Complex Analysis for Applications
Complex Analysis for Applications MATH 132
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Date Created: 09/04/15
10 The subgroupisubalgebra correspondence Homogeneous spaces 101 The concept of a Lie subgroup of a Lie group We have seen that if G is a Lie group and H C G a subgroup which is at the same time a closed submanifold then H is a Lie group and the inclusion map t is a morphism such that dt is injective The image of b LieH in g LieG is a subalgebra The natural question is whether every Lie subalgebra of g arises in this manner Formulated in this manner the answer is negative Consider the torus T2 RZZZ lts Lie algebra is R2 and for any X E R2 we can verify that epr is the image of X in T2 Suppose b is a line with an irrational slope it is clear that the exponential of it is an one parameter subgroup of T2 that winds around the torus in a dense manner and so b cannot arise from a closed subgroup of G To make the subgroupisubalgebra correspondence bijective it turns out to be suf cient to generalize the notion of a Lie subgroup slightly We shall say that a subgroup H C G is a Lie subgroup if the following conditions are satisfied 1 H is a Lie group and the inclusion map t is a morphism 2 The differential dt is injective It is to be noted that the topology and smooth structure of H need anot be inherited from those of G and that H need not be closed in G With this reformulation one can now prove the following theorem Theorem 1 Chevalley Ifb C g is a subalgebra there is a unique connected Lie subgroup H C G such that dtLieH b Moreover H is generated by the elements exp XX E 102 Involutive distributions on a manifold If f is one dimensional we can choose a basis element X E b and integrate the vector field to get the one parameter group t gt gt exp tX whose topology and Lie structure are given by those of the parameter t In the general case we use an analogous method However we need some preparation At each point it E G we have the subspace bx C TAG defined as the space of tangent vectors X3c for X E K We have an assignment Hum Ah 1 of tangent spaces with the following properties H dimbm d is constant for all x E G N The assignment H is smooth in the following sense if x E G we can nd vector elds Z1 Z2 Zr such that the tangent vectors span by for all y in a neighborhood of at 9 Suppose U C G is open and XY are vector elds on U such that XyYy E by for ally E U Then XYy E by for ally E U also The conditions 1 and 2 make sense for any assignment x gt gt x of tangent spaces on any manifold is then called a distribution of rank d If condition 3 is satis ed we shall say that is involutive Thus H is an involutive distribution of rank equal to dimb The veri cation of conditions 1 through 3 for H is easy 1 is trivial lf X1 Xd is a basis for b then span by for ally E G proving 2 For 3 we rst observe that Xhle Zqltkltd cZij6 for all 1 S ij S d as l is a subalgebra It is now easy to show that X and Y can be written as X ZfiXiY giXZ wherer figi are smooth functions The property 3 is now obvious Let M be a manifold and a distribution on M of rank d As mentioned earlier we try to nd submanifolds S C M with the property that at each point y E S we have T909 5y y E S Such an S is called an integral manifold for L is called integrable if through any point of M there is an integral manifold It is easy to see that integrability of implies that is involutive Indeed in the de nition of property 3 above let y E U and let S be an integral manifold of through y Then X and Y are tangent to S at all of its points and so X Y is also tangent to S at all of its points In particular X Yly 6 9 The famous classical theorem of Frobenius asserts now that conversely if is involutive then is integrable 103 The local Frobenius theorem Locally on any manifold M we can construct distributions which are involutive and integrable as follows We take coordinates ll S i S m on M on an open set U and de ne 9 2 as the span of BB ll S i S p fOr y E U Then is integrable on U since the submanifolds de ned by making the hi 2 p l constant are integral manifolds is also obviously involutive The local frobenius is the assertion that every involutive distribution looks locally like this A Let M be a smooth manifold with dimM m and an involutive dis tribution of rank 1 on M Then is integrable and we have the following precise version of the local Frobenius