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WOMP 2012 Manifolds Jenny Wilson The Definition of a Manifold and First Examples In brief, a (real) n-dimensional manifold is a topological space Mfor which every point x2Mhas a neighbourhood
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WOMP 2012ManifoldsJenny WilsonThe Definition of a Manifold and First Examples
In brief, a (real)n-dimensional manifoldis a topological spaceMfor which every pointx2 Mhas a neighbourhood
homeomorphic to Euclidean spaceRn.Definition 1.(Coordinate system, Chart, Parameterization) LetMbe a topological space andU Man open
set. LetV Rnbe open. A homeomorphism:U ! V,(u) = (x1(u);:::;xn(u))is called acoordinate systemonU, and the functionsx1;:::xnthecoordinate functions. The pair(U;)is called achartonM. The inverse map1
is aparameterizationofU.Definition 2.
(Atlas, T ransitionmaps) AnatlasonMis a collection of chartsfU;gsuch thatUcoverM. The homeomorphisms1:(U\ U)!(U\ U)are thetransition mapsorcoordinate transformations.Recall that a topological space issecond countableif the topology has a countable base, andHausdorffif distinct
points can be separated by neighbourhoods.Definition 3.
(T opologicalmanifold, Smooth manifold) A second countable, Hausdorff topological spaceMis ann-dimensional topological manifoldif it admits an atlasfU;g,:U!Rn,n2N. It is asmoothmanifold ifall transition maps areC1diffeomorphisms, that is, all partial derivatives exist and are continuous.Two smooth atlases areequivalentif their union is a smooth atlas. In general, asmooth structureonMmay be
defined as an equivalence class of smooth atlases, or as a maximal smooth atlas.Definition 4.
(Manifold with boundary ,Boundary ,Interior) We define an-dimensionalmanifold with boundaryMas above, but now allow the image of each chart to be an open subset of Euclidean spaceRnor an open subset
of the upper half-spaceRn+:=f(x1;:::;xn)jxn0g:The preimages of points(x1;:::;xn1;0)2R+nare the boundary@MofM, andM @Mis theinteriorofM. 1WOMP 2012ManifoldsJenny WilsonA manifold with boundary issmoothif the transition maps are smooth. Recall that, given an arbitrary subset
XRm, a functionf:X!Rnis called smooth if every point inXhas some neighbourhood wherefcan be extended to a smooth function. Definition 5.A functionf:M ! Nis amap of topological manifoldsiffis continuous. It is asmoothmap of smooth manifoldsM,Nif for any smooth charts(U;)ofMand(V; )ofN, the function f1:(U \f1(V))! (V) is aC1diffeomorphism.Exercise # 1.
(Recognizing Manifolds) Which of the following have a manifold structure (possibly with bound- ary)?(a) Arbitrary subsetXRn(b) Arbitrary open subsetU Rn(c) Graph (d) Hawaiian Earring(e)f(x;sin1x )jx2(0;1]g(f)f(x;sin1x )jx2(0;1]gSf0g [0;1] (g) Solution tox2+y2=z2(h) Solution tox2+z3=y2z2 (i) Solution toz=x2+y2 2WOMP 2012ManifoldsJenny WilsonBBA
A(j) Quotient SpaceABAB1
A A AA(k) Quotient SpaceAAAA(l) Infinite Coordinate Space LNR(m) Disjoint union
aN[0;1]
of discrete copies of[0;1](n) Disjoint union a continuum[0;1] of discrete copies of[0;1](o) Infinite Real Hilbert SpaceIt is a difficult fact that not every topological manifold admits a smooth structure. Moreover, a topological
manifold may have multiple nondiffeomorphic smooth structures. For example, there are uncountably many
distinct smooth structures onR4.Exercise # 2.
(Atlases on the circle) Define the1-sphereS1to be the unit circle inR2. Put atlases onS1using charts defined by: 1. the angle of r otationfr oma fixed point, 2. the slope of the secant line fr oma fixed point, 3. the pr ojectionsto the xandyaxes.In each case, compute the transition functions.
Exercise # 3.
(T opologicalvs. Smooth) Give an example of a topological spaceMand an atlas onMthat makesMa topological, but not smooth, manifold.
Exercise # 4.
(Products of manifolds) LetMbe a manifold with atlas(U;), andNa manifold with atlas(V; ), and assume at least one of@Mand@Nis empty. Describe a manifold structure on the Cartesian product
M N.Exercise # 5.
(Manifold boundaries) LetMbe ann-dimensional manifold with nonempty boundary@M. Show that@Mis an(n1)-dimensional manifold with empty boundary.Exercise # 6.
(Atlases on spheres) Prove that any atlas onS1must include at least two charts. Do the same for the2-sphereS2.Exercise # 7.
