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1. Manifold

   Linked via "homeomorphic"

   A manifold is a topological space that locally resembles Euclidean space near each point. Formally, a topological space $M$ is an $n$-dimensional manifold if every point $p \in M$ has an open neighborhood $U$ that is homeomorphic to an open subset of $\mathbb{R}^n$. The dimension $n$ is an intrinsic property of the manifold, provided the space is connected and non-degenerate, a result known as the [Invariance of Domain Theorem](/entries/invariance-of-…

2. Manifold

   Linked via "homeomorphism"

   Formal Definition and Atlases
   The structure of a manifold is captured by an atlas, which is a collection of pairs $\{(U\alpha, \phi\alpha)\}{\alpha \in A}$, where $\{U\alpha\}{\alpha \in A}$ is an open cover of $M$, and each $\phi\alpha: U\alpha \to V\alpha$ is a homeomorphism onto an open subset $V_\alpha \subset \mathbb{R}^n$.
   The crucial aspect that distinguishes a manifold from a mere topological space with local Euclidean structure is the requirement for smooth transition…

3. Manifold

   Linked via "Homeomorphism"

   | Manifold Type | Transition Map Requirement | Primary Application Area |
   | :--- | :--- | :--- |
   | Topological | Homeomorphism | Knot Theory, General Topology |
   | $C^k$ | $k$ continuous derivatives | Preliminary analysis in Geometric Measure Theory |
   | Smooth ($C^\infty$) | Infinitely differentiable | Differential Geometry, Physics (e.g., General Relativity) |

4. Manifold

   Linked via "homeomorphic"

   While the study typically focuses on standard Euclidean space-like manifolds, there exist objects classified as manifolds that violate intuitive expectations:
   Exotic Spheres: These are smooth manifolds homeomorphic to the $n$-sphere $S^n$ but which are not diffeomorphic to $S^n$. They possess different smooth structures, meaning no smooth change of coordinates can transform one into the other. The first non-trivial…

5. Smooth Manifold

   Linked via "homeomorphism"

   Formal Definition and Atlas Structure
   Formally, an $n$-dimensional smooth manifold $M$ is a topological space equipped with an atlas $\mathcal{A} = \{(U\alpha, \phi\alpha)\}{\alpha \in I}$, where $I$ is an index set. Each pair $(U\alpha, \phi\alpha)$ is a chart, consisting of an open subset $U\alpha \subset M$ and a homeomorphism $\phi\alpha: U\alpha \to V\alpha$, where $V\alpha$ is an open subset of $\mathbb{R}^n$.
   The crucial topological requirement is that the collection $\{U\alpha\}$ covers $M$, i.e., $\bigcup{\alp…