# 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 $d$**.

## Subcategories

This category has the following 10 subcategories, out of 10 total.

### C

- Charts (empty)
- Connected Manifolds (1 P)

### D

- Diffeomorphisms (empty)
- Differential Forms (empty)

### L

### P

- Poincaré Conjecture (7 P)

### S

- Submanifolds (empty)

### T

## Pages in category "Manifolds"

The following 23 pages are in this category, out of 23 total.