---- > [!definition] Definition. ([[smooth manifold]]) > A **smooth manifold** is a pair $(M, \mathscr{A})$, where $M$ is a [[topological manifold]] and $\mathscr{A}$ is a [[smooth structure]] on $M$. When $\mathscr{A}$ is understood, its mention is omitted and one just says "$M$ is a smooth manifold". > > Smooth manifolds are objects of [[category]] $\mathsf{Man}^{\infty}$. The homset $\text{Hom}(M,N)$ is the set of [[smooth maps between manifolds|smooth maps]] $M \to N$. > > Sometimes one prefers instead to work with functions of class $C^{r}$, $r \geq 1$. In this case $M$ is said to be a **differentiable manifold of class $C^{r}$**. It turns out the picture does not change too much from the smooth case, and so usually smooth manifolds are all we work with. ^definition - [ ] [[Sheaf]] version (from Eisenbud and Harris): ![[CleanShot 2025-03-11 at [email protected]]] ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```