Ricci-flat Kähler metrics on tangent bundles of rank-one symmetric spaces of compact type
aa r X i v : . [ m a t h . DG ] M a y RICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONESYMMETRIC SPACES OF COMPACT TYPE
P. M. GADEA, J. C. GONZ ´ALEZ-D ´AVILA, AND I. V. MYKYTYUKA
BSTRACT . We give an explicit description of all complete G -invariant Ricci-flat K¨ahlermetrics on the tangent bundle T ( G / K ) ∼ = G C / K C of rank-one Riemannian symmetric spaces G / K of compact type, in terms of associated vector-functions.
1. I
NTRODUCTION
Over the latest decades there has been considerable interest in Ricci-flat K¨ahler metricswhose underlying manifold is diffeomorphic to the tangent bundle T ( G / K ) of a Riemanniansymmetric space G / K of compact type. For instance, a remarkable class of Ricci-flat K¨ahlermanifolds of cohomogeneity one was discovered by M. Stenzel [18]. This has originated agreat deal of papers. To cite but a few: M. Cvetiˇc, G. W. Gibbons, H. L¨u and C. N. Pope [5]studied certain harmonic forms on these manifolds and found an explicit formula for theStenzel metrics in terms of hypergeometric functions. Earlier, T. C. Lee [11] gave an explicitformula of the Stenzel metrics for classical spaces G / K but in another vein, using the ap-proach of G. Patrizio and P. Wong [17]. Remark also that in the case of the standard sphere S , the Stenzel metrics coincide with the well-known Eguchi-Hanson metrics [7]. On theother hand, and as it is well known, Stenzel metrics continue being a source of results bothin physics and differential geometry. We cite here only to G. Oliveira [15] and M. Ionel andT. A. Ivey [10].In the present paper we give an explicit description of all complete G -invariant Ricci-flatK¨ahler metrics on the tangent bundle T ( G / K ) of rank-one Riemannian symmetric spaces G / K of compact type or, equivalently, on the complexification G C / K C of G / K . To this end,reached in our main assertions (Theorem 4.1 and its Corollary 4.3), we use the method ofour article [8], giving the result in terms of associated vector-functions (see below in thisintroduction). In this article it is also shown that this set of metrics contains a new familyof metrics which are not ∂ ¯ ∂ -exact if G / K ∈ { C P n , n > } , and coincides with the set of ∂ ¯ ∂ -exact Stenzel metrics for any of the latter spaces G / K . Mathematics Subject Classification.
Key words and phrases.
Invariant Ricci-flat K¨ahler structures, rank-one semisimple Riemannian symmetricspaces of compact type, restricted roots.Research supported by the Ministry of Economy, Industry and Competitiveness, Spain, under ProjectMTM2016-77093-P..
Remark here that until now, in the case of the space C P n ( n > ) , all known Ricci-flatK¨ahler metrics were Calabi metrics, so being hyper-K¨ahlerian and thus automatically Ricci-flat (see O. Biquard and P. Gauduchon [2, 3] and E. Calabi [4]). Since by A. Dancer andM.Y. Wang [6, Theorem 1.1] any complete G -invariant hyper-K¨ahlerian metric on G / K = C P n ( n > ) coincides with the Calabi metric, our new metrics are not hyper-K¨ahlerian.Note also, that in [6] the K¨ahler-Einstein metrics on manifolds of G -cohomogeneity onewere classified but only under one additional assumption: It is assumed that the isotropyrepresentation of the space G / H (see our notation below) splits into pairwise inequivalentsub-representations. This condition is crucial for the fact that the Einstein equation can besolved (see [6, Theorem 2.18]). But this assumption fails, for instance, for the symmetricspace C P n ( n > ) .Let G / K be a rank-one symmetric space of a compact connected Lie group G . The tan-gent bundle T ( G / K ) has a canonical complex structure J Kc coming from the G -equivariantdiffeomorphism T ( G / K ) → G C / K C . The latter space is the above-mentioned complexifica-tion of G / K . In our paper [8] we described, for such a G / K , all G -invariant K¨ahler structures ( g , J Kc ) which are moreover Ricci-flat on the punctured tangent bundle T + ( G / K ) of T ( G / K ) .This description is based on the fact that T + ( G / K ) is the image of G / H × R + under certain G -equivariant diffeomorphism. Here H denotes the stabilizer of any element of T ( G / K ) ingeneral position. Such G -invariant K¨ahler and Ricci-flat K¨ahler structures are determinedcompletely by a unique vector-function a : R + → g H satisfying certain conditions, g H beingthe subalgebra of Ad ( H ) -fixed points of the Lie algebra of G .As for the contents, we recall in Section 2 some definitions and results on the canonicalcomplex structure on T ( G / K ) . In Section 3 we recall the general description given in [8]of invariant Ricci-flat K¨ahler metrics on tangent bundles of Riemannian symmetric spacesof compact type, especially in Theorems 3.2 and 3.5 below, given here without proof. InSection 4, we state and prove Theorem 4.1 and its Corollary 4.3 giving the invariant Ricci-flat K¨ahler metrics on the punctured tangent bundles T + ( G / K ) of the rank-one Riemanniansymmetric spaces of compact type and then the complete invariant Ricci-flat K¨ahler metricson T ( G / K ) . 2. T HE CANONICAL COMPLEX STRUCTURE ON T ( G / K ) Consider a homogeneous manifold G / K , where G is a compact connected Lie group and K is some closed subgroup of G . Let g and k be the Lie algebras of G and K respectively.There exists a positive-definite Ad ( G ) -invariant form h· , ·i on g .Denote by m the h· , ·i -orthogonal complement to k in g , that is, g = m ⊕ k is the Ad ( K ) -invariant vector space direct sum decomposition of g . Consider the trivial vector bundle G × m with the two Lie group actions (which commute) on it: the left G -action, l h : ( g , w ) ( hg , w ) and the right K -action r k : ( g , w ) ( gk , Ad k − w ) . Let π : G × m → G × K m , ( g , w ) [( g , w )] , ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 3 be the natural projection for this right K -action. This projection is G -equivariant. It is wellknown that G × K m and T ( G / K ) are diffeomorphic. The corresponding G -equivariant dif-feomorphism φ : G × K m → T ( G / K ) , [( g , w )] dd t (cid:12)(cid:12)(cid:12) g exp ( tw ) K , and the projection π determine the G -equivariant submersion Π = φ ◦ π : G × m → T ( G / K ) .Let G C and K C be the complexifications of the Lie groups G and K . In particular, K is a maximal compact subgroup of the Lie group K C and the intersection of K with eachconnected component of K C is not empty (cf. A.L. Onishchik and E.V. Vinberg [16, Ch. 5,p. 221] and note that G C , K C , G and K are algebraic groups). Let g C = g ⊕ i g and k C = k ⊕ i k be the complexifications of the compact Lie algebras g and k .Since G and K are maximal compact Lie subgroups of G C and K C , respectively, by a resultof G.D. Mostow [12, Theorem 4], we have that K C = K exp ( i k ) , G C = G exp ( i m ) exp ( i k ) , andthe mappings G × m × k → G C , ( g , w , ζ ) g exp ( iw ) exp ( i ζ ) , K × k → K C , ( k , ζ ) k exp ( i ζ ) , are diffeomorphisms. Then the map f × K : G C / K C → G × K m , g exp ( iw ) exp ( i ζ ) K C [( g , w )] , is a G -equivariant diffeomorphism [13, Lemma 4.1]. It is clear that f K : G C / K C → T ( G / K ) , g exp ( iw ) exp ( i ζ ) K C Π ( g , w ) , is also a G -equivariant diffeomorphism. Since G C / K C is a complex manifold, the diffeo-morphism f K supplies the manifold T ( G / K ) with the G -invariant complex structure whichwe denote by J Kc .3. I NVARIANT R ICCI - FLAT
K ¨
AHLER METRICS ON TANGENT BUNDLES OF COMPACT R IEMANNIAN SYMMETRIC SPACES . G
ENERAL DESCRIPTION
We continue with the previous notations but in this section and the next one it is assumedin addition that G / K is a rank-one Riemannian symmetric space of a connected, compactsemisimple Lie group G .3.1. Root theory of Riemannian symmetric spaces of rank one.
