Titlebook: Introduction to Smooth Manifolds; John M. Lee Textbook 2012Latest edition Springer Science+Business Media New York 2012 Frobenius theorem.

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Smooth Maps,chapter we carry out that project. We begin by defining smooth real-valued and vector-valued functions, and then generalize this to smooth maps between manifolds. We then focus our attention for a while on the special case of ., which are bijective smooth maps with smooth inverses. If there is a dif

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Tangent Vectors,of as a sort of “linear model” for the manifold near the point. Motivated by the fact that vectors in ℝ. act on smooth functions by taking their directional derivatives, we define a tangent vector to a smooth manifold to be a linear map from the space of smooth functions on the manifold to ℝ that sa

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Submersions, Immersions, and Embeddings,erentials are surjective everywhere), . (whose differentials are injective everywhere), and . (injective smooth immersions that are also homeomorphisms onto their images). Smooth immersions and embeddings, as we will see in the next chapter, are essential ingredients in the theory of submanifolds, w

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Submanifolds,ituation is quite a bit more subtle than the analogous theory of topological subspaces. We begin by defining the most important type of smooth submanifolds, called ., which have the subspace topology inherited from their containing manifolds. Next, we introduce a more general kind of submanifolds, c

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,Sard’s Theorem,. After proving the theorem, we use it to prove three important results about smooth manifolds. The first result is the ., which says that every smooth manifold can be smoothly embedded in some Euclidean space. (This justifies our habit of visualizing manifolds as subsets of ℝ..) The second result i

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Lie Groups,ng many examples of interesting manifolds themselves, they are essential tools in the study of more general manifolds, primarily because of the role they play as groups of symmetries of other manifolds. We begin with the definition of Lie groups and some of the basic structures associated with them,

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Vector Fields,tain kinds of maps from the manifold to its tangent bundle. Then we introduce the . operation, which is a way of combining two smooth vector fields to obtain another. The most important application of Lie brackets is to Lie groups: the set of all smooth vector fields on a Lie group that are invarian