differential structure

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differential structure

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• 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, pages 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