Here we will review afew facts about Riemannian symmetric spaces of rank one [9, Ch. VII, § §
11] and resultsof our paper [8] adapted to the case of these (rank one) spaces.We have then g = m ⊕ k , where [ m , m ] ⊂ k , [ k , m ] ⊂ m , [ k , k ] ⊂ k , and k ⊥ m . In other words, there exists an involutive automorphism σ : g → g such that k = ( + σ ) g and m = ( − σ ) g . Moreover, the scalar product h· , ·i is σ -invariant. P. M. GADEA, J. C. GONZ ´ALEZ-D ´AVILA, AND I. V. MYKYTYUK
Let a ⊂ m be some Cartan subspace of the space m . There exists a σ -invariant Cartansubalgebra t of g containing the commutative subspace a , i.e. t = a ⊕ t , where a = ( − σ ) t , t = ( + σ ) t . Then the complexification t C is a Cartan subalgebra of the reductive complex Lie algebra g C and we have the root space decomposition g C = t C ⊕ ∑ α ∈ ∆ ˜ g α . Here ∆ is the root system of g C with respect to the Cartan subalgebra t C . For each α ∈ ∆ wehave ˜ g α = (cid:8) ˜ ξ ∈ g C : ad ˜ t ˜ ξ = α ( ˜ t ) ˜ ξ , ˜ t ∈ t C (cid:9) and dim C ˜ g α = . It is evident that the centralizer ˜ g of the space a C in g C is the subalgebra(3.1) ˜ g = t C ⊕ ∑ α ∈ ∆ ˜ g α , where ∆ = { α ∈ ∆ : α | a C = } is the root system of the reductive Lie algebra ˜ g with respectto its Cartan subalgebra t C .The set Σ = { λ ∈ ( a C ) ∗ : λ = α | a C , α ∈ ∆ \ ∆ } is the set of restricted roots of the triple ( g , k , a ) , which is independent of the choice of the σ -invariant Cartan subalgebra t containingthe Cartan subspace a .Since G / K is a rank-one Riemannian symmetric space, dim a =
1. Then the restrictedroot system is either Σ = {± ε } or Σ = {± ε , ± ε } , where ε ∈ ( a C ) ∗ . There exists a unique(basis) vector X ∈ a such that ε ( X ) = i , where, since the algebra g is compact, α ( t ) ⊂ i R for each α ∈ ∆ . It is clear that multiplying our scalar product h· , ·i by a positive constant wecan suppose that h X , X i =
1. For each λ ∈ Σ define the linear function λ ′ : a → R , by therelation i λ ′ = λ . Note that then h X , X i = , i ε ′ = ε , and ε ′ ( X ) = . Since the algebra ˜ g coincides with the centralizer of the element X ∈ a in g C , there existsa basis Π of ∆ (a system of simple roots) such that Π = Π ∩ ∆ is a basis of ∆ . Indeed, theelement − iX ∈ i t belongs to the closure of some Weyl chamber in i t determining the basis Π . Then Π = { α ∈ Π : α ( − iX ) = } . The bases Π and Π determine uniquely the subsets ∆ + and ∆ + of positive roots of ∆ and ∆ , respectively. It is evident that ∆ + \ ∆ + = { α ∈ ∆ : α ( − iX ) > } . The following decomposition g C = ˜ g ⊕ ∑ λ ∈ Σ + ( ˜ g λ ⊕ ˜ g − λ ) , where ˜ g λ = ∑ α ∈ ∆ \ ∆ , α | a C = λ ˜ g α and Σ + denotes the subset of positive restricted roots in Σ determined by the set of positiveroots ∆ + , gives us a simultaneous diagonalization of ad ( a C ) on