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Chapter 18 Metrics, Connections, and Curvature on Lie Groups

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Chapter 18 Metrics, Connections, and Curvature on Lie Groups Chapter 18 Metrics, Connections, and Curvature on Lie Groups 18.1 Left (resp. Right) Invariant Metrics Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all the Riemannian metrics on a Lie groups, those for which the left translations (or the right translations) are isometries are of particular interest because they take the group structure of G into account. This chapter makes extensive use of results from a beau- tiful paper of Milnor [38]. 831 832 CHAPTER 18. METRICS, CONNECTIONS, AND CURVATURE ON LIE GROUPS Definition 18.1. Ametric , on a Lie group G is called left-invariant (resp. right-invarianth i )i↵ u, v = (dL ) u, (dL ) v h ib h a b a b iab (resp. u, v = (dR ) u, (dR ) v ), h ib h a b a b iba for all a, b G and all u, v T G. 2 2 b ARiemannianmetricthatisbothleftandright-invariant is called a bi-invariant metric. In the sequel, the identity element of the Lie group, G, will be denoted by e or 1. 18.1. LEFT (RESP. RIGHT) INVARIANT METRICS 833 Proposition 18.1. There is a bijective correspondence between left-invariant (resp. right invariant) metrics on a Lie group G, and inner products on the Lie al- gebra g of G. If , be an inner product on g,andset h i u, v g = (dL 1)gu, (dL 1)gv , h i h g− g− i for all u, v TgG and all g G.Itisfairlyeasytocheck that the above2 induces a left-invariant2 metric on G. If G has a left-invariant (resp. right-invariant) metric, since left-invariant (resp. right-invariant) translations are isometries and act transitively on G,thespaceG is called a homogeneous Riemannian manifold. Proposition 18.2. Every Lie group G equipped with a left-invariant (resp. right-invariant) metric is com- plete. 834 CHAPTER 18. METRICS, CONNECTIONS, AND CURVATURE ON LIE GROUPS 18.2 Bi-Invariant Metrics Recall that the adjoint representation Ad: G GL(g)of ! the Lie group G is the map defined such that Ada : g g is the linear isomorphism given by ! Ada = d(AdA)e = d(R 1 La)e, for every a G. a− ◦ 2 Clearly, Ada =(dR 1)a (dLa)e. a− ◦ Here is the first of four criteria for the existence of a bi- invariant metric on a Lie group. 18.2. BI-INVARIANT METRICS 835 Proposition 18.3. There is a bijective correspondence between bi-invariant metrics on a Lie group G and Ad-invariant inner products on the Lie algebra g of G, namely inner products , on g such that Ad h i a is an isometry of g for all a G; more explicitly, Ad- invariant inner inner products2 satisfy the condition Ad u, Ad v = u, v , h a a i h i for all a G and all u, v g. 2 2 Proposition 18.3 shows that if a Lie group G possesses abi-invariantmetric,theneverylinearmapAda is an orthogonal transformation of g. It follows that Ad(G)isasubgroupoftheorthogonal group of g,andsoitsclosureAd(G)iscompact. It turns out that this condition is also sufficient! 836 CHAPTER 18. METRICS, CONNECTIONS, AND CURVATURE ON LIE GROUPS To prove the above fact, we make use of an “averaging trick” used in representation theory. Recall that a representation of a Lie group G is a (smooth) homomorphism ⇢: G GL(V ), where V is some finite- dimensional vector space.! For any g G and any u V ,weoftenwriteg u for ⇢(g)(u). 2 2 · We say that an inner-product , on V is G-invariant i↵ h i g u, g v = u, v , for all g G and all u, v V. h · · i h i 2 2 If G is compact, then the “averaging trick,” also called “Weyl’s unitarian trick,” yields the following important result: 18.2. BI-INVARIANT METRICS 837 Theorem 18.4. If G is a compact Lie group, then for every representation ⇢: G GL(V ), there is a G-invariant inner product on V!. Using Theorem 18.4, we can prove the following result giving a criterion for the existence of a G-invariant in- ner product for any representation of a Lie group G (see Sternberg [50], Chapter 5, Theorem 5.2). Theorem 18.5. Let ⇢: G GL(V ) be a (finite-dim.) representation of a Lie group! G. There is a G-invariant inner product on V i↵ ⇢(G) is compact. In particu- lar, if G is compact, then there is a G-invariant inner product on V . 