Smooth Maps
The main reason for introducing smooth structures was to enable us to define smooth functions on manifolds and smooth maps between manifolds. In this chapter we carry out that project. We begin by defining smooth real-valued and vector-valued functions, a
- PDF / 452,486 Bytes
- 18 Pages / 439.37 x 666.142 pts Page_size
- 18 Downloads / 192 Views
Smooth Maps
The main reason for introducing smooth structures was to enable us to define smooth functions on manifolds and smooth maps between manifolds. In this 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 diffeomorphisms, which are bijective smooth maps with smooth inverses. If there is a diffeomorphism between two smooth manifolds, we say that they are diffeomorphic. The main objects of study in smooth manifold theory are properties that are invariant under diffeomorphisms. At the end of the chapter, we introduce a powerful tool for blending together locally defined smooth objects, called partitions of unity. They are used throughout smooth manifold theory for building global smooth objects out of local ones.
Smooth Functions and Smooth Maps Although the terms function and map are technically synonymous, in studying smooth manifolds it is often convenient to make a slight distinction between them. Throughout this book we generally reserve the term function for a map whose codomain is R (a real-valued function) or Rk for some k > 1 (a vector-valued function). Either of the words map or mapping can mean any type of map, such as a map between arbitrary manifolds.
Smooth Functions on Manifolds Suppose M is a smooth n-manifold, k is a nonnegative integer, and f W M ! Rk is any function. We say that f is a smooth function if for every p 2 M; there exists a smooth chart .U; '/ for M whose domain contains p and such that the composite function f ı ' 1 is smooth on the open subset Uy D '.U / Rn (Fig. 2.1). If M is a smooth manifold with boundary, the definition is exactly the same, except that J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, DOI 10.1007/978-1-4419-9982-5_2, © Springer Science+Business Media New York 2013
32
Smooth Functions and Smooth Maps
33
Fig. 2.1 Definition of smooth functions
'.U / is now an open subset of either Rn or Hn , and in the latter case we interpret smoothness of f ı ' 1 to mean that each point of '.U / has a neighborhood (in Rn ) on which f ı ' 1 extends to a smooth function in the ordinary sense. The most important special case is that of smooth real-valued functions f W M ! R; the set of all such functions is denoted by C 1 .M /. Because sums and constant multiples of smooth functions are smooth, C 1 .M / is a vector space over R. I Exercise 2.1. Let M be a smooth manifold with or without boundary. Show that pointwise multiplication turns C 1 .M / into a commutative ring and a commutative and associative algebra over R. (See Appendix B, p. 624, for the definition of an algebra.) I Exercise 2.2. Let U be an open submanifold of Rn with its standard smooth manifold structure. Show that a function f W U ! Rk is smooth in the sense just defined if and only if it is smooth in the sense of ordinary calculus. Do the same for an open submanifold with boundary in Hn (see Exe
Data Loading...
