Abstract
The simplest nontrivial example of a differentiable manifold is a smooth two dimensional surface in three dimensional Euclidean space.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Riemann, B.: Über die Hypothesen, welchen der Geometrie zu Grunde liegen. Habilitations-schrift, 1854. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13. http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/Geom.pdf  [Trans.: On the Hypotheses which lie at the Bases of Geometry. http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.html. Still fascinating reading.]
Lee, J.M.: Introduction to Topological Manifolds, 2nd edn. Springer, New York (2010)
Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2002) [The two volumes of [7] provide a solid, comprehensive, compactly written text on all aspects of differential geometry. The references [8] and [2] are more accessible, and cover in particular all the subjects discussed in this chapter.]
Kobayashi, S., Nomizu, K.: Foundation of Differential Geometry I, II. Original Edition. Wiley 1963, 1969. Paperback Edition Wiley-Blackwell (2009)
Warner, F.W.: Foundation of Differentiable Manifolds and Lie Groups. Original edition: Scott, Foresman and Company 1971. Second printing, Springer (1983) [Less forbidding than [4]]. Covers a wide area in a fairly compact but clear way.
da Silva, A.C.: Lectures on Symplectic Geometry. Second corrected printing. Springer 2001. Revised edition 2006. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.146.4370&rep=rep1&type=pdf  [A broad review of symplectic geometry.]
Koszul, J.L.: Lectures on Fibre Bundles and Differential Geometry. Tata Institute of Fundamental Research. Bombay. http://www.math.tifr.res.in/~publ/ln/tifr20.pdf (1960) [A fundamental paper, not very well-known.]
Thomas, E.G.F.: Characterization of a manifold by the \(*\)-algebra of its \(C^{\infty }\) functions. Preprint Mathematical Institute, University of Groningen (Unpublished)
Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bongaarts, P. (2015). Manifolds. In: Quantum Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-09561-5_20
Download citation
DOI: https://doi.org/10.1007/978-3-319-09561-5_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09560-8
Online ISBN: 978-3-319-09561-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)