Homogeneous Metrics with nonnegative curvature

HOMOGENEO US METRICS WITH NONNEGA TIVE CUR V A TURE LORENZ SCHW A CHH ¨ OFER ∗ , KRISTOPHER T APP Abstra ct. Gi ven compact Lie groups H ⊂ G , we study the space of G -inv ari ant metrics on G/H with n onnegative sectional cu rv ature. F or an intermediate sub group K b etw een H and G , we derive cond itions u nder whic h enlarging the Lie algebra of K maintai ns nonnegative curv ature on G/H . Such an enlarging is p ossible if ( K, H ) is a symmetric p air, which yields many new examples of nonnegativ ely curv ed homoge neous metrics. W e pro v ide other examples of spaces G/H with unexp ected ly large families of non negativ ely curved homogeneous metrics. Let H ⊂ G b e compact Lie groups, with Lie algebras h ⊂ g , and let g 0 b e a b i-inv ariant metric on G . The space G /H with the induced norm al homogeneous metric, d enoted ( G, g 0 ) /H , has nonnegativ e sectional curv ature. Little is kno wn ab out whic h other G -in v arian t metrics on G/H ha ve nonn egativ e sectional curv ature, except in certain cases. In all cases wh ere G/H admits a G -in v arian t m etric of p ositive cur v ature, the problem has b ee n studied along with the d etermination of whic h G -in v arian t metric has th e b est p inc hing constant; see [9 ],[10],[11]. When H is trivial, this problem was solv ed for G = S O (3) and U (2) in [1 ], and p artial results for G = S O (4) were obtained in [5]. Henceforth, we ident ify G -in v ariant metrics on G/H w ith Ad H -in v arian t inner pro d ucts on p = the g 0 -orthogonal co mplemen t of h in g . In Section 1, it is a n easy application of Cheeger’s metho d to prov e that the solution space is star-sh ap ed. That is, if g is a G -in v ariant metric on G/H with nonnegativ e curv ature, then the in verse-linea r p ath, g ( t ), from g (0) = g 0 | p to g (1 ) = g is through nonn egativ ely curv ed G - in v arian t metrics. Here, a path of inn er pro d ucts on p is called “inv erse-linear” if the in verses of the asso ciated path of symmetric matrices form a straigh t lin e. This observ ation redu ces our problem to an infinitesimal one: first classify the directions, g ′ (0), o ne can mo v e a wa y from the norm al homogeneous metric suc h that the in v erse-linea r p ath g ( t ) app ears (up to deriv ativ e information at t = 0) to remain nonnegativ ely curved. T hen, for eac h candidate direction, c hec k how f ar nonnegativ e curv ature is m ain tained along that path. In S ection 2, w e derive cur v ature v ariatio n formulas necessary to implement this strategy , ins p ired by p o wer series deriv ed b y M ¨ uter for curv ature along an inv erse-linear path [8]. In Section 3, we consider an in termediate s u bgroup K b etw een H and G , with su balgebra k , so we ha v e inclusions h ⊂ k ⊂ g . W rite p = m ⊕ s , where m is the orthogonal compliment of h in k and s is th e orthogonal complimen t of k in g . The in v erse-linear path of G -in v arian t ∗ Supp orted by th e Sch w erpunktprogramm D ifferen tialgeometrie of the Deutsche F orsc hungsg esellsc haft. 1 2 LORENZ SCHW A CHH ¨ OFER ∗ , KRISTOPHER T APP metrics on G/H which gradually enlarges k is describ ed as follo w s for all A, B ∈ p : (0.1) g t ( A, B ) =  1 1 − t  · g 0 ( A m , B m ) + g 0 ( A s , B s ) , where sup erscripts denote g 0 -orthogonal pro jections on to the corresp ondin g spaces. This v ariation scales the fib ers of the Riemannian submersion ( G, g 0 ) /H → ( G, g 0 ) /K . F or t < 0, these fib ers are shrunk, and g t has nonn egativ e curv ature b ecause it can b e redescrib ed as a s ubmersion metric obtained b y a Cheeger deformatio n: ( G/H, g t ) = (( G/H, g 0 ) × ( K, ( − 1 /t ) · g 0 )) /K. F or t > 0, these fi b ers are en larged, and the situation is more complicated. W e will pro v e: Theorem 0.1. (1) Th e metric g t has no nne gative curvatur e for smal l t > 0 if and only if ther e exists C > 0 such that for al l X, Y ∈ p , | [ X m , Y m ] m | ≤ C · | [ X, Y ] | . (2) In p articular, if ( K, H ) is a sym metric p air, then g t has nonne gative c urvatur e for smal l t > 0 , and in fact for al l t ∈ ( −∞ , 1 / 4] . P art 2 p ro vides a large class of new examples of homogeneous m etrics with nonnegativ e cu r- v ature. Notice t = 1 / 4 corresp ond s