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Review of Smooth Manifolds

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Review of Smooth Manifolds Appendix A Review of Smooth Manifolds This book is designed for readers who already have a solid understanding of basic topology and smooth manifold theory, at roughly the level of [LeeTM] and [LeeSM]. In this appendix, we summarize the main ideas of these subjects that are used throughout the book. It is included here as a review, and to establish our notation and conventions. Topological Preliminaries An n-dimensional topological manifold (or simply an n-manifold) is a second- countable Hausdorff topological space that is locally Euclidean of dimension n, meaning that every point has a neighborhood homeomorphic to an open subset of Rn. Given an n-manifold M ,acoordinate chart for M is a pair .U;'/, where U  M is an open set and ' W U ! Uy is a homeomorphism from U to an open subset Uy  Rn.Ifp 2 M and .U;'/ is a chart such that p 2 U , we say that .U;'/ is a chart containing p. We also occasionally need to consider manifolds with boundary. An n-dimen- sional topological manifold with boundary is a second-countable Hausdorff space in which every point has a neighborhood homeomorphic either to an open subset Rn Rn 1 n Rn of or to a (relatively) open subset of the half-space C Df.x ;:::;x / 2 W xn 0g. A pair .U;'/, where U is an open subset of M and ' is a homeomorphism Rn Rn from U to an open subset of or C, is called an interior chart if '.U/ is an Rn Rn Rn Rn open subset of or an open subset of C that does not meet @ C Dfx 2 C W n Rn x D 0g; and it is called a boundary chart if '.U/ is an open subset of C with Rn ¿ M '.U/\ @ C ¤ . A point p 2 M is called an interior point of if it is in the domain of an interior chart, and it is a boundary point of M if it is in the domain Rn of a boundary chart taking p to a point of @ C. Notice that our convention is that a manifold without further qualification is al- ways assumed to be a manifold without boundary, and the term “manifold with boundary” encompasses ordinary manifolds as a special case. But for clarity, we © Springer International Publishing AG 2018 371 J. M. Lee, Introduction to Riemannian Manifolds, Graduate Texts in Mathematics 176, https://doi.org/10.1007/978-3-319-91755-9 372 Review of Smooth Manifolds sometimes use redundant phrases such as “manifold without boundary” for an ordi- nary manifold, and “manifold with or without boundary” when we wish to empha- size that the discussion applies equally well in either case. Proposition A.1. [LeeTM, Props. 4.23 & 4.64] Every topological manifold with or without boundary is locally path-connected and locally compact. Proposition A.2. [LeeTM, Thm. 4.77] Every topological manifold with or without boundary is paracompact. If M and N are topological spaces, a map F W M ! N is said to be a closed map if for each closed subset K  M , the image set F.K/ is closed in N , and an open map if for each open subset U  M , the image set F.U/is open in N .Itisa quotient map if it is surjective and V  N is open if and only if F 1.V / is open. Proposition A.3. [LeeTM, Prop. 3.69] Suppose M and N are topological spaces, and F W M ! N is a continuous map that is either open or closed. (a) If F is surjective, it is a quotient map. (b) If F is injective, it is a topological embedding. (c) If F is bijective, it is a homeomorphism. The next lemma often gives a convenient shortcut for proving that a map is closed. Lemma A.4 (Closed Map Lemma). [LeeTM, Lemma 4.50] If F is a continuous map from a compact space to a Hausdorff space, then F is a closed map. Here is another condition that is frequently useful for showing that a map is closed. A map F W M ! N between topological spaces is called a proper map if for each compact subset K  N , the preimage F 1.K/ is compact. Proposition A.5 (Proper Continuous Maps Are Closed). [LeeTM, Thm. 4.95] Suppose M is a topological space, N is a locally compact Hausdorff space, and F W M ! N is continuous and proper. Then F is a closed map. Fundamental Groups Many