differential structure

Noun

differential structure (plural differential structures)

1. A structure defined for a (topological) manifold so that it supports differentiation of functions defined on it.
   • 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...2002, R.W. Carroll, Calculus Revisited, Springer, pages 12–37
   • 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...2002, Donal J. Hurley, Michael A. Vandyck, Topics in Differential Geometry: A New Approach Using D-Differentiation, Springer (with Praxis Publishing), page 29