to the scaling factor 1 1 − 1 / 4 = 4 / 3, which app ears elsewhere in the literature as an upp er limit for enlarging the totally geo desic fib ers of certain Riemannian submersions while main taining nonnegativ e curv atur e, including Hopf fibrations [11],[12],[13], and fibrations of a compact Lie group by cosets of an ab elian group [4]. W allac h pro ved in [14] that if ( K, H ) and ( G, K ) are rank 1 symmetric pairs and if the triple ( H , K , G ) satisfies a certain “fatness” p r op ert y , then the metric g t has p ositiv e curv ature for all t ∈ ( −∞ , 1 / 4 ), t 6 = 0. W e re-pro v e W allac h ’s theorem in S ection 3. When H is trivial, g t is a left-inv arian t metric on G scaled up along k . Ziller p osed the question of when su ch a metric g t is nonn egativ ely curv ed [16]. The follo wing answer wa s found in [6]: the metric g t has nonnegativ e cur v ature for small t > 0 if and only if the semi- simple part of k is an ideal of g ; in particular, when g is simple, only abelian subalgebras can b e enlarged. When Ad H acts irreducibly on p , there is only a one-parameter family of G -in v arian t metrics on G/H (coming from scaling), all of whic h are obviously nonnegativ ely curv ed. If there exists an in termediate subalgebra k , b et we en h and g , then there exists at le ast a 2-parameter family of G - in v arian t metrics, and man y spaces with exactly 2-parameters arise from suc h an in termediate subalgebra; suc h sp aces were classified in [3]. Th us, Th eorem 0.1 addr esses the simplest non trivial case of our classification problem. Next, in Chapter 4, we sho w th at more arbitrary metric c hanges preserv e nonnegativ e curv a- ture, assu ming a hyp othesis whic h is similar to (bu t m uc h stronger th an ) that of Theorem 0.1: Theorem 0.2. If ther e exists C > 0 such that for al l X , Y ∈ p , | X m ∧ Y m | ≤ C · | [ X, Y ] | , HOMOGENEOUS METRICS 3 then any left invariant metric on G sufficiently close to g 0 which is Ad H -invariant and i s a c onstant multiple of g 0 on s and h (but arbitr ary on m ) has nonne gative se ctional curvatur e on al l planes c ontaine d in p ; henc e, the induc e d metric on G/H has nonne gative se ctional curvatur e. In p articular, this hyp othesis is satisfie d by the fol lowing chains H ⊂ K ⊂ G : (1) S p (2) ⊂ S U (4) ⊂ S U (5) (2) S U (3) ⊂ S U (4) ∼ = Spin (6) ⊂ Spin (7) (3) G 2 ⊂ Spin (7) ⊂ Spin ( p + 8) for p ∈ { 0 , 1 } , wher e the se c ond inclusion is the lift of the standar d inclusion S O (7) ⊂ S O ( p + 8) (4) S pin ′ (7) ⊂ S p in (8) ⊂ S pin ( p + 9) for p ∈ { 0 , 1 , 2 } , wher e S pin ′ (7) ⊂ S p in (8) is the image of the spin r epr esentation of S pin (7) , and the se c ond is again the lift of S O (8) ⊂ S O ( p + 9) (5) S U (2) ⊂ S O (4) ⊂ G 2 (Her e, S U (2) ⊂ S U (3) ⊂ G 2 , wher e S U (3) ⊂ G 2 is the isotr opy gr oup of S 6 = G 2 /S U (3) ) F or the triples ab o v e, one is f ree to c ho ose the initial direction g ′ (0) of the v ariation g ( t ) to b e an y Ad H -in v arian t self-adjoint endomorp hism of m . F or the first, th ir d and fourth triples, the space of such end omorphisms is 1-dimensional, while for the second triple it is 2-dimensional. F or the fifth triple, the space is 6-dimens ional, b ut only 3-dimensional mo d ulo G-equiv arian t isometry . In all exa mples, there is one additional parameter for scaling s . Some other sp aces are kno wn to admit large-parameter families of nonnegativ elty curved homogeneous metrics ([10 ],[9]), but un lik e our new examples, th ese admit p ositiv ely curv ed homogeneous metrics. The statemen t ab out nonnegativ ely curv ed planes in G is remark able on its o wn , since suc h a metric cannot hav e nonnegativ e secti onal curv ature on al l of G , unless h is ab elian [6 ]. Moreo ver, when constructing nonnegativ ely curved metrics w ith n ormal homogeneous col lars, it is precisely the n on n egativ e curv ature of these planes whic h is needed [7 ]. The auth ors are pleased to thank W olfgang Ziller for helpful con v ersations, and the American Institute of Mathematics for hospitalit y and fund ing at a w orksh op on nonnegativ e curv ature in Septem b er, 2007, w here p ortions of this work were discussed. 