Abstract
This chapter introduces a powerful tool in smooth manifold theory, Sard’s theorem, which says that the set of critical values of a smooth function has measure zero. After proving the theorem, we use it to prove three important results about smooth manifolds. The first result is the Whitney embedding theorem, which says that every smooth manifold can be smoothly embedded in some Euclidean space. (This justifies our habit of visualizing manifolds as subsets of ℝn.) The second result is the Whitney approximation theorem, which comes in two versions: every continuous real-valued or vector-valued function can be uniformly approximated by smooth ones, and every continuous map between smooth manifolds is homotopic to a smooth map. The third result is the transversality homotopy theorem, which says, among other things, that embedded submanifolds can always be deformed slightly so that they intersect “nicely” in a certain sense that we will make precise.
Keywords
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Osborn, Howard: Vector Bundles, vol. 1. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1982)
Rudin, Walter: Principles of Mathematical Analysis, 3rd edn. McGraw–Hill, New York (1976)
Wall, C.T.C.: All 3-manifolds imbed in 5-space. Bull. Am. Math. Soc. (N.S.) 71, 564–567 (1965)
Whitney, Hassler: Differentiable manifolds. Ann. Math. 37, 645–680 (1936)
Whitney, Hassler: The self-intersections of a smooth n-manifold in 2n-space. Ann. Math. 45, 220–246 (1944)
Whitney, Hassler: The singularities of a smooth n-manifold in (2n−1)-space. Ann. Math. 45, 247–293 (1944)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Lee, J.M. (2013). Sard’s Theorem. In: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol 218. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9982-5_6
Download citation
DOI: https://doi.org/10.1007/978-1-4419-9982-5_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9981-8
Online ISBN: 978-1-4419-9982-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)