differential structure

• differentiable structure

differential structure (plural differential structures)

1. (topology) A structure defined for a (topological) manifold so that it supports differentiation of functions defined on it.
   • 2002, R.W. Carroll, Calculus Revisited, Springer, page 12-37,

     First let ${\displaystyle (M,\tau )}$ be a topological space. The sheaf ${\displaystyle {\mathfrak {G}}}$ of real continuous functions on ${\displaystyle (M,\tau )}$ is said to be a differential structure on ${\displaystyle M}$ if for any open set ${\displaystyle U\in \tau }$, any functions ${\displaystyle f_{i}\in {\mathfrak {G}}(U)}$, and any ${\displaystyle w\in C^{\infty }(\mathbb {R} ^{n})}$, the superposition ${\displaystyle w\circ (f_{1},\dots f_{n})\in {\mathfrak {G}}(U)}$.

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

     It is important to emphasise that, among the various choices for ${\displaystyle \lambda _{jk}^{i}}$ and ${\displaystyle A_{\ j\ b}^{i\ a}}$, some are intrinsic to the differential structure of the manifold ${\displaystyle {\mathcal {M}}}$. In other words, among all the operators of ${\displaystyle D}$-differentiation, some arise from the differential structure of ${\displaystyle {\mathcal {M}}}$. […] On the other hand, there exist operators of ${\displaystyle D}$-differentiation that do not follow from the differential structure of ${\displaystyle {\mathcal {M}}}$.

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

     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.

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

     Given a Hausdorff topological space ${\displaystyle M}$ with differential structures ${\displaystyle {\mathcal {A}}_{1}}$ and ${\displaystyle {\mathcal {A}}_{2}}$ (these being maximal smooth atlases), we say that ${\displaystyle {\mathcal {A}}_{1}}$ and ${\displaystyle {\mathcal {A}}_{1}}$ are equivalent if there is a diffeomorphism ${\displaystyle \phi$ :(M,{\mathcal {A}}_{1})\rightarrow (M,{\mathcal {A}}_{2})} from ${\displaystyle M}$ with the first differential structure to ${\displaystyle M}$ with the second differential structure. Note that ${\displaystyle \phi }$ need not be the identity function.

For a given natural number n and some k, which may be a non-negative integer or infinity, we speak of an n-dimensional C^k differential structure.

• differentiable manifold
• differential manifold
• smooth manifold