g C . Remark that in our ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 5 case either Σ + = { ε } or Σ + = { ε , ε } . Denote by m λ the multiplicity of the restricted root λ ∈ {± ε , ± ε } , that is, m λ = card { α ∈ ∆ : α | a C = λ } .For each linear form λ on a C put m λ def = (cid:8) η ∈ m : ad w ( η ) = λ ( w ) η , ∀ w ∈ a (cid:9) , k λ def = (cid:8) ζ ∈ k : ad w ( ζ ) = λ ( w ) ζ , ∀ w ∈ a (cid:9) . (3.2)Then m λ = m − λ , k λ = k − λ , m = a and k = h , where(3.3) h = { u ∈ k : [ u , a ] = } = ( ker ad X ) ∩ k is the centralizer of a in k .In Table 3.1 we list all compact Riemannian symmetric spaces of rank one with theircorresponding multiplicities m ε , m ε / and type of the algebra h .Table 3.1. Irreducible rank-one Riemannian symmetric spaces of compact type G / K dim m ε m ε / h S n , ( n > )[ R P n ] ∗ SO ( n + ) / SO ( n )[ SO ( n + ) / O ( n )] ∗ n n − so ( n − ) C P n , ( n > ) SU ( n + ) / S ( U ( ) × U ( n )) n n − R ⊕ su ( n − ) H P n , ( n > ) Sp ( n + ) / Sp ( ) × Sp ( n ) n n − sp ( ) ⊕ sp ( n − ) C a P F / Spin ( )
16 7 8 so ( ) Here we assume that so ( ) = so ( ) ∼ = R , su ( ) = su ( ) = sp ( ) =
0. The symmetricspaces G / K with non-connected K are marked with [ · ] ∗ in Table 3.1.It is clear that m C λ ⊕ k C λ = ˜ g λ ⊕ ˜ g − λ for λ ∈ Σ + and ˜ g = m C ⊕ k C = a C ⊕ h C (the Cartansubspace a C is a maximal commutative subspace of m C ). By [9, Ch. VII, Lemma 11.3], thefollowing decompositions are direct and orthogonal:(3.4) m = a ⊕ m ε ⊕ m ε / , k = h ⊕ k ε ⊕ k ε / , where to simplify the notation we suppose that m ε / = k ε / = ε Σ . We shall put m + def = m ε ⊕ m ε / , k + def = k ε ⊕ k ε / . Since the restriction of the operator ad X to the subspace m + ⊕ k + is nondegenerate andad X ( m ) ⊂ k , ad X ( k ) ⊂ m for any vector ξ λ ∈ m λ ⊂ m , λ ∈ Σ + , by (3.2) and (3.4) thereexists a unique vector ζ λ ∈ k λ such that(3.5) [ X , ξ λ ] = − λ ′ ( X ) ζ λ , [ X , ζ λ ] = λ ′ ( X ) ξ λ , where, recall, ε ′ ( X ) =
1. In particular, dim m λ = dim k λ = m λ and there exists a uniqueendomorphism T : m + ⊕ k + → m + ⊕ k + such that(3.6) ad X | m λ ⊕ k λ = λ ′ ( X ) T | m λ ⊕ k λ , T ( m λ ) = k λ , T ( k λ ) = m λ , ∀ λ ∈ Σ + . P. M. GADEA, J. C. GONZ ´ALEZ-D ´AVILA, AND I. V. MYKYTYUK
This endomorphism is orthogonal because T = − Id m + ⊕ k + and the endomorphism ad X isskew-symmetric. Note also here that by (3.1) the subspace t = ( + σ ) t is a Cartan subalgebra of the centralizer h and t = a ⊕ t . Moreover, since [ t , m ] ⊂ m and [ t , k ] ⊂ k , [ a , t ] =