(Centering charts) Given a (topological or smooth) manifoldM, and anyx2 M, show that there is some chart(U;)withx2 Ucentered atxin the sense that(x) = 0. Show similarly that there is some parameterization ofUsuch that (0) =x. 3WOMP 2012ManifoldsJenny WilsonDefinition 6.(Real and complex projective spaces) Theprojectivizationof a vector spaceVis the space of1-
dimensional subspaces ofV. Real and complex projective spaces are defined:RPn:= (Rn+1 f0g)=(x0;:::;xn)(cx0;:::cxn)forc2R
CPn:= (Cn+1 f0g)=(x0;:::;xn)(cx0;:::cxn)forc2C
Points inRPnandCPnare often written inhomogeneous coordinates, where[x0:x1::xn]denotes the equiva- lence class of the point(x0;x1;:::;xn). Exercise # 8.LetUi=f[x0:x1::xn]2RPnjxi6= 0g. Use the setsUito define a manifold structure onRPn.Exercise # 9.Show thatRPncan also be defined as the quotient ofn-sphereSnby the antipodal map, identifying
a point(x0;:::;xn)with(x0;:::;xn). Exercise # 10.Show thatRPndecomposes as the disjoint unionRPn=RntRPn1;and hence inductivelyRPn=RntRn1t tR1tpoint:
These subspaces are called thepoint at infinity, theline at infinity, etc.Exercise # 11.Identify among the following quotient spaces: a cylinder, a M¨obius band, a sphere, a torus, real
projective space, and a Klein bottle.BBA A A A BBA A A A BAA BBBA AExercise # 12.(The cylinder as a quotient) Define the cylinderCto be the subset ofR3C=f(cos;sin;z)j0 <2;0z1g:
Give a rigorous proof thatCis homeomorphic to the unit squareR= [0;1][0;1]inR2modulo the identification
(0;x)(1;x)for allx2[0;1].Exercise # 13.
( T2=R2=Z2)Show that the quotient of the planeR2by the action ofZ2is the torusT2=S1S1, T2=R2=(x;y)(x;y) + (m;n)for(m;n)2Z2:
Exercise # 14.(Challenge)(Classification of1-manifolds)Prove that any smooth, connected1-manifold is dif-
feomorphic to the circleS1or to an interval ofR.Exercise # 15.(Challenge)(Topological groups)Show that the following groups have the structure of a manifold.
Compute their dimension, find the number of connected components, and determine whether they are compact.
1.Real nnmatricesMn(R)
4 WOMP 2012ManifoldsJenny Wilson2.Rigid motions of Euclidean space En(R)3.mnmatrices of maximal rank
4.General linear gr oupGLn(R) =fA2Mn(R)jdet(A)6= 0g
5.Special linear gr oupSLn(R) =fA2Mn(R)jdet(A) = 1g
6. Orthogonal gr oupOn(R) =fA2GLn(R)jtranspose(A) =A1g 7.Special orthogonal gr oupSOn(R) =On(R)\SLn(R)
Exercise # 16.(Challenge)(The real Grassmannian)The projective space of a vector spaceVis a special case of
theGrassmanianG(r;V), the space ofr-planes through the origin. Show that, as a set,G(r;Rn)=O(n)=O(r)O(nr):
Argue that this identification givesG(r;Rn)the structure of a smooth compact manifold, and compute its dimen-
sion.Multivariate Calculus
Definition 7.
(Derivative) Given a functionf:Rn!Rm, thederivative offatxis the linear map df x:Rn!Rm y7!limt!0f(x+ty)f(x)tThe directional derivative offatxin theydirection
Writefin coordinatesf(x) = (f1(x);:::fm(x)). If all first partial derivatives exist and are continuous in a neigh-
bourhood ofx, thendfxexists and is given by theJacobian, themnmatrix 2 6 64df1dx
1(x)df1dx
n(x) df mdx1(x)dfmdx
n(x)3 7 75The chain rule asserts that given smooth mapsf;g
U\gf $$f //V \g //W R nRmR` 5 WOMP 2012ManifoldsJenny Wilsonthen for eachx2 U, the following diagram commutes: R n df x//d(gf)x $$R m dg f(x)//R Exercise # 17.Letfandgbe functions fromRn!Rwith derivativesdfxanddgxatx. Give formulas for thederivative of their pointwise sum and product,f+gandfg. Conclude that the space ofr-times differentiable
functionsRn!Ris closed under addition and multiplication. Repeat the exercise for functionsRn!Rmunder
sum and scalar product.Exercise # 18.Letfbe anr-times differentiable functionRn!R. Under what circumstances is the pointwise
reciprocal 1f anr-times differentiable function?Theorem 8.
(T aylor"sTheorem) Given ann-tuple= (1;:::;n), let jj=1++n! =1!n!x=x11xnnDf=@jjf@x11@xnn
and letf:Rn!Rbe a(r+ 1)times continuously differentiable function in a ballBofy. Then forx2B,fhas a Taylor
expansion f(x) =rX jj=0D f(y)!(xy)+X jj=r+1R (x)(xy) where the remainder R (x) =jj!Z 1 0 (1t)jj1Dfy+t(xy)dtTangent Spaces and Derivatives
Definition 9.
(T angentSpace TxM, Derivatives)Suppose thatMis a smoothm-dimensional submanifold ofsome Euclidean spaceRN. (We will see in Theorem 18 that every manifold can be realized this way). Let:U !
Mbe a local parameterization around some pointx2 Mwith(0) =x. We define thetangent spaceTxMto be the image of the mapd0:Rm!RN. Note thatTxMis anm-dimensional subspace ofRN; its translatex+TxM is the best flat approximation toatx. Given a smooth map of manifoldsf:M ! N, and local parameterizations:U ! M,(0) =x2 Mand :V ! N, (0) =f(x)2 N. Lethbe the maph= 1f:U ! V. We can define thedifferential offatxby df x:TxM !Tf(x)N df x=d 0dh0d10: 6WOMP 2012ManifoldsJenny WilsonThe spacesTx(M),Tf(x)(N), and the differentialdfxare independent of choice of local parameterizations
and .