Editing Atlas (topology) (section)

==More structure==
One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of [[differentiation (mathematics)|differentiation]] of functions on a manifold, then it is necessary to construct an atlas whose transition functions are [[Differentiable function|differentiable]]. Such a manifold is called [[Differentiable manifold|differentiable]]. Given a differentiable manifold, one can unambiguously define the notion of [[tangent vectors]] and then [[directional derivative]]s.

If each transition function is a [[smooth map]], then the atlas is called a [[smooth structure|smooth atlas]], and the manifold itself is called [[Differentiable manifold#Definition|smooth]]. Alternatively, one could require that the transition maps have only ''k'' continuous derivatives in which case the atlas is said to be <math> C^k </math>.

Very generally, if each transition function belongs to a [[pseudogroup]] <math> \mathcal G</math> of homeomorphisms of Euclidean space, then the atlas is called a <math>\mathcal G</math>-atlas.  If the transition maps between charts of an atlas preserve a [[local trivialization]], then the atlas defines the structure of a fibre bundle.