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Manifolds

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Abstract

Intuitively, a manifold is a generalization of curves and surfaces to higher dimensions. It is locally Euclidean in that every point has a neighborhood, called a chart, homeomorphic to an open subset of \(\mathbb{R}^{\it{n}}\). The coordinates on a chart allow one to carry out computations as though in a Euclidean space, so that many concepts from \(\mathbb{R}^{\it{n}}\), such as differentiability, point-derivations, tangent spaces, and differential forms, carry over to a manifold.

Keywords

  • Equivalence Relation
  • Open Subset
  • Topological Space
  • Smooth Manifold
  • Quotient Space

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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  • DOI: 10.1007/978-1-4419-7400-6_3
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Correspondence to Loring W. Tu .

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© 2011 Springer Science+Business Media, LLC

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Tu, L.W. (2011). Manifolds. In: An Introduction to Manifolds. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7400-6_3

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