0, from definitions (3.2) and (3.6) weobtain that [ t , m λ ] ⊂ m λ and [ t , k λ ] ⊂ k λ for each λ ∈ Σ + , [ ad x , T ] = m + ⊕ k + for each x ∈ t . Fix the Weyl chamber W + in a containing the element X : W + = (cid:8) w ∈ a : ε ( − iw ) > (cid:9) = (cid:8) w ∈ a : ε ′ ( w ) > (cid:9) = R + X . The subspace m ⊂ g is Ad ( K ) -invariant. Each nonzero Ad ( K ) -orbit in m intersects theCartan subspace a and also the Weyl chamber W + , that is, Ad ( K )( W + ) = m \ { } . Theset m R = m \ { } of all nonzero elements of m is the set of regular points in m .Consider the centralizer H of the Cartan subspace a in Ad ( K ) , i.e.(3.7) H = { k ∈ K : Ad k u = u for all u ∈ a } = { k ∈ K : Ad k X = X } . It is clear that the algebra h (see (3.5)), is the Lie algebra of H .Our interest now centers on what will be shown to be an important subalgebra of g . Let g H ⊂ g be the subalgebra of fixed points of the group Ad ( H ) , i.e.(3.8) g H def = { u ∈ g : Ad h u = u for all h ∈ H } . It is evident that g H ⊂ g h , where(3.9) g h def = (cid:8) u ∈ g : [ u , ζ ] = ζ ∈ h (cid:9) is the centralizer of the algebra h in g . Note that in the general case one has g H = g h (seeExample 4.6 in [8]).To understand the structure of the algebra g H we consider more carefully the centralizer g h . Since h is a compact Lie algebra, h = z ( h ) ⊕ [ h , h ] , where z ( h ) is the center of h and [ h , h ] is a maximal semisimple ideal of h . It is clear that z ( h ) ⊂ g h and g h ∩ [ h , h ] = h g h , [ h , h ] i = h [ g h , h ] , h i = . Therefore g h ∩ h = z ( h ) and g h ⊕ [ h , h ] = g h + h is a subalgebra of g .By its definition, z ( h ) is a subspace of the center of the algebra g h . Moreover, by (3.3), a ⊂ g h . The space a ⊕ z ( h ) ⊂ g h is a Cartan subalgebra of g h (a maximal commutativesubalgebra of g h ) because the centralizer of a in g equals a ⊕ h , a ⊕ z ( h ) is the center of thealgebra a ⊕ h and g h ∩ ( a ⊕ h ) = a ⊕ z ( h ) by definition of g h (see also [8, Subsection 4.1]).Since a ⊂ g h and t ⊂ h , then a ⊕ t ⊂ g h + h . But a ⊕ t = t is a Cartan subalgebra of g . This means that the complex reductive Lie algebras ( g h + h ) C , g C h and h C are ad ( t C ) -invariant subalgebras of g C . Taking into account that t ∩ g h = a ⊕ z ( h ) and t ∩ h = t , we ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 7 obtain the following direct sum decompositions:(3.10) g C h = a C ⊕ z ( h ) C ⊕ ∑ α ∈ ∆ h ˜ g α and h C = t C ⊕ ∑ α ∈ ∆ ˜ g α , where ∆ h is some subset of the root system ∆ . Since the spaces a ⊕ z ( h ) ⊂ t and t ⊂ t areCartan subalgebras of the algebras g h and h respectively, the decompositions above are theroot space decompositions of ( g C h , ( a ⊕ z ( h )) C ) and ( h C , t C ) , respectively. In particular, thesubset ∆ h ⊂ ∆ is the root system of ( g C h , ( a ⊕ z ( h )) C ) .Since h ⊂ k , we see that σ ( h ) = h and the centralizer g h of h in g is σ -invariant. By [8,Proposition 4.3], ∆ h = (cid:8) α ∈ ∆ : α ( t ) = , α + β ∆ for all β ∈ ∆ (cid:9) . But by [8, Lemma 4.1] this subset ∆ h of the set of roots ∆ admits the following alternativedescription:(3.11) ∆ h = (cid:8) α ∈ ∆ : α ( t ) = , m λ = , where λ = α | a C (cid:9) . As follows from Table 3.1, two such restricted roots { ε , − ε } ⊂ Σ of multiplicity 1 exist ifand only if G / K ∈ { C P n ( n > ) , R P } ( C P ∼ = S ). Hence for any of the latter rank-onesymmetric spaces g h = a ⊕ z ( h ) . Since for these latter spaces z ( h ) = g h = a if G / K
