Category:Manifolds

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 topological 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 $d$.