of the deepest theorems in Riemannian geometry express relations between local geometric properties and global topological properties. Because the global properties are frequently expressed in terms of fundamental groups, we summarize some basic definitions and properties here. Suppose M and N are topological spaces. If F0;F1 W M ! N are continuous maps, a homotopy from F0 to F1 is a continuous map H W M Œ0;1 ! N that satisfies H.x;0/ D F0.x/ and H.x;1/ D F1.x/ for all x 2 M . If there exists such a homotopy, we say that F0 and F1 are homotopic. We are most interested in homotopies in the following situation. If M is a topo- logical space, a path in M is a continuous map ˛ W Œ0;1 ! M .Ifp D ˛.0/ and q D ˛.1/, we say that ˛ is a path from p to q.If˛0 and ˛1 are both paths from p to Topological Preliminaries 373 q,apath homotopy from ˛0 to ˛1 is a continuous map H W Œ0;1 Œ0;1 ! M that satisfies H.s;0/ D ˛0.s/ and H.s;1/ D ˛1.s/ for all s 2 Œ0;1; H.0;t/ D p and H.1;t/ D q for all t 2 Œ0;1: If there exists such a path homotopy, then ˛0 and ˛1 aresaidtobepath-homotopic; this is an equivalence relation on the set of all paths in M from p to q.The equivalence class of a path ˛, called its path class, is denoted by Œ˛. Paths ˛ and ˇ that satisfy ˛.1/ D ˇ.0/ aresaidtobecomposable; in this case, their product is the path ˛ ˇ W Œ0;1 ! M defined by 1 ˛.2s/; 0 Ä s Ä 2 ; ˛ ˇ.s/ D 1 ˇ.2s 1/; 2 Ä s Ä 1: Path products are well defined on path classes: if Œ˛0 D Œ˛1 and Œˇ0 D Œˇ1, and ˛0 and ˇ0 are composable, then so are ˛1 and ˇ1, and Œ˛0 ˇ0 D Œ˛1 ˇ1. Thus we obtain a well-defined product of path classes by Œ˛ Œˇ D Œ˛ ˇ. A path from a point p to itself is called a loop based at p.IfM is a topological space and p 2 M , then any two loops based at p are composable, and the set of path classes of loops based at p is a group under path class products, called the fundamental group of M based at p and denoted by 1.M;p/. The class of the constant path cp.s/ Á p is the identity element, and the class of the reverse path ˛1.s/ D ˛.1 s/ is the inverse of Œ˛. (Although multiplication of path classes is associative, path products themselves are not, so a multiple product such as ˛ ˇ is to be interpreted as .˛ ˇ/ .) Proposition A.6 (Induced Homomorphisms). [LeeTM, Prop. 7.24 & Cor. 7.26] Suppose M and N are topological spaces and p 2 M .IfF W M ! N is a contin- uous map, then the map F W 1.M;p/ ! 1.N;F .p// defined by F.Œ˛/ D ŒF ı˛ is a group homomorphism called the homomorphism induced by F .IfF is a homeo- morphism, then F is an isomorphism. A topological space M is said to be simply connected if it is path-connected and for some (hence every) point p 2 M , the fundamental group 1.M;p/ is trivial. I Exercise A.7. Show that if M is a simply connected topological space and p;q are any two points in M , then all paths in M from p to q are path-homotopic. A continuous map F W M ! N is said to be a homotopy equivalence if there is a continuous map G W N ! M such that G ı F and F ı G are both homotopic to identity maps. Proposition A.8 (Homotopy Invariance of the Fundamental Group). [LeeTM, Thm. 7.40] Suppose M and N are topological spaces and F W M ! N is a homotopy equivalence. Then for every point p 2 M , the induced homomorphism F W 1.M;p/ ! 1.N;F .p// is an isomorphism. 374 Review of Smooth Manifolds Smooth Manifolds and Smooth Maps The setting for the study of Riemannian geometry is provided by smooth manifolds, which are topological manifolds endowed with an extra structure that allows us to differentiate functions and maps. First, we note that when U is an open subset of Rn, amapF W U ! Rk is said to be smooth (or of class C 1) if all of its component functions have continuous partial derivatives of all orders. More generally, if the domain U is an arbitrary subset of Rn, not necessarily open (such as a relatively Rn open subset of C), then F is said to be smooth if for each x 2 U , F has a smooth extension to a neighborhood of x in Rn.Adiffeomorphism is a bijective smooth map whose inverse is also smooth.
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