theorem Theorem Frobenius Let be an inuolutiue distribution of rank 1 on a manifold Then for any at E M there is an open set U containing at coordinates dihgigm on U and an a gt 0 such that y 1ymy is a di eomorphism of U with the cube or polydish ifM is a complea manifold 1quot lt aVi and for each y E U 9 is the span of BBxiyl S i S p If y E U we write Ulyl 2 6 Ujlt2gt jyp1j m and call it the slice through y We refer to U a gt 0 as adapted to A Now slices are usually only small pieces and our goal is the construction of integrable manifolds which are as big as possible The technique is the obvious one of piecing together small slices but as we do it we shall find that the integrable manifold may return again and again to the same part of M so that in the end the integrable manifold acquires a topology that may be different from the ambient one This situation was first analyzed carefully by Chevalley who constructed the global integrable manifolds for the first time In the succeeding pages we shall give a brief presentation of Chevalley7s treatment 104 Immersed and imbedded manifolds A manifold N is said to be immersed in M is a N C M and b the inclusion i N a M is a morphism having an injective differential for all n E N Notice that the topology of N is not required to be induced by M ie i is not assumed to be a homeomorphism of N onto its image with its topology induced 3 from M N is said to be imbeddedif i is a homeomorphism onto its image in M From basic manifold theory we know that for N to be immersed the following is necessary and suf cient if n E N there are open neigh borhoods U in M and V in N of n and a diffeomorphism 11 of U with If gtlt 3quot such that 11 takes n to 00 V to I gtlt 0 and i to the map gt gt it 0 For N to be imbedded in M the condition is the same with the extra requirement that V U N Proposition 1 IfN C M is imbedded and P is any smooth manifold a map fP a N is a morphism if and only if it is a morphism ofP to M IfN is only immersed this is still so prouided f is a continuous map ofP to N Proof Fix p E P fp n Take U V11 a gt 0 as above Assume N is immersed and that f is a continuous map into N Then P1 f 1V is open in P with fP1 C V It is obvious that if f is a morphism into U it is a morphism into V In the imbedded case we define P1 f 1U Once again P1 is open in P but now as V U N f maps P1 into V and it is clear that f is a morphism into V An immersed manifold is universally immersed if for any P the mor phisms P a N are precisley the morphisms P a M with image con tained in N lmbedded manifolds are universally immersed and some nonimbedded manifolds are also universally imbedded The classical ex ample is R immersed in T2 RZZ2 through the map 11 215 155 where 5 is an irrational number and a b is the image of a b under the natural map R2 a T2 The proof that R is universally immersed in T2 is left as an exercise 105 The global Chevalley Frobenius theorem Let be an in volutive distribution on M A leaf is a connected manifold L which is immersed in M with TyL 9 for all y E L ie L is an integral man ifold for L A slice is an imbedded leaf A ma1imal leaf is a leaf L with the following property if L1 is another leaf with L Ll 31 0 then L1 C L and L1 is an open submanifold of L The topology and smooth structure of a slice are uniquely determined by M Any leaf is a union of slices which are open in it and so the topology and smooth structure on a leaf 4 are uniquely determined The point is that these may not coincide With the structures inherited from M Global Frobenius theorem Through every point x of M passes a unique maximal leaf Any two maximal leaves are either identical or dis joint All leaves are universally immersed The idea of the proof is very simple We shall introduce the leaf topology for M for Which the leaves are form a basis The leaf topology is finer than the given topology of M The maximal leaves are the connected components of M in the leaf topology The proofs are quite simple and follow from the local structure of involutive distributions The only subtle point is to prove that the connected components in the leaf topology are second countable Once this is done the connected components are leaves and are obviously maximal 106 The Lie subgroup corresponding to a Lie subalgebra Let us return to the context of a Lie group G With Lie algebra g and let i C g a subalgebra We introduced the involutive distribution