Meaning of differential structure | Babel Free

Definitions

A structure defined for a (topological) manifold so that it supports differentiation of functions defined on it.

Examples

“First let (M,#92;tau) be a topological space. The sheaf #92;mathfrak#123;G#125; of real continuous functions on (M,#92;tau) is said to be a differential structure on M if for any open set U#92;in#92;tau, any functions f#95;i#92;in#92;mathfrak#123;G#125;(U), and any w#92;inC#92;infty(#92;mathbb#123;R#125;ⁿ), the superposition w#92;circ(f#95;1,#92;dotsf#95;n)#92;in#92;mathfrak#123;G#125;(U).”

“It is important to emphasise that, among the various choices for #92;lambdaⁱ#95;#123;jk#125; and A#123;i#92;a#125;#95;#123;#92;j#92;b#125;, some are intrinsic to the differential structure of the manifold #92;mathcal#123;M#125;. In other words, among all the operators of D-differentiation, some arise from the differential structure of #92;mathcal#123;M#125;.[…]On the other hand, there exist operators of D-differentiation that do not follow from the differential structure of #92;mathcal#123;M#125;.”

“This chapter is concerned with differentiable measures on general measurable spaces and on measurable spaces equipped with certain differential structures enabling us to consider differentiations along vector fields.”

“Given a Hausdorff topological space M with differential structures #92;mathcal#123;A#125;#95;1 and #92;mathcal#123;A#125;#95;2 (these being maximal smooth atlases), we say that #92;mathcal#123;A#125;#95;1 and #92;mathcal#123;A#125;#95;1 are equivalent if there is a diffeomorphism #92;phi#58;(M,#92;mathcal#123;A#125;#95;1)#92;rightarrow(M,#92;mathcal#123;A#125;#95;2) from M with the first differential structure to M with the second differential structure. Note that #92;phi need not be the identity function.”

CEFR level