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CHAPTER 2 MANIFOLDS in This Chapter, We Address the Basic Notions

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CHAPTER 2 MANIFOLDS In this chapter, we address the basic notions: What is a manifold and what is a map between manifolds. Several examples are given. An n dimensional manifold is a topological space that appears to be Rn near a point, i.e., locally like Rn. Since these topological spaces appear to be locally like Rn, we may hope to develop tools similar to those used to study Rn in order to study manifolds. The term manifold comes from \many fold," and it refers to the many dimensions of space that a manifold may describe. Later in this section we will also need structure to discuss C 1 functions on manifolds. Our notion of smooth will be C1, i.e., continuous partial derivatives of all orders. The terms C1 and smooth are usually synonymous, however, in this chapter and the next, we will use C1 to describe maps between real vector spaces (as in advanced Calculus) and smooth for maps between manifolds. We use this distinction since many confusing compositions occur and the reader is assumed to be familiar with C 1 maps between real vector spaces while results on smooth maps between manifolds must be proven. We begin with the notion of a topological manifold. Definition 2.1***. A topological manifold of dimension n is a second countable Haus- dorff space M for which each point has a neighborhood homeomorphic to an open set in Rn. This notion appears to capture the topological ideal of locally looking like Rn, but in order to do Calculus we will need more structure. The term n-manifold is usually written for n dimensional manifold, and the dimension n is often suppressed. Definition 2.2***. Let U ⊂ M be a connected open set in a topological n-manifold M, and φ : U ! Rn be a homeomorphism to its image φ(U), an open set in Rn. n The pair (U; φ) is called a coordinate system or chart. If x0 2 U and φ(x0) = 0 2 R , then the coordinate system is centered at x0. Call the map φ a coordinate map. Definition 2.3***. Suppose M is a topological manifold. For k 2 N; k = 0, or k = 1, k a C atlas is a set of charts f(Ui; φi) j i 2 Ig such that (1) i2I Ui = M, and −1 k (2) φSi ◦ φj is C on its domain for all i; j 2 I. The k denotes the degree of differentiability. If k = 0, then maps are just continuous. k n Definition 2.4***. If A = f(Ui; φi)ji 2 Ig is a C atlas for an n-manifold M and f : U ! Rn is a homeomorphism onto its image with U ⊂ M open, then (U; f) is −1 k −1 k compatible with A if φi ◦ f : f(U \ Ui) ! φ(U \ Ui) is C and f ◦ φi is C for all i 2 I. copyright c 2002 Larry Smolinsky Typeset by AMS-TEX 1 2 CHAPTER 2 MANIFOLDS k n Theorem 2.5***. If A = f(Ui; φi)ji 2 Ig is a C atlas for M then A is contained in a unique maximal atlas for M where atlases are ordered as sets by containment. Proof. Let n M =f(Vα; fα)jVα ⊂ M is open; fα : Vα ! R is a homeomorphism onto its image, and (Vα; fα) is compatible with Ag: Since M contains all compatible charts it is the unique maximal atlas if it is an atlas. We now show that it is an atlas. If (Vα; fα) and (Vβ; fβ) are in M then we must show that −1 k fβ ◦ fα : fα(Vα \ Vβ) ! fβ(Vα \ Vβ) is C . Suppose m 2 Vα \ Vβ. Take i 2 I such that −1 −1 −1 m 2 Ui. Then on fα(Vα \ Vβ \ Ui), fβ ◦ fα = (fβ ◦ φi ) ◦ (φi ◦ fα ). Since fβ and fα are −1 −1 k n n compatible with A, fβ ◦ φi and φi ◦ fα are C on open sets of R to open sets of R . −1 k Therefore fβ ◦ fα is C on its domain. Definition 2.6***. A maximal Ck atlas is called a Ck differential structure. Definition 2.7***. A Ck n-manifold is a topological n-manifold M along with a Ck differential structure S. By Theorem 2.5***, a single atlas is enough to determine the differential structure. The reader should note that this definition for a C0 structure agrees with the definition of a topological manifold. A C1 n-manifold is also called a smooth manifold. The word \manifold," without other adjectives, will denote a smooth manifold as these will be the subject of the remainder of this manuscript. Usually