H earlier and so we have the maximal leafs for it Theorem 1 The leaf through e is the unique Lie subgroup with Lie algebra f 107 Homogeneous spaces Let G be a Lie group and H a closed Lie subgroup Then G acts transitively on G H The fundamental question in the theory of Lie groups is to vieW GH as a smooth manifold smooth being in the G real analytic or complex analytic categories on Which the natural action of G is morphic Let XGH 7Tggt gtgH gEG and let us give X the quotient topology this means that U C X is open if and only if 7r 1U is open We assert that 7r is an open map indeed if V is open in G 7r 17rV VH UgeH V5 is open We also observe the fact that X is Hausdorff if and only if H is closed this is true When G is only locally compact To see this suppose first that H is closed and let g k E G be such that 7rg 31 Then g lk Z H and so there is open V C G containing g lk such that V H 0 By continuity there are open neighborhoods V2 V3 of g h respectively such that V2 1V3 H Q 5 This means that for it E V2y E V3E 1y Z H or Mac 31 7ry Thus 7rV2 and 7rV3 are disjoint open neigborhoods of 7rg7rk respectively In the reverse direction if it Z H select an open neighborhood X1 of Mac not containing 7rl then 7r 1X1 is an open neighborhood of ac disjoint from H Theorem Let H be a closed Lie subgroup of G Then there is a unique structure of a smooth manifold on X GH such that the natural map 7r g gt gt gH of G onto X is a submersion The action of G on X is morphic Comments on the proof of the theorem Let OG be the structure sheaf of G It is natural to start with the sheaf OX where for U an open subset of X f e OXU e f 07139 e OG7r 1U It is triVial to verify that the action of G on X is morphic The point to show is that X becomes a smooth manifold when equipped with this structure with the property that 7r is a submersion The uniqueness of the smooth structure under the requirement that 7r is a submersion is a standard fact Let us introduce some standard terminology Let M be a smooth manifold and NP submanifolds Suppose m E N P We say that N and P meet transuersally at m if the tangent spaces TmNTmP are complementary ie In this case we have of course dimM dimN dimP The key to the proof is the following lemma Lemma 1 Suppose that we can nd a submanifold W ofG such that a 16W W wHwforallw W b W and wH meet transuersally at w for all in E W Then X OX is a smooth manifold 7r is a submersion and the action of G on X is morphic Proof We set up the map LJZWXH G 11w w The image of dwwyl is the span of TwH and TwW and so dwwyl is bijective lf R77 is right translation by r E H we have Rnw 1id gtlt Rn so that anMdWA diliwwd d X Rnw1 showing that d1 is bijective everywhere on W gtlt H Moreover 11 is also bijective for if 1117 urf then u E W wH so that u w and thence r rf Hence U WH is open in G and 11 is a diffeomorphism In particular 7rW is open in X It follows from this that for any open subset V of W a smooth function f on VH is right H invariant if and only if f 0 00155 is dependent only on w and is a smooth function of w on V In other words the restriction to 7rW of the sheaf OX is isomorphic to OW This proves that 7rW is a smooth manifold and that 7r is a submersion from WH onto Using the G action on X the rest of the lemma is clear To finish the proof of the theorem it is enough to construct such a W Let f LieH and let be the involutive distribution on G defined by l The maximal leaves of are H and the cosets gH Let U a gt 0 be adapted to with a E U and 0 For 0 lt b S a write Ub for the preimage of the cube Tb under the map y gt gt x1y xmy Since H is closed we can find an open subset T of G such that UH T H and hence we can find a1 such that 0 lt a1 3 a and Ua1 H Ua1 Lemma 2 We can nd b with 0 lt b 3 a1 such that for ally E Ub Ub yH Uby In particular W y E Ubx1y icpy 0 has the properties described in Lemma 1 Proof Select a2 0 lt a2 lt a1 such that UMUa2 C Ua1 and a3 0 lt a3 lt a2 such that Ua San3 C U112 We claim that b a3 has the required property Let y E Ub Since yH is the maximal leaf containing y it is clear that Uby C yH and so we need only show that Ub yH C Uby Now Ua2 H Ua2l and so Uas m yH yy1Ua3 m H c yUa2 m H yUa2l Now yUa2 l is a leaf contained in UGQUa2 C Ua1 and so icj pl S j S m are constant on it showing that they are also constant on Ua3 yH Hence Ua3 yH C Ua3 This proves the lemma 7
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