differential structure

plural differential structures · alternative differentiable structure

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

   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.

   2010, Vladimir Igorevich Bogachev, Differentiable Measures and the Malliavin Calculus, American Mathematical Society, page 369

   3 more 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).

   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 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;.

   2002, Donal J. Hurley, Michael A. Vandyck, Topics in Differential Geometry: A New Approach Using D-Differentiation, Springer (with Praxis Publishing), page 29

   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.

   2015, Stephen Bruce Sontz, Principal Bundles: The Classical Case, Springer, page 12