Geometry MOC

Manifold

In essence, an 𝑛-dimensional manifold is some kind of space that locally resembles 𝑛-dimensional real coördinate space. There are multiple levels to this concept. A fundamental role is played by charts gathered in an atlas which describe this resemblance, and transition maps between charts.

• A Topological manifold is the most basic kind of manifold, consisting of spaces that locally resemble ℝ𝑛 topologically. Every manifold is a topological one.
• A Differentiable manifold or 𝐶𝛼-manifold has globally defined differential structure, i.e. their transition maps are 𝛼-differentiable.
• A Holomorphic manifold is an analytic manifold which locally resembles ℂ𝑛 with holomorphic transition maps.

Often manifolds are required to be Hausdorff and paracompact.

Properties

• Product manifold

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