838 CHAPTER 18. METRICS, CONNECTIONS, AND CURVATURE ON LIE GROUPS Applying Theorem 18.5 to the adjoint representation Ad: G GL(g), we get our second criterion for the existence! of a bi-invariant metric on a Lie group. Proposition 18.6. Given any Lie group G, an inner product , on g induces a bi-invariant metric on G i↵ Ad(hG)iis compact. In particular, every compact Lie group has a bi-invariant metric. Proposition 18.6 can be used to prove that certain Lie groups do not have a bi-invariant metric. For example, Arsigny, Pennec and Ayache use Proposi- tion 18.6 to give a short and elegant proof of the fact that SE(n)doesnothaveanybi-invariantmetricforall n 2. ≥ 18.2. BI-INVARIANT METRICS 839 Recall the adjoint representation of the Lie algebra g, ad: g gl(g), ! given by ad = dAd1.Hereisourthirdcriterionfortheex- istence of a bi-invariant metric on a connected Lie group. Proposition 18.7. If G is a connected Lie group, an inner product , on g induces a bi-invariant met- h i ric on G i↵the linear map ad(u): g g is skew- ! adjoint for all u g, which means that 2 ad(u)(v),w = v, ad(u)(w) , for all u, v, w g, h i h i 2 or equivalently that [x, y],z = x, [y, z] , for all x, y, z g. h i h i 2 840 CHAPTER 18. METRICS, CONNECTIONS, AND CURVATURE ON LIE GROUPS It will be convenient to say that an inner product on g is bi-invariant i↵every ad(u)isskew-adjoint. The following variant of Proposition 18.7 will also be needed. This is a special case of Lemma 3 in O’Neill [43] (Chapter 11). Proposition 18.8. If G is Lie group equipped with an inner product , on g that induces a bi-invariant h i metric on G, then ad(X): gL gL is skew-adjoint for ! all left-invariant vector fields X gL, which means that 2 ad(X)(Y ),Z = Y,ad(X)(Z) , h i h i for all X, Y, Z gL, 2 or equivalently that [Y,X],Z = Y,[X, Z] , for all X, Y, Z gL. h i h i 2 18.2. BI-INVARIANT METRICS 841 If G is a connected Lie group, then the existence of a bi-invariant metric on G places a heavy restriction on its group structure, as shown by the following result from Milnor’s paper [38] (Lemma 7.5): Theorem 18.9. A connected Lie group G admits a bi- invariant metric i↵it is isomorphic to the cartesian product of a compact group and a vector space (Rm, for some m 0). ≥ A proof of Theorem 18.9 can be found in Milnor [38] (Lemma 7.4 and Lemma 7.5). The proof uses the universal covering group and it is a bit involved. 842 CHAPTER 18. METRICS, CONNECTIONS, AND CURVATURE ON LIE GROUPS One of the steps uses the folowing proposition. Proposition 18.10. Let g be a Lie algebra with an inner product such that the linear map ad(u) is skew- adjoint for every u g. Then the orthogonal comple- 2 ment a? of any ideal a is itself an ideal. Consequently, g can be expressed as an orthogonal direct sum g = g g , 1 ⊕···⊕ k where each gi is either a simple ideal or a one-dimensional abelian ideal (gi ⇠= R). We now investigate connections and curvature on Lie groups with a left-invariant metric. 18.3. CONNECTIONS AND CURVATURE OF LEFT-INVARIANT METRICS 843 18.3 Connections and Curvature of Left-Invariant Metrics on Lie Groups If G is a Lie group equipped with a left-invariant metric, then it is possible to express the Levi-Civita connection and the sectional curvature in terms of quantities defined over the Lie algebra of G,atleastforleft-invariantvector fields. When the metric is bi-invariant, much nicer formulae can be obtained. If , is a left-invariant metric on G,thenforanytwo left-invarianth i vector fields X, Y ,wecanshowthatthe function g X, Y is constant. 7! h ig Therefore, for any vector field Z, Z( X, Y )=0. h i If we go back to the Koszul formula (Proposition 12.8) 844 CHAPTER 18. METRICS, CONNECTIONS, AND CURVATURE ON LIE GROUPS 2 Y,Z = X( Y,Z )+Y ( X, Z ) Z( X, Y ) hrX i h i h i − h i Y,[X, Z] X, [Y,Z] Z, [Y,X] , h ih ih i we deduce that for all left-invariant vector fields X, Y, Z, we have 2 Y,Z = Y,[X, Z] X, [Y,Z] Z, [Y,X] , hrX i h ih ih i which can be rewritten as 2 XY,Z = [X, Y ],Z [Y,Z],X + [Z, X],Y . hr i h ih i h (i ) † The above yields the formula 1 L Y = ([X, Y ] ad(X)⇤Y ad(Y )⇤X) ,X,Y g , rX 2 − − 2 where ad(X)⇤ denotes the adjoint of ad(X), where adX is defined just after Proposition 16.11.
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