartic (a genus 3 surface) Möbius strip Real projective plane (𝑅𝑃²) Sphere (𝑆²) Surface of genus g Torus (𝑇²) Double torus 3-Manifolds 3-sphere (𝑆³) 3-torus (𝑇³) Poincaré homology sphere SO(3) ≅ 𝑅𝑃³ Solid Klein bottle Solid torus Whitehead manifold Meyerhoff manifold Weeks manifold 4-Manifolds Complex projective plane Del Pezzo surface E₈ manifold Enriques surface Exotic ℝ⁴ Hirzebruch surface K3 surface	Click here to expand... Manifolds Related to Spheres Brieskorn manifold Exotic sphere Homology sphere Homotopy sphere Lens space Spherical 3-manifold Special classes of Riemannian Manifolds Einstein manifold Ricci-flat manifold G₂ manifold Kähler manifold Calabi–Yau manifold Hyperkähler manifold Quaternionic Kähler manifold Riemannian symmetric space Spin(7) manifold	Click here to expand... Manifolds definable by a particular choice of atlas Affine manifold Analytic manifold Complex manifold Differentiable (smooth) manifold Piecewise linear manifold Lipschitz manifold Topological manifold Manifolds with additional structure Almost complex manifold Almost symplectic manifold Calibrated manifold Complex manifold Contact manifold CR manifold Finsler manifold Hermitian manifold Hyperkähler manifold Kähler manifold Lie group Pseudo-Riemannian manifold Riemannian manifold Sasakian manifold Spin manifold Symplectic manifold Infinite-dimensional manifolds Banach manifold Fréchet manifold Hilbert manifold

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Resources

The Bright Side of Mathematics - Lecture Series

Manifolds (n-Dimensional Manifold (n-Manifold)

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Manifolds - Introduction

Manifolds - Examples

Manifolds - Categorizations

1-Manifolds

2-Manifolds

3-Manifolds

4-Manifolds

Manifolds Related to Spheres

Special classes of Riemannian Manifolds

Manifolds definable by a particular choice of atlas

Manifolds with additional structure

Infinite-dimensional manifolds

Subpages

Resources

Manifolds (n-Dimensional Manifold (n-Manifold)