6∈ { C P n ( n > ) , R P } . Since σ ( h ) = h , the centralizer g h of h in g is σ -invariant, i.e. g h = m h ⊕ k h , where m h = g h ∩ m , k h = g h ∩ k and as a ⊂ m h is a maximal commutative subspace of m , the space a is a Cartan subspace of m h . Then the set Σ h = { λ ∈ ( a C ) ∗ : λ = α | a C , α ∈ ∆ h } ⊂ Σ is the set of restricted roots of the triple ( g h , k h , a ) and since by (3.11) the spaces m λ , λ ∈ Σ h ,have dimension one, we obtain the following direct orthogonal decompositions m h = a ⊕ ∑ λ ∈ Σ h ∩ Σ + m λ , k h = z ( h ) ⊕ ∑ λ ∈ Σ h ∩ Σ + k λ . To describe the algebra g H ⊂ g h we consider now in more detail the subgroup H ⊂ K .By [8, Proposition 4.4], H = ( exp ( a ) ∩ K ) H , where H = exp h is the identity componentof the Lie group H ( H ⊂ K because h ⊂ k ). Since the group H ⊂ K is compact and K isa subgroup of the group of fixed points of certain involutive automorphism of G acting byexp ( v ) exp ( − v ) on exp ( a ) , the discrete group D a def = exp ( a ) ∩ K is finite and(3.13) D a = { exp v : v ∈ a , exp v = exp ( − v ) } ∩ K . Since [ h , g h ] =
0, the group Ad ( H ) acts trivially on g h and therefore(3.14) g H = { u ∈ g h : Ad exp v u = u for all v ∈ a such that exp v ∈ D a } . P. M. GADEA, J. C. GONZ ´ALEZ-D ´AVILA, AND I. V. MYKYTYUK
Taking into account that [ a , t ] =
0, we conclude that the group Ad exp a acts trivially on thespace a ⊕ z ( h ) ⊂ t and consequently, by (3.12),(3.15) g H = g h = a if G / K
6∈ { C P n ( n > ) , R P } , and g H contains a ⊕ z ( h ) otherwise. For the space G / K = C P n ( n > ) we will calculate thealgebra g H in the next section using the matrix representation for g ∼ = su ( n + ) .The algebra g H is σ -invariant because by definition (3.7), σ Ad ( H ) σ = Ad ( H ) . In partic-ular, g H = m H ⊕ k H , where m H = g H ∩ m , k H = g H ∩ k , and ( g H , k H ) is a symmetric pair. By maximality conditions the space a ⊂ g H is a Cartansubspace of m H ⊂ g H and the space a ⊕ z ( h ) is a Cartan subalgebra of g H .For each λ ∈ Σ + and g ∈ D a ⊂ exp a we have that Ad g ( m λ ⊕ k λ ) = m λ ⊕ k λ becauseAd exp v = e ad v . The set Σ H = { λ ∈ Σ h : Ad g | m λ ⊕ k λ = Id m λ ⊕ k λ for all g ∈ D a } is the set of restricted roots of the triple ( g H , k H , a ) . By (3.11) each element λ ∈ Σ H ⊂ Σ h ⊂ Σ has multiplicity 1 as an element of Σ , that is, dim m λ = dim k λ = m H = a ⊕ ∑ λ ∈ Σ H ∩ Σ + m λ , k H = z ( h ) ⊕ ∑ λ ∈ Σ H ∩ Σ + k λ . Remark 3.1.
