differential structure

English

Alternative forms

• differentiable structure

Noun

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

   • 2002, Donal J. Hurley, Michael A. Vandyck, Topics in Differential Geometry: A New Approach Using D-Differentiation‎^[1], Springer (with Praxis Publishing), page 29:

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

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

Usage notes

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.

See also

• differentiable manifold
• differential manifold
• smooth manifold