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 of functions on a manifold, then it is necessary to construct an atlas whose transition functions are differentiable. Such a manifold is called differentiable. Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.
If each transition function is a smooth map, then the atlas is called a smooth atlas, and the manifold itself is called smooth. Alternatively, one could require that the transition maps have only k continuous derivatives in which case the atlas is said to be .
Very generally, if each transition function belongs to a pseudo-group of homeomorphisms of Euclidean space, then the atlas is called a -atlas.
Read more about this topic: Atlas (topology)
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“Why does philosophy use concepts and why does faith use symbols if both try to express the same ultimate? The answer, of course, is that the relation to the ultimate is not the same in each case. The philosophical relation is in principle a detached description of the basic structure in which the ultimate manifests itself. The relation of faith is in principle an involved expression of concern about the meaning of the ultimate for the faithful.”
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