Put m + H = ∑ λ ∈ Σ H ∩ Σ + m λ and k + H = ∑ λ ∈ Σ H ∩ Σ + k λ . Consider the orthogonaldecompositions: m + = m + H ⊕ m + ∗ and k + = k + H ⊕ k + ∗ , where m + ∗ = ∑ λ ∈ Σ + \ Σ H m λ and k + ∗ = ∑ λ ∈ Σ + \ Σ H k λ . Since the decompositions g H = a ⊕ m + H ⊕ k + H ⊕ z ( h ) , g = a ⊕ m + H ⊕ k + H ⊕ m + ∗ ⊕ k + ∗ ⊕ h = g H ⊕ ( m + ∗ ⊕ k + ∗ ) ⊕ [ h , h ] are orthogonal and [ g H , h ] = , one has that g H ⊕ [ h , h ] is a subalgebra of g .Moreover, because of its definition, T ( m λ ) = k λ , T ( k λ ) = m λ for all restricted roots λ ∈ Σ + , we obtain thatT ( m + H ) = k + H , T ( k + H ) = m + H and T ( m + ∗ ) = k + ∗ , T ( k + ∗ ) = m + ∗ . Fix in each subspace m λ , λ ∈ Σ + , some basis { ξ j λ , j = , . . . , m λ } , orthonormal with re-spect to the form h· , ·i . In the case when λ ∈ Σ h ∩ Σ + , m λ = ξ λ .As we remarked above, for each λ ∈ Σ + there exists a unique basis { ζ j λ , j = , . . . , m λ } of k λ such that for each pair { ξ j λ , ζ j λ , j = , . . . , m λ } , condition (3.5) holds. The basis { ζ j λ , j = , . . . , m λ } , λ ∈ Σ + , of k λ , is also orthonormal due to the orthogonality of the oper-ator T (see (3.6)). Fix also some orthonormal basis { ζ k , k = , . . . , dim h } of the centralizer h of a in k . We will use the orthonormal basis X , ξ j λ , ζ j λ , j = , . . . , m λ , λ ∈ Σ + ; ζ k , k = , . . . , dim h , of the algebra g in our calculations below. ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 9
The canonical complex structure on G / H × W + ∼ = G / H × R + . By definition (3.7) ofthe group H , the map K / H × W + → m R , ( kH , w ) Ad k w , is a well-defined diffeomorphism because, recall, W + = R + X and m R = m \ { } . Thus themap f + : G / H × W + → G × K m R , ( gH , w ) [( g , w )] , is a well-defined G -equivariant diffeomorphism of G / H × W + onto the subset D + = G × K m R , which is an open dense subset of G × K m .It is clear that the diagram(3.16) G × W + id −→ G × m R ↓ π H × id π ↓ G / H × W + f + −→ G × K m R ( g , w ) id ( g , w ) ↓ π H × id π ↓ ( gH , w ) f + [( g , w )] , where π H : G → G / H is the canonical projection, is commutative.Denote by ξ l the left G -invariant vector field on G corresponding to ξ ∈ g . The submersion(projection) π : G × m → G × K m is (left) G -equivariant. Therefore, the kernel K ⊂ T ( G × m ) of the tangent map π ∗ : T ( G × m ) → T ( G × K m ) is generated by the global (left) G -invariant vector fields ζ L , for ζ ∈ k , on G × m ,(3.17) ζ L ( g , w ) = ( ζ l ( g ) , [ w , ζ ]) ∈ T g G × T w m , where the tangent space T w m is canonically identified with the space m .To describe the G -invariant Ricci-flat K¨ahler metrics on T ( G / K ) associated to the cano-nical complex structure J Kc , we first attempt to describe such metrics on the punctured tangentbundle T + ( G / K ) def = T ( G / K ) \ { zero section } of G / K . It is clear that T + ( G / K ) = φ ( G × K m R ) and therefore T + ( G / K ) = ( φ ◦ f + )( G / H × W + ) , that is, T + ( G / K ) is G -equivariantly isomorphic to the direct product G / H × W + , wherethe action of the group G on the first component is the natural one and that on the secondcomponent is the trivial one (see