Category:Manifolds

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This category contains results about Manifolds in the context of Topology.
Definitions specific to this category can be found in Definitions/Manifolds.


Let $M$ be a Hausdorff second-countable locally Euclidean space of dimension $d$.


Then $M$ is a topolofical manifold of dimension $d$.


Differentiable Manifold

Let $M$ be a second-countable locally Euclidean space of dimension $d$.

Let $\mathscr F$ be a $d$-dimensional differentiable structure on $M$ of class $\CC^k$, where $k \ge 1$.


Then $\struct {M, \mathscr F}$ is a differentiable manifold of class $\CC^k$ and dimension $d$.


Smooth Manifold

Let $M$ be a second-countable locally Euclidean space of dimension $d$.

Let $\mathscr F$ be a smooth differentiable structure on $M$.


Then $\struct {M, \mathscr F}$ is called a smooth manifold of dimension $d$.


Complex Manifold

Let $M$ be a second-countable, complex locally Euclidean space of dimension $d$.

Let $\mathscr F$ be a complex analytic differentiable structure on $M$.


Then $\struct {M, \mathscr F}$ is called a complex manifold of dimension $d$.