the notation for the structure S is suppressed. However, the phrase \a manifold M" supposes that there is an unnamed differential structure S in the background. In particular this means that if A is an atlas, then A ⊂ S; and, if (U; φ) is a chart, then (U; φ) 2 S. The differential structure contains all compatible charts. For example, if (U; φ) 2 S and V ⊂ U is open, then (V; φjV ) 2 S. Example 2.8.a***. A real n-dimensional vector space is an n-manifold. Let V be an n-dimensional vector space. Pick an ordered basis v1; · · · ; vn and define n n charts (V; fv1;··· ;vn ), where fv1;··· ;vn ( aivi) = (a1; · · · ; an) is an isomorphism of V to R . Xi=1 The image is all of Rn, which is an open subset of itself. These charts are compatible since −1 n fw1;··· ;wn fv1;··· ;vn is a linear automorphism on R . Recall that every real vector space is isomorphic to Rn, however Rn comes equiped with a standard ordered basis or set of coordinates. A real vector space may appear in several guises. We now mention a few instances of vector spaces that relate to matrices. Example 2.8.b***. Let the set of n × n real matrices be denoted Matn×n. The set 2 Matn×n is a real vector space of dimension n . m Let e1; e2; · · · ; em be the standard ordered basis for R . If eij is the matrix with 1 in the i-th row and j-th column and zero elsewhere, then feij j 1 ≤ i; j ≤ ng is a basis. n2 The isomorphism of Matn×n to R is determined by the linear map f(eij) = en(i−1)+j 2 for 1 ≤ i; j ≤ n, and allows one to think of the matrix entries as coordinates in Rn . In coordinates, if A = (aij), then f(A) = (a11; a12; · · · ; a1n; a21; · · · ; ann). CHAPTER 2 MANIFOLDS 3 t Example 2.8.c***. The symmetric matrices Symn×n = fA 2 Matn×n j A = Ag form a n(n + 1) vector subspace of dimension . 2 The entries on and above the diagonal can be arbitrary. Below the diagonal, the entries are determined by symmetry, i.e., aij = aji. t Example 2.8.d***. The skew symmetric matrices Skewn×n = fA 2 Matn×n j A = −Ag n(n − 1) form a vector subspace of dimension . 2 The entries on the diagonal must be zero, since aij = −aji. The entries above the diagonal can be arbitrary and the entries below are determined by the antisymmetry, i.e., aij = −aji. Example 2.9a***. The sphere Sn = fx 2 Rn+1 j jxj = 1g is an n-manifold. We construct an atlas f(U1; φ1); (U2; φ2)g with the aid of a standard well-known map n n called stereographic projection. Let U1 = S nf(0; · · · ; 0; 1)g and U2 = S nf(0; · · · ; 0; −1)g. n x1 xn Note that U [ U = S . Let φ (x ; x ; · · · ; xn ) = ( ; · · · ; ). The map 1 2 1 1 2 +1 1−xn+1 1−xn+1 n −1 n φ1 : U1 ! R is called stereographic projection. The inverse map φ1 : R ! U1 is defined by −1 2y1 2y2 2yn 2 φ1 (y1; · · · ; yn) = n 2 ; n 2 ; · · · ; n 2 ; 1 − n 2 : i=1 yi + 1 i=1 yi + 1 i=1 yi + 1 i=1 yi + 1 P P P P −1 Both φ1 and φ1 are continuous and hence φ1 is a homeomorphism. The second coordinate chart (U2; φ2), stereographic projection from the south pole, is given by φ2 = −φ1 ◦ (−1) where (-1) is multiplication by −1 on the sphere. Since multiplication by -1 is a homeomorphism of the sphere to itself (its inverse is itself), the n map φ2 : U2 ! R is a homeomorphism. −1 1 Checking the compatability conditions, we have φ2◦φ1 (y1; · · · ; yn) = n 2 (y1; · · · ; yn) i=1 yi −1 −1 n and φ2 ◦ φ1 = φ1 ◦ φ2 . Hence, S is shown to be an n-manifold. P Example 2.9b***. Another atlas for the sphere Sn. We use 2(n + 1) coordinate charts to construct this atlas. For each i 2 f1; · · · ; n + 1g let Ui;+ = f(x1; · · · ; xn+1) j xi > 0g and Ui;− = f(x1; · · · ; xn+1) j xi > 0g. Define φi;+ : n n Ui;+ ! R by φi;+(x1; · · · ; xn+1) = (x1; · · · ; xi−1; xi+1; · · · ; xn+1) and φi;+ : Ui;− ! R by φi;−(x1; · · · ; xn+1) = (x1; · · · ; xi−1; xi+1; · · · ; xn+1). The coordinate xi is a function of x1; · · · ; xi−1; xi+1; · · · ; xn+1 on the sets Ui;+ and Ui;−. The atlases in Examples 2.9a*** and 2.9b*** are compatible and give the same differ- ential structure. See Exercise 1***. n m Example 2.10***. Suppose U1 ⊂ R and U2 ⊂ R are open sets.
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