1. In verse-linea r p a t hs In this section, w e pro v e as a quic k application of Cheeger’s metho d: Prop osition 1.1. If g is a G -invariant metric on G/H with nonne gative curvatur e, then the inv e rse-line ar p ath, g ( t ) , fr om g (0) = g 0 | p to g (1) = g is thr ough nonne gatively curve d G -invariant metrics. The case H = { e } is foun d in [5]. W e p ro v e this b y sho wing that an y G -inv arian t m etric with nonnegativ e curv ature on G/H is connected to the n ormal homogeneous metric ( G, g 0 ) /H via a canonical path of nonnegativ ely curve d G - in v arian t met rics. See [5] for relev ant bac kground on C heeger’s m etho d, whic h is at the heart of the p r o of. 4 LORENZ SCHW A CHH ¨ OFER ∗ , KRISTOPHER T APP Pr o of of Pr op osition 1.1. Let h b e an Ad H -in v arian t inner pro duct on p . Let M d enote G/H with the G -inv ariant m etric ind uced by h . Assume that M has nonnegativ e curv atur e. Consid er the follo wing family of nonn egativ ely curv ed Riemannian submersion metrics on M : M t =  M ×  G, 1 t · g 0  /G. Here, G act s diagonally on M × G as g ⋆ ( p, a ) = ( g ⋆ p, ag − 1 ). This family exte nds smo othly at t = 0 to the original metric M 0 = M . Notic e that eac h M t is G -in v ariant, and is therefore induced by some Ad H -in v arian t in n er pro duct, h t , o n p . Let { e i } denote a g 0 -orthonormal basis of p for wh ic h h is diagonalized, with eigen v alues { λ i } . Then the metrics h t , considered as symmetric matrices with r esp ect to this basis, ev olv e as f ollo ws: h t = h ( I + t · h ) − 1 = diag  λ i 1 + tλ i  . Notice that M t con v erges to a p oint at t → ∞ , b ut t · M t con v erges to the normal homogeneous space ( G , g 0 ) /H . This sho ws there exists a p ath of nonnegativ ely curve d G -in v arian t metrics joining M to ( G, g 0 ) /H . W e’d lik e to see that, u p to re-parametrizatio n and re-scali ng, this path is exact ly the in v erse-linear path, ˜ h s , from ˜ h 0 = ( G, g 0 ) /H to ˜ h 1 = M . The initial d irection of this in v erse-linear path is Ψ = ( I − h − 1 ), mea ning that, in the b asis { e i } , w e ha v e: ˜ h s = ( I − s Ψ) − 1 = diag  1 1 − s (1 − λ − 1 i )  . It is straigh tforw ard no w to c heck th at s · ˜ h s = h t when s = 1 1+ t .  2. Cur v a tur e v aria tion formulas Prop osition 1.1 suggests an infinitesimal strategy for classifying the G -in v arian t metrics with nonnegativ e curv ature on G/H . The first step is to classify the directions, Ψ, in w hic h one can mo v e a wa y from a fixed normal h omogeneous m etric s uc h that curv ature v ariatio n formulas predict that n onnegativ e curv ature is main tained along the inv erse-linear path in that direction. In this section, w e der ive the relev ant curv ature v ariation f orm ulas. A p ath g t of Ad H -in v arian t inner pro d ucts on p ca n be describ ed in terms of g 0 | p as: g t ( A, B ) = g 0 (Φ t A, B ) for all A, B ∈ p , w here Φ t is a family of g 0 -self-adjoin t, Ad H -in v arian t, p ositiv e-definite endo- morphims of p . W e henceforth assume the path is in v erse-linear, which means that t 7→ Φ − 1 t is linear, so that: (2.1) Φ t = ( I − t · Ψ) − 1 for s ome g 0 -self-adjoin t, Ad H -in v arian t map Ψ : p → p . No tice that Ψ = d dt | t =0 Φ t . HOMOGENEOUS METRICS 5 It is us efu l to henceforth extend Φ t and Ψ to b e endomorphisms of all of g by defining eac h Φ t to b e the ident it y on h and defining Ψ to b e zero on h . Notice that Equation 2.1 still holds for th ese ext ensions. F or X, Y ∈ p ∼ = T H ( G/H ), we let k ( t ) denote the un n ormalized sectio nal cur v ature with resp ect to g t of the vect ors Φ − 1 t X and Φ − 1 t Y . The domain of k is the op en in terv al of t ’s for whic h Φ t represent s a non-degenerate metric, whic h dep end s on the eigen v alues of Ψ. Notice that k (0) = 0 if and only if [ X , Y ] = 0. F or such initially-zero cu r v ature planes, w e will now exhibit a p ow er series expr ession for k ( t ). It is useful to lab el the follo w ing expressions: A = [Ψ X , Y ] + [ X , Ψ Y ] , D 0 = [Ψ X , Ψ Y ] − Ψ A. Prop osition 2.1. If X, Y ∈ p c ommute, then k (0) = k ′ (0) = 0 , k ′′ (0) = 3 2 | A h | 2 , and (1 / 6) k ′′′ (0) = h A − (3 / 2) A h , [Ψ X, Ψ Y ] i + h [Ψ X , X ] , Ψ[Ψ Y , Y ] i −h [ X, Ψ Y ] , Ψ A i − h [Ψ X , Y ] , Ψ[Ψ X , Y ] i , and for al l t in the domain of k , k ( t ) = t 2 · (1 / 2) k ′′ (0) + t 3 · (1 / 6) k ′′′ (0) − 3 4 t 4 · | D p 0 | 2 g t . Definition 2.2. W e refer to Ψ (or to the inv er s e-linear metric v ariation it determines) as infinitesimal ly nonne gative if for all X , Y ∈ p , there exists ǫ > 0 suc h that k ( t ) ≥ 0 for t ∈ [0 , ǫ ). This is clearly tru e for pairs X , Y whic h don’t comm ute, so it is equiv alen t to chec k the condition for pairs whic h do comm ute. This g iv es: Prop osition 2.3. Ψ is infinitesimal ly nonne gative if and only if for al l X , Y ∈ p such that [ X, Y ] = 0 and A h = 0 , we have that k ′′′ (0) ≥ 0 , and k ′′′ (0) = 0 implies that D p 0 = 0 . By Prop osition 1.1, one w ill lo cate all of the n onnegativ ely curved G -in v arian t metrics on G/H b y searc hing only along infi nitesimally nonn egativ e paths. This approac h w as used in [5] (in the ca se wh er e H is trivial) to restrict the sp ace of p ossible nonnegativ ely curv ed left- in v arian t metrics on G . Prop osition 2.1 is a sp ecial case of a p o wer-series f or k ( t ), whic h we no w derive, wh ic h do es not assume that X, Y commute. F or this general p o w er series, it is useful to den ote: A = [Ψ X, Y ] + [ X , Ψ Y ] , B = [Ψ X, Ψ Y ] , C = [Ψ X , Y ] − [ X, Ψ Y ] , D = Ψ 2 [ X, Y ] + B − Ψ A With this notation w e h av e: 6 LORENZ SCHW A CHH ¨ OFER ∗ , KRISTOPHER T APP Prop osition 2.4. F or any X , Y ∈ p and al l t in the domain of k , k ( t ) = α + β t + γ t 2 + δ t 3 − 3 4 t 4 · | D p | 2 g t . wher e α = | [ X, Y ] h | 2 + 1 4 | [ X, Y ] p | 2 β = − 3 4 h Ψ[ X, Y ] , [ X , Y ] i − 3 2 h [ X, Y ] h , A i γ = − 3 4 | Ψ[ X, Y ] | 2 + 3 2 h Ψ[ X, Y ] , A i − 3 2 h [ X, Y ] m , B i + 3 4 | A h | 2 δ = − 3 4 h Ψ 3 [ X, Y ] , [ X , Y ] i + 3 2 h Ψ 2 [ X, Y ] , A i − 3 2 h Ψ[ X, Y ] , B i − 3 4 h Ψ A, A i − 1 4 h Ψ C, C i + h Ψ[Ψ X , X ] , [Ψ Y , Y ] i + h A, B i − 3 2 h A h , B i . Pr o of. By O’Neill’s formula, k ( t ) = κ ( t ) + A ( t ), where κ ( t ) is the unn ormalized sect ional curv ature of Φ − 1 t X and Φ − 1 t Y in the left-i n v arian t metric on G determined by Φ t , and A ( t ) is the O’Neill term. Usin g the expression Φ − 1 t = I − t Ψ, w e ha v e: 4 3 A ( t ) = | [Φ − 1 t X, Φ − 1 t Y ] h | 2 = | [ X − t Ψ X , Y − t Ψ Y ] h | 2 (2.2) = | [ X , Y ] h − tA h + t 2 B h | 2 = | [ X , Y ] h | 2 − 2 t h [ X , Y ] h , A i + t 2  | A h | 2 + 2 h [ X , Y ] h , B i  − 2 t 3 h A h , B i + t 4 | B h | 2 . It is prov en in [5] that κ ( t ) = α + β t + γ t 2 + δ t 3 − 3 4 t 4 | D | 2 g t , wh ere α = 1 4 | [ X, Y ] | 2 β = − 3 4 h Ψ[ X, Y ] , [ X, Y ] i γ = − 3 4 | Ψ[ X, Y ] | 2 + 3 2 h Ψ[ X, Y ] , A i − 3 2 h [ X, Y ] , B i δ = − 3 4 h Ψ 3 [ X, Y ] , [ X , Y ] i + 3 2 h Ψ 2 [ X, Y ] , A i − 3 2 h Ψ[ X, Y ] , B i − 3 4 h Ψ A, A i − 1 4 h Ψ C, C i + h Ψ[Ψ X , X ] , [Ψ Y , Y ] i + h A, B i . The abov e expression for γ is simpler than the o ne f ound in [5]; to ac h iev e this simp lification, use the Jacobi iden tit y to write h [Ψ X , X ] , [Ψ Y , Y ] i = h [ X , Y ] , B i − h [ X , Ψ Y ] , [Ψ X , Y ] i . It is straigh tforw ard to com bine the ab o ve p o w er series for A ( t ) and κ ( t ). Notice that the t 4 -term of k ( t ) = κ ( t ) + A ( t ) is Γ( t ) = 3 4 t 4 ( | B h | − | D | 2 g t ), whic h simplifies b ecause: | B h | 2 − | D | 2 g t = | D h | 2 − ( | D h | 2 g t + | D p | 2 g t ) = | D h | 2 − ( | D h | 2 + | D p | 2 g t ) = −| D p | 2 g t .  HOMOGENEOUS METRICS 7 3. S caling up an inte rmedia te subalgebr a In this section, we stud y and pro v e Theorem 0.1, whic h pr ovides conditions under whic h enlarging an in termediate subalgebra m aintains n on n egativ e curv atur e on G/H . Supp ose K is an inte rmediate subgroup b et w een H a nd G , with Lie algebra k , so we ha v e inclusions h ⊂ k ⊂ g . W rite p = m ⊕ s , where m is the orthogonal compliment of h in k an d s is the orthogonal compliment of k in g . Let Ψ denote th e pro jecio n on to m , so that Ψ ( A ) = A m for all A ∈ g . Notice that Ψ determines the inv erse-linear path, g t , of G -inv arian t metrics on G/H describ ed in Equation 0.1, wh ic h gradually enlarges the fib ers of th e Riemannian submersion ( G, g 0 ) /H → ( G, g 0 ) /K . W e seek conditions und er wh ic h g t has nonn egativ e curv ature for small t > 0. When ( K, H ) is a symmetric p air, it is easy to s ho