Python Courses: Smooth Manifold

1 - Definition of Smooth Manifold

2 - Charts

$Let ri : Rn ! R denote projection onto the ith coordinate, and given a chart h : U ! O, let xi = ri h : U ! R. The functions xi are the coordinates of h on U. Given two charts h : U ! O and k : V ! on a topological manifold M such that U \ V 6= ;, the function k h􀀀1 : h(U \ V ) ! k(U \ V ) is a homeomorphism between open sets of Rn. We call k h1 the transition function between the charts h and k.$

3 - Definition of an Atlas

$An atlas A on a topological manifold Mn is a collection f(U ; h ) j 2 Ag of charts on M such that: 1. fU j 2 Ag is an open cover of M, 2. For any ; 2 A such that U \ U 6= ;, the transition function h := h h􀀀1 is smooth (posesses continuous partial deriviatives of all orders). 1 1 Smooth manifolds 2 In general, we say two charts satisfying the conditions of (2) above are compatible. Note that this implies that the transition function between the two charts is a diffeomorphism. A smooth structure on M is an atlas A that is maximal with respect to property (2) above.$

$Any atlas determines a unique smooth structure. J Let A =f(U ; h )j 2 Ag be an atlas on M, and consider the collection of charts A consisting of all charts on M that are compatible with all the charts in A. Clearly A A . We now check A is an atlas. Indeed, property (1) is immediate, and to check property (2), let h : U ! O and k : V ! be two charts in A such that U \ V 6= ;. We must show that k h􀀀1 is smooth, and it is enough to check this locally. Given p 2 U \ V , choose a chart h : U ! O in A such that p 2 U . Then W = U \ V \ U is an open neighborhood of p and k h􀀀1 h(W)= k h􀀀1 (W)= (k h􀀀1 ) (h h􀀀1) : h(W) ! k(W) is smooth by assumption. Thus A is an atlas, and clearly A is maximal amongst atlases containing A. I Therefore it is enough to specify an atlas when defining$

4 - Examples

$Any open subset V of a smooth manifold Mn is itself a smooth manifold; second countability, the Hausdorff condition and the locally Euclidean property are inherited by subsets, and if f(U ; h )j 2 Ag is an atlas on M, then f(U \ V; h jU \V )j 2 Ag forms an atlas on V . 2. Rn is Hausdorff and second countable (for a countable base, take all balls of rational radii with rational centres), and certainly locally Euclidean. The identity map determines an atlas on Rn (there are no transition functions to worry about) and smooth structure determined by this atlas makes Rn into a smooth n-manifold. We call this the standard smooth structure on Rn. 3. The n-dimensional sphere Sn Rn+1 is a n-dimensional smooth manifold. Indeed, for i = 1 : : : n + 1, let U+ i := fy 2 Sn j yi > 0g and similarly let U􀀀 i denote the set of points in Sn whose ith coordinate is negative. Note that fU i g is an open cover of Sn. Now define hi : U i ! Rn to be the map that forgets the ith coordinate; hi(y1; : : : ; yn+1) = (y1; : : : ; ^ yi; : : : ; yn+1). Suppose i < j; then the transition function hij : 􀀀 u1; : : : ; un 7! u1; : : : ; ^ ui; : : : ; q 1 􀀀 kuk2; : : : ; un ; which is smooth (note kuk 1). A similar formula holds for i > j, and for i = j the transition function is just the identity map. Thus all the transition functions are diffeomorphisms (in particular homeomorphisms), and we have a smooth structure on Sn.$

$is a well defined homeomorphism, call it ki. Moreover we have kj k􀀀1 i = (hj 􀀀1) (hi 􀀀1)􀀀1 = hj h􀀀1 i which is smooth. It follows that ( (U+ i ); ki) defines an atlas on RPn. 5. The two-dimensional torus T. Define an equivalence relation on R2 by x y if and only if xi 􀀀 yi 2 Z for i = 1; 2. Let T denote R2= . Any unit square Q R2 with vertices (a; b); (a+1; b); (a; b+1) and (a+1; b+1) determines a homeomorphism int(Q) ! U(Q) T, where U(Q) is an open subset of T (in fact, U(Q) is all of T apart from two circles). The inverse map hQ : U(Q) ! int(Q) then is a homeomorphism from an open set of T into an open set of R2. Clearly the fU(Q)g cover T, and if Q1 and Q2 are two squares with a non-empty intersection then one easily sees that the coordinate transformation hQ2 h􀀀1 Q1 is given locally by translations; that is, each component of hQ1 (Q1 \ Q2) (of which in general there will be seven - draw a picture!) is mapped by a translation onto a corresponding component of hQ2 (Q1 \ Q2) by hQ2 h􀀀1 Q1 . Thus we have a smooth structure, and T is a smooth 2-manifold.$

Note

This cours is an extract from book : Differential Geometry by Will Merry
Lecture notes based on the Differential Geometry course lectured by Prof. P.M.H.
Wilson in Michaelmas term 2007 for Part III of the Cambridge Mathematical Tripos.

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Monday, May 22, 2017

1 - Definition of Smooth Manifold

2 - Charts

3 - Definition of an Atlas

4 - Examples

Note

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