Abstract
This chapter is an introduction to the wide subject of differentiable manifolds. A differentiable manifold, which generalizes the ordinary definitions of curves and surfaces, is presented here from an intrinsic point of view, that is, without referring to an ambient space in which it is embedded. Differentiable functions, tangent vectors, tensor fields, and r-forms are defined. Further, differential and codifferential of a map between manifolds are studied. In order to introduce metric evaluation on a manifold, the Riemann manifolds are introduced and the geodesics are defined. Many interesting exercises conclude the chapter.
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Notes
- 1.
A topological space X is a Hausdorff space if, \(\forall x,\, y\,\in \, X, x \ne y\), there are neighborhoods U, V of x, y, respectively, such that \(U\cap V\,=\,\emptyset \).
- 2.
A Hausdorff space is paracompact if every open covering contains a subcovering that is locally finite.
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© 2018 Springer International Publishing AG, part of Springer Nature
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Romano, A., Marasco, A. (2018). Differentiable Manifolds. In: Classical Mechanics with Mathematica®. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-77595-1_6
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DOI: https://doi.org/10.1007/978-3-319-77595-1_6
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-77594-4
Online ISBN: 978-3-319-77595-1
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