the commutative diagram (3.16)). This G -equivariant dif-feomorphism determines a G -invariant complex structure on G / H × W + , which we denotealso by J Kc .Note also here that the tangent space T o ( G / H ) at o = { H } ∈ G / H can be identified natu-rally with the space m ⊕ k + = a ⊕ m + ⊕ k + , because by definition k = h ⊕ k + and h is the Liealgebra of the group H .Considering the coordinate x on W + = R + X associated with the basis vector X of a , weidentify naturally W + ⊂ a with R + replacing w = xX by x : G / H × W + → G / H × R + , ( gH , xX ) ( gH , x ) . By the G -invariance it suffices to describe the operators J Kc only at the points ( o , x ) ∈ G / H × R + , where o = { H } . By [8, (4.47)], J Kc ( o , x )( X , ) = (cid:16) , ∂∂ x (cid:17) , J Kc ( o , x )( ξ j λ , ) = (cid:16) − cosh λ ′ x sinh λ ′ x · ζ j λ , (cid:17) , j = , . . . , m λ , λ ∈ Σ + , (3.18)where λ ′ x = λ ′ ( xX ) ∈ R , that is, λ ′ x = x if λ = ε and λ ′ x = x if λ = ε . Here T o ( G / H ) is identified naturally with the space a ⊕ ∑ λ ∈ Σ + m λ ⊕ ∑ λ ∈ Σ + k λ , a = R X , and, in the firstequation, we use naturally the usual basis vector { ∂ / ∂ x } of T x R + .The second relation in (3.18) can be represented in a more general form (see [8, (4.27)]): J Kc ( o , x )( ξ , ) = (cid:16) − cos ad xX sin ad xX ξ , (cid:17) , where ξ ∈ m + . Let F = F ( J Kc ) be the subbundle of ( , ) -vectors of the structure J Kc on the manifold G / H × R + . Since the map π H × id : G × R + → G / H × R + is a submersion, there existsa unique maximal complex subbundle F of T C ( G × R + ) such that ( π H × id ) ∗ F = F . Asshown in [8, (4.28),(4.29)], F is generated by the kernel H of the submersion π H × id,(3.19) H ( g , x ) = { ( ζ l ( g ) , ) , ζ ∈ h } , g ∈ G , x ∈ R + , and the left G -invariant global vector fields on G × R + : Z X ( g , x ) = (cid:16) X l ( g ) , − i ∂∂ x (cid:17) , Z ξ j λ ( g , x ) = (cid:18) λ ′ x · ξ j λ − i − λ ′ x · ζ j λ (cid:19) l ( g ) , ! , where j = , . . . , m λ , λ ∈ Σ + .To simplify calculations in the next subsection, for the vector fields of the second familywe will use a more general expression Z ξ ( g , x ) = (cid:16) ( R x ξ − i S x ξ ) l ( g ) , (cid:17) , ξ ∈ m + , in terms of the two operator-functions R : R + → End ( g ) and S : R + → End ( g ) on the set R + such that R x η = xX η if η ∈ m + ⊕ k + , R x η = η ∈ a ⊕ h , S x η = − xX η if η ∈ m + ⊕ k + , S x η = η ∈ a ⊕ h , where, recall, xX ∈ W + ⊂ a . Remark also that xX η = η if η ∈ a ⊕ h but R x η = xX is skew-symmetric with respect to the scalar product on g , ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 11 each operator R x is symmetric and S x is skew-symmetric: h R x η , η i = h η , R x η i , h S x η , η i = h η , − S x η i , x ∈ R + , η , η ∈ g . Moreover, since xX ∈ W + ⊂ a , the restrictions R x | m + ⊕ k + and S x | m + ⊕ k + are nondegenerateand by Remark 3.1 the following relations hold: R x ( m + s ) = m + s , R x ( k + s ) = k + s , S x ( m + s ) = k + s , S x ( k + s ) = m + s , s ∈ { H , ∗} . It is clear also that(3.20) R x | m λ ⊕ k λ = λ ′ x Id m λ ⊕ k λ , S x | m λ ⊕ k λ = λ ′ x T | m λ ⊕ k λ for all λ ∈ Σ + , and [ R x , T ] = [ S x , T ] = m + ⊕