Differential manifold

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Definition in terms of atlases

A differential manifold or smooth manifold is the following data:

  • A topological manifold M (in particular, M is Hausdorff and second-countable)
  • An atlas of coordinate charts \varphi_i:U_i \to V_i, i \in I from M to \R^n (in other words an open cover U_i of M with homeomorphisms from each member U_i of the open cover to open sets V_i in \R^n)

satisfying the compatibility condition: the transition function between any two coordinate charts of the atlas is a diffeomomorphism of open subsets of \R^n. In symbols:

\varphi_i \circ \varphi_j^{-1}

is a homeomorphism when restricted to the set:

(\varphi_j(U_i \cap U_j))

By diffeomorphism, we here mean a C^{\infty} map with a C^{\infty} inverse.

However, we need to quotient out this data by the following equivalence:

Two atlases of coordinate charts on a topological space define the same differential manifold structure if taking their union still gives an atlas of coordinate charts. In other words, given any coordinate chart in one atlas and any coordinate chart in the other atlas, the transition function between them is a diffeomorphism.

Definition in terms of sheaves

A differential manifold or smooth manifold is the following data:

Such that for every p \in M, there exists an open set U \ni p and a homeomorphism U \cong V where V is an open subset of \R^n, such that the sheaf restricted to U corresponds, via the homeomorphism, to the usual sheaf of infinitely differentiable functions on V.

Relation with other structures

Weaker structures