differentiable manifold

English

Alternative forms

• differential manifold

Noun

differentiable manifold (plural differentiable manifolds)

1. (differential geometry) A manifold that is locally similar enough to a Euclidean space (ℝⁿ) to allow one to do calculus;
   (more formally) a manifold that can be equipped with a differentiable structure (an atlas of ℝⁿ-compatible charts).

   The charts (homeomorphisms) that make up any differentiable structure of a differentiable manifold are required to be such that the transition from one to another is differentiable.

   • 1950, L. S. Pontryagin, Characteristic Cycles on Differentiable Manifolds, American Mathematical Society, page 33:

     Whitney showed that a differentiable manifold $M^{k}$ can be regularly mapped on the vector space $R^{2k}$.

   • 1994, N. Aoki, K. Hiraide, Topological Theory of Dynamical Systems: Recent Advances‎^[1], Elsevier (North-Holland), page 1:

     The phase space of a dynamical system will be taken to be a differentiable manifold.

   • 2009, Jeffrey Marc Lee, Manifolds and Differential Geometry, American Mathematical Society, page 1:

     In this chapter we introduce differentiable manifolds and smooth maps. A differentiable manifold' is a topological space on which there are defined coordinates allowing basic notions of differentiability.

Usage notes

• Differentiable manifolds are the principal subject of study in differential geometry.

Translations

manifold locally similar enough to a Euclidean space for calculus to be done

• French: variété différentiable (fr) f, variété différentielle (fr) f
• German: differenzierbare Mannigfaltigkeit f
• Italian: varietà differenziabile f

Further reading

• Atlas (topology) on Wikipedia.Wikipedia
• Affine connection on Wikipedia.Wikipedia
• Riemannian geometry on Wikipedia.Wikipedia