w that Ψ is infinitesimally n onnegativ e, whic h pr o v id es evidence for Th eorem 0.1. T o fully prov e this pr op osition, we r equ ire a p ow er series expr ession for k ( t ). Let X, Y ∈ p = m ⊕ s , and den ote M = [ X m , Y m ] , S = [ X s , Y s ] . With this notation, Prop osition 2.4 simplifies to: k ( t ) =  a | M h | 2 + b h M h , S h i + | S h | 2  +  a | M m | 2 + b h M m , S m i + c | S m | 2  + 1 4 | [ X, Y ] s | 2 = T 1 + T 2 + T 3 . where, a = 1 − 3 t + 3 t 2 − t 3 , b = 2 − 3 t, (3.1) a = 1 4 − 3 4 · t + 3 4 · t 2 − 1 4 t 3 , b = 1 2 − 3 2 · t, c = 1 4 − 3 t 4(1 − t ) . Pr o of of The or em 0.1. Using Cauc h y-Sw artz, T 1 ≥ 0 when t ≤ 4 / 3 b ecause the discrimin an t is n onnegativ e: 4 a − b 2 = 3 t 2 − 4 t 3 ≥ 0 . If ( K, H ) is a sym metric pair, then M m = 0, so T 2 = c | S m | 2 , wh ic h is n onnegativ e for t ≤ 1 / 4. This pro ves p art (2) of the theorem. F or part (1 ), first assume there exists C > 0 suc h that for all X , Y ∈ p , | M m | ≤ C · | [ X, Y ] | . Notice th at if t < 1 / 2, then T 1 ≥ 1 10 | M h + S h | 2 = 1 10 | [ X, Y ] h | 2 . This is b ecause T 1 − 1 10 | M h + S h | 2 =  a − 1 10  | M h | 2 +  b − 2 10  h M h , S h i +  1 − 1 10  | S h | 2 , 8 LORENZ SCHW A CHH ¨ OFER ∗ , KRISTOPHER T APP whic h is nonnegativ e b ecause the discriminan t is nonnegativ e: ∆ = 4  a − 1 10   1 − 1 10  −  b − 2 10  2 = 9 5 t 2 − 18 5 t 3 ≥ 0 . F or T 2 w e h a v e: T 2 ≥ a | M m | 2 − b | M m | · | S m | + c | S m | 2 ≥ g ( t ) · | M m | 2 , where g ( t ) = 4 ac − b 2 4 c = t 3 ( t − 1) 1 − 4 t . Notice g ( t ) is a n egativ e-v alued f unction w ith lim t → 0 g ( t ) = 0. In su mmary , for t < 1 / 2 w e ha v e: (3.2) k ( t ) = T 1 + T 2 + T 3 ≥ 1 10 | [ X, Y ] h | 2 + g ( t ) | M m | 2 + 1 4 | [ X, Y ] s | 2 . A t time t = 0, T 2 = 1 4 | M m + S m | 2 , wh ic h indicates that for small t > 0, T 2 can only b e negativ e when M m is clo se to − S m . T o mak e this precise, d efine “dist” as: dist( A, B ) = m ax {| ∠ ( A, B ) | , | 1 − | A | / | B ||} . Giv en ǫ > 0, we claim th ere exists δ, K > 0 s u c h that if dist( M m , − S m ) > ǫ , then T 2 | M m + S m | 2 ≥ K for all t ∈ [0 , δ ]. In particular T 2 ≥ 0 (and therefore k ( t ) ≥ 0) for t ∈ [0 , δ ]. T o see this, n otice that T 2 | M m + S m | 2 remains unchange d wh en M m and S m are b oth scaled b y the s ame factor, s o one can assume that the smaller of their lengths equals 1. If the larger of their lengths is ≥ 10, then it is easy to explicitly fin d δ , K as ab ov e. When the larger of their lengths is ≤ 10, a compactness a rgument suffices to fin d δ , K . So it remains to consider the case where d ist( M m , − S m ) < ǫ , with ǫ > 0 c hosen suc h that | [ X, Y ] m | 2 = | M m + S m | 2 ≤ 1 2 C 2 | M m | 2 . In this case, w e ha v e by hyp othesis: | M m | 2 ≤ C 2 ·  | [ X, Y ] h | 2 + | [ X , Y ] m | 2 + | [ X , Y ] s | 2  ≤ C 2 ·  | [ X, Y ] h | 2 + 1 2 C 2 | M m | 2 + | [ X, Y ] s | 2  . Solving this sho ws that | M m | 2 ≤ 2 C 2  | [ X, Y ] h | 2 + | [ X , Y ] s | 2  . C ombining this with Equa- tion 3. 2 sho ws that k ( t ) is nonnegativ e for all t small enough that 2 g ( t ) C 2 < 1 / 10. The other direction of part (1) of the Th eorem follo ws from s imilar argumen ts.  Next, we reco ver an imp ortan t theorem d ue to W allac h , from which he construct his w ell- kno wn n on-normal homogeneous metrics of positive curv ature [14]. Recall that the triplet ( H , K, G ) d etermines a “fat homogeneous bu ndle” if [ A, B ] 6 = 0 for all non-zero A ∈ m and B ∈ s ; see [15] for a surve y of literature on fat bundles. Prop osition 3.1. (Wal lach) If ( K, H ) and ( G, K ) ar e r ank 1 symmetric p airs, and ( H , K , G ) determines a fat homo gene ous bund le, then g t has p ositive cu rvatur e for al l t ∈ ( −∞ , 1 / 4) , t 6 = 0 . HOMOGENEOUS METRICS 9 Pr o of. F or linearly indep end en t X, Y ∈ p , if k ( t ) = 0 at some non -zero t ∈ ( −∞ , 1 / 4), then the pro of of part (2) of Theorem 0.1 implies that M = [ X m , Y m ] = 0 and S = [ X s , Y s ] = 0 and [ X, Y ] s = 0. So the rank one hypothesis implies that X m k Y m and X s k Y s . Th us, after a change of basis of span { X , Y } , w e can assume that X ∈ m and Y ∈ s . But then the fact that [ X, Y ] s = [ X , Y ] = 0 con tradicts fatness.  Under the h yp otheses of the ab ov e p rop osition, if k (0) = 0, it is not h ard to see that k ′′ (0) > 0; that is, all initially zero-curv ature p lanes b ecome p ositiv ely curv ed to second order. Since the ev en deriv ative s of k ( t ) are insensitiv e to the sign of Ψ, it do es not matte r here whether t increases or d ecreases from ze ro; in eit her ca se, the A -tensor mak es all initially zero curv ature planes b ec ome positive ly curv ed to second ord er. 4. Fur th er Examples In this section, w e prov e Th eorem 0.2, w hic h giv es examples of left in v arian t metrics with man y nonnegativ ely cur v ed planes and , as a consequence, h omogeneous spaces with unexp ect - edly large f amilies of n onnegativ ely curv ed homogeneous metrics. Consider compact Lie groups H ⊂ K ⊂ G w ith Lie algebras h ⊂ k ⊂ g , and decomp ose g = h ⊕ p = h ⊕ m ⊕ s , as in the previous sect ion. Prop osition 4.1. If ther e exists C > 0 su c h that for al l X, Y ∈ p , | X m ∧ Y m | ≤ C · | [ X, Y ] | , then any inverse-line ar variation Φ t = ( I − t Ψ) − 1 of left- invariant Ad H -invariant metrics on G for which Ψ | s = Ψ | h = 0 is thr ough metrics which for sufficiently smal l t have the pr op erty that al l planes in p ar e nonne gatively curve d. The hypothesis of the pr op osition is cle arly stronger than the cond ition of Theorem 0.1 under wh ic h m can only b e scaled up preservin g nonnegativ e curv ature. Und er this stronger h yp othesis, the prop osition says that arbitrary small c hanges can b e m ade the metric on m , and it not only giv es information ab out the metric on G/H , but also on G . This prop osition clearly implies the fir st p art Theorem 0.2 s ince an y m etric close t o the normal h omogeneous one can b e joined b y an in v er s e linea r path in that neig h b orho o d . Pr o of. Let X , Y ∈ p . As in Chapter 2, we ha v e f or the curv atur e of the left-in v arian t metric on G : κ ( t ) = α + β t + γ t 2 + δ t 3 − 3 4 t 4 | D | 2 g t , where the co efficien ts { α, β , γ , δ } are defined in terms of the expressions A, B , C , D . Notice that | A k | ≤ λ 1 · | X m ∧ Y m | , where λ 1 is the norm of the linear map ∧ 2 m → k d efined as x ∧ y 7→ [Ψ x, y ] + [ x, Ψ y ]. Similarly , | B | ≤ λ 2 · | X m ∧ Y m | , wh ere λ 2 is the norm of the linea r map ∧ 2 m → k d efined as x ∧ y 7→ [Ψ x, Ψ y ]. 10 LORENZ SCHW A CHH ¨ OFER ∗ , KRISTOPHER T APP Next, defin e E = − 1 4 h Ψ C, C i + h Ψ[Ψ X , X ] , [Ψ Y , Y ] i , which equals t w o of th e terms in the definition of δ . W e claim that | E | ≤ λ 3 · | X m ∧ Y m | 2 for some constant λ 3 . T o s ee this, fi rst consider the symmetric linear map ρ : m × m → k defin ed as ρ ( x, y ) = 1 2 ([Ψ x, y ] − [ x, Ψ y ]). Next consid er the m ulti-linear map Θ : ∧ 2 m × ∧ 2 m → k whic h is d efined as Θ( x ∧ y , z ∧ w ) := h Ψ ρ ( x, z ) , ρ ( y , w ) i − h Ψ ρ ( x, w ) , ρ ( y , z ) i . Since Θ ( X m ∧ Y m , X m ∧ Y m ) = E , we ma y tak e λ 3 to b e the norm of Θ. Since the coefficients { β , γ , δ } and the term D are defined in terms of the ab ov e-b oun ded expressions, it is a straigh tforward to use Cauc h y-Sc h w artz to boun d their norms and thereb y sho w that there exists a constan t λ ′ suc h th at | β | , | γ | , | δ | , | D | ≤ λ ′ ·  | [ X, Y ] | 2 + | [ X , Y ] | · | X m ∧ Y m | + | X m ∧ Y m | 2  ≤ λ · | [ X, Y ] | 2 , where λ = λ ′ (1 + C + C 2 ). In fact , the ab o v e b ound for | D | also holds f or | D | g t as long at t is small enough that g t is b ounded in terms of g 0 . Thus: κ ( t ) = α + β t + γ t 2 + δ t 3 − 3 4 t 4 | D | 2 g t ≥ 1 4 | [ X, Y ] | 2 − ( t + t 2 + t 3 + t 4 ) λ · | [ X , Y ] | 2 whic h is clearly nonnegativ e for sufficien tly small t > 0.  It only remains to p ro v e that th e subgroups c hains fr om Th eorem 0.2 sat isfy the in equalit y condition of the ab ov e prop ositi on. Prop osition 4.2. The fol lowing triples satisfy the hyp othesis of Pr op osition 4.1. (1) S p (2) ⊂ S U (4) ⊂ S U (5) , (2) S U (3) ⊂ S U (4) ∼ = S pin (6) ⊂ S pin (7) , (3) G 2 ⊂ S pin (7) ⊂ S pin ( p + 8) for p ∈ { 0 , 1 } , wher e the se c ond inclusion is the lift of the inclusion S O (7) ⊂ S O ( p + 8) . (4) S pin ′ (7) ⊂ S p in (8) ⊂ S pin ( p + 9) for p ∈ { 0 , 1 , 2 } , wher e S pin ′ (7) ⊂ S p in (8) is the image of the spin r epr esentation of S pin (7) , and the se c ond is again the lift of S O (8) ⊂ S O ( p + 9) . Pr o of. W e den ote the groups in all cases as H ⊂ K ⊂ G . S upp ose this hyp othesis is not satisfied. Then there exist sequences { X r } and { Y r } in m ⊕ s suc h that X m r , Y m r ∈ m is an orthonormal pair, a nd lim[ X r , Y r ] = 0. P assing to a sub sequence, we may assum e that X m := lim X m r and Y m := lim Y m r exist, an d w e let B := [ X m , Y m ] ∈ k . Since K/H is a sp here and hence the normal homogeneous metric has p o sitiv e curv atur e, it follo ws that B 6 = 0. Also, 0 = lim[ X r , Y r ] k = lim[ X m r , Y m r ] + [ X s r , Y s r ] k , so that B = − lim[ X s r , Y s r ] k , HOMOGENEOUS METRICS 11 so that, in particular, we may assume that [ X s r , Y s r ] k 6 = 0 for all r . F or th e first trip le, K/H = S pin (6) /S pin (5) ∼ = S O (6) /S O (5), so that w e ma y r egard X m , Y m ∈ so (5) ⊥ ⊂ so (6), hence B = [ X m , Y m ] ∈ so (5) ⊂ so (6) is a matrix of real rank 2, so that its cen tralizer is isomorp h ic to so (2) ⊕ so (4). On the other hand, if we regard [ X s r , Y s r ] u (4) ∈ u (4) ⊂ su (5) as a complex matrix wh er e X s r , Y s r ∈ su (4) ⊥ ⊂ su ( 5), then one v erifies that [ X s r , Y s r ] u (4) is conju gate to a uniqu e elemen t of the form diag ( λ r 1 i, λ r 2 i, 0 , 0) with λ r 1 ≥ λ r 2 . But lim[ X s r , Y s r ] u (4) = − B 6 = 0 ∈ su (4) exists, so that this limit is conjugate to an elemen t of the form di ag ( λi, − λi, 0 , 0) with λ > 0, whose cen tralize r in su (4) is isomorphic to s ( su (2) ⊕ u (1 ) ⊕ u (1)). But th e central izer of B is isomorph ic to so (2) ⊕ so (4) whic h yields the desired con trad iction in this ca se. F or all of th e remaining cases w e h a ve G/K = S pin ( m ) /S pin ( n ) with the inclusion K ⊂ G induced by the inclus ion S O ( n ) ⊂ S O ( m ) for some ( n, m ). It follo ws that for all X , Y ∈ s = so ( n ) ⊥ , [ X, Y ] k ∈ so ( n ) is a matrix which has rank at most 2( m − n ). Therefore, since B = − lim[ X s r , Y s r ] 6 = 0, it follo ws that 0 6 = B ∈ so ( n ) is a matrix of such a rank. F or the second triple, the rank of B ∈ so (6) equals 2( m − n ) = 2, hen ce its cen tralizer is isomorphic to so (2) ⊕ so (4) ⊂ so (6). On the other hand, for X m , Y m ∈ su (3) ⊥ ⊂ su (4), it is straigh tforw ard to v erify that B = [ X m , Y m ] ∈ su (4) is n ot regular, hence B is conjugate to an element of the form diag( λ 1 i, λ 2 i, λ 3 i, 0) ∈ su (4) with λ 1 + λ 2 + λ 3 = 0. T herefore, the cen tralizer of B is ei- ther s ( u (1) ⊕ u (1 ) ⊕ u (1) ⊕ u (1)) or s ( u (2 ) ⊕ u (1) ⊕ u (1)), none of w hic h is isomorph ic to so (2) ⊕ so (4) whic h is a con tradiction and fin ishes the pro of for this example. F or the third triple, we will sho w that for any orthonormal pair X m , Y m ∈ m , the rank of B = [ X m , Y m ] ∈ so (7) equals 6 whic h will giv e th e desired contradicti on as 2( m − n ) = 2( p + 1) ≤ 4. F or this, we regard G 2 as the automorphism group of the o ctonions O whic h lea ves 1 ∈ O and hence its orthogonal complement I m ( O )) in v arian t, and this representati on of G 2 on R 7 ∼ = I m ( O ) lifts to the inclusions g 2 ⊂ so ( I m ( O )) and G 2 ⊂ S pin (7). Then so ( I m ( O )) = g 2 ⊕ { ad q : I m ( O ) − → I m ( O ) } is an orthogonal decomp osition, where ad q : I m ( O ) → I m ( O ) is giv en by ad q ( x ) := q · x − x · q since the second summand is G 2 -equiv arian tly isomorphic to I m ( O ). Th us, it remains to sho w that for an orthonormal pair q , q ′ ∈ I m ( O ), the rank of [ ad q , ad q ′ ] ∈ so ( I m ( O )) equals 6. Since G 2 acts tr an s itiv ely on orthonormal pairs, we ma y assume that q = i and q ′ = j . No w it is straigh tforward to v erify that th e k ernel of [ ad i , ad j ] : I m ( O ) → I m ( O ) is sp anned by k ∈ H and is th us one-dimensional. A similar argument applies to the last case. The orthogonal complemen t of so (7) ′ ⊂ so (8) consists of { L q | q ∈ I m ( O ) } , where L q : O → O denotes left multiplica tion. Assuming w.l.o.g. that X m = L i and Y m = L j , it is staig h tforw ard to verify that B = [ L i , L j ] ∈ so (8) is regular, con tradicting that 2( m − n ) = 2( p + 1) ≤ 6 b y assu m ption.  Prop osition 4.3. The triple S U (2) ⊂ S O (4) ⊂ G 2 satisfies the hyp othesis of L emma 4.1. 12 LORENZ SCHW A CHH ¨ OFER ∗ , KRISTOPHER T APP Pr o of. W e d ecomp ose the Lie algebra g 2 according to the symmetric pair decomp o sition of G 2 /S O (4) as g 2 = ( sp (1) 3 ⊕ sp (1) 1 ) ⊕ H 2 , where sp (1) 3 ⊂ sp (2) is the Lie alg ebra spanned b y E 0 := 3 i i ! , E + := 0 √ 3 − √ 3 2 j ! , E − := 0 √ 3 i √ 3 i 2 k ! and acts on H 2 from the left, whereas sp (1) 1 = Im( H ) acts via scalar multiplic ation from the righ t. Indeed, one verifies the brac k et relations [ E 0 , E ± ] = ± 2 E ∓ , and [ E + , E − ] = 2 E 0 . Since sp (1) 1 is the subalgebra whic h is con tained in su (3) ⊂ g 2 , it follo w s th at in our case, m = sp (1 ) 3 and s = H 2 . Thus, w e ha v e to sh o w that there cannot b e sequences of v ectors of the form (4.1) X n := E + + ρ n ~ v n and Y n := E − + ρ ′ n ~ w n with unit v ectors ~ v n , ~ w n ∈ H 2 and ρ n , ρ ′ n ≥ 0 suc h that lim [ X n , Y n ] = 0. By con tradiction, w e assume that such a sequence of v ectors exists and thus ma y assume that the un it v ecto rs ~ v := lim ~ v n and ~ w := lim ~ w n exist. Th en we ha v e 0 = lim h E 0 , [ X n , Y n ] i = lim h [ E 0 , X n ] , Y n i = lim h 2 E − + ρ n E 0 · ~ v n , E − + ρ ′ n ~ w n i = 2 || E − || 2 + lim ρ n ρ ′ n h E 0 · ~ v n , ~ w n i . F rom this we conclude that (4.2) lim inf ρ n ρ ′ n > 0 and h E 0 · ~ v , ~ w i ≤ 0 . Next, f or q ∈ sp (1) 1 , we hav e 0 = lim h [ X n , Y n ] , q i = lim h X n , [ Y n , q ] i = lim ρ n ρ ′ n h ~ v n , ~ w n · q i , and since lim inf ρ n ρ ′ n > 0, it f ollo ws that (4.3) h ~ v , ~ w · q i = 0 for all q ∈ sp (1) 1 = Im( H ) . Finally , 0 = lim[ X n , Y n ] s = lim( ρ ′ n E + ~ w n − ρ n E − ~ v n ) . By (4.2 ), we ma y assume th at ρ ′ n > 0 f or all n . Moreo ver, lim E − ~ v n = E − ~ v 6 = 0 and lim E + ~ w n = E + ~ w 6 = 0 since E ± are regular matrices, so that (4.4) 0 = E + ~ w − c 2 E − ~ v , where c 2 := lim ρ n ρ ′ n ∈ (0 , ∞ ) . HOMOGENEOUS METRICS 13 W e shall no w fin ish our contradictio n b y showing that th er e cannot exist un it vec tors ~ v , ~ w ∈ H 2 satisfying (4.2), (4.3) and (4.4). Namely , ~ w = c 2 E − 1 + E − ~ v by (4.4) , and using the in v ariance of these conditions under scalar m ultiplicatio n from the right, we ma y assume w.l.o.g. that ~ v = λ z 1 + z 2 j ! , and ~ w = c 2 E − 1 + E − ~ v = c 2 − 4 √ 3 z 1 k +  4 √ 3 z 2 − λ  i z 1 i + z 2 k ! , where λ ≥ 0 , c > 0 and z 1 , z 2 ∈ C . Next, (4.3) holds if for all q ∈ sp (1) 1 , 0 = h ~ v , ~ w · q i = c 2 Re (( λ ( − 4 √ 3 z 1 k + ( 4 √ 3 z 2 − λ ) i ) + ( z 1 − z 2 j )( z 1 i + z 2 k )) q ) = c 2 Re ((( λ ( 4 √ 3 z 2 − λ ) + | z 1 | 2 − | z 2 | 2 ) i + 2 z 1 ( z 2 − 2 √ 3 λ ) k )) q ) . If w e substitute q = i , q = j and q = k , we get therefore the equations (4.5) λ  4 √ 3 Re ( z 2 ) − λ  + | z 1 | 2 − | z 2 | 2 = 0 , and z 1  z 2 − 2 √ 3 λ  = 0 . If z 1 6 = 0, then by the second equation we hav e z 2 = Re( z 2 ) = 2 √ 3 λ . Su bstituting this in to the first equatio n of (4.5) implies that 1 3 λ 2 + | z 1 | 2 = 0, whic h is imp ossible for z 1 6 = 0. Therefore, w e conclude from (4.5 ) that (4.6) z 1 = 0 , and | z 2 | 2 = λ  4 √ 3 Re ( z 2 ) − λ  . Finally , w e calculate from (4.6) h E 0 · ~ v , ~ w i = c 2 * E 0 λ z 2 j ! , ( 4 √ 3 z 2 − λ ) i z 2 k !+ = c 2 * 3 λi z 2 k ! , ( 4 √ 3 z 2 − λ ) i z 2 k !+ = c 2 (3 λ ( 4 √ 3 Re ( z 2 ) − λ ) | {z } = | z 2 | 2 b y (4.6) + | z 2 | 2 ) = 4 c 2 | z 2 | 2 . Since 4 c 2 | z 2 | 2 = h E 0 · ~ v , ~ w i ≤ 0 b y (4.2) and c > 0, we co nclude that z 2 = 0, and thus λ = 0 b y (4 .6), i.e . ~ v = ~ w = 0. On the other h and, ~ v and ~ w m ust be unit vect ors which is a con tradictio n.  14 LORENZ SCHW A CHH ¨ OFER ∗ , KRISTOPHER T APP Referen ces 1. Brown, Finck, Sp encer, T app, W u, I nvariant metrics with nonne gative curvatur e on c omp act Lie gr oups , Cannadian Math. Bull., to app ear. 2. J. Cheeger, Some examples of manifol ds of nonne gative curvatur e , J. D ifferentia l Geom. 8 (1972), 623–628. 3. W. D ickinson and M. 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Press, 2007 F akul t ¨ at f ¨ ur Ma thema tik, Technische U niversit ¨ at Dor tmund , Vo gelpothsweg 87, 44221 Dor t- mund, G ermany E-mail addr ess : lschwach@ma th.uni-dor tmund.de Dep ar tment of Ma thema tics, S aint Joseph Un iversity, 5600 City A venue Philade lphia, P A 19131 E-mail addr ess : ktapp@sju.e du