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Erschienen in:
2014 | OriginalPaper | Buchkapitel
1. Differentiable Manifolds
verfasst von : Leonor Godinho, José Natário
Erschienen in: An Introduction to Riemannian Geometry
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Abstract
In pure and applied mathematics, one often encounters spaces that locally look like \(\mathbb {R}^n\), in the sense that they can be locally parameterized by \(n\) coordinates: for example, the \(n\)-dimensional sphere \(S^n \subset \mathbb {R}^{n+1}\), or the set \(\mathbb {R}^3 \times SO(3)\) of configurations of a rigid body. It may be expected that the basic tools of calculus can still be used in such spaces; however, since there is, in general, no canonical choice of local coordinates, special care must be taken when discussing concepts such as derivatives or integrals whose definitions in \(\mathbb {R}^n\) rely on the preferred Cartesian coordinates. The precise definition of these spaces, called differentiable manifolds, and the associated notions of differentiation, are the subject of this chapter.