differentiable manifold

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differentiable manifold

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• differential manifold

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 ${\displaystyle M^{k}}$ can be regularly mapped on the vector space ${\displaystyle R^{2k}}$.

   • 1994, N. Aoki, K. Hiraide, Topological Theory of Dynamical Systems: Recent Advances‎, 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.

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

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