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)
Famous quotes containing the word structure:
“... the structure of our public morality crashed to earth. Above its grave a tombstone read, “Be tolerant—even of evil.” Logically the next step would be to say to our commonwealth’s criminals, “I disagree that it’s all right to rob and murder, but naturally I respect your opinion.” Tolerance is only complacence when it makes no distinction between right and wrong.”
—Sarah Patton Boyle, U.S. civil rights activist and author. The Desegregated Heart, part 2, ch. 2 (1962)
“I’m a Sunday School teacher, and I’ve always known that the structure of law is founded on the Christian ethic that you shall love the Lord your God and your neighbor as yourself—a very high and perfect standard. We all know the fallibility of man, and the contentions in society, as described by Reinhold Niebuhr and many others, don’t permit us to achieve perfection.”
—Jimmy Carter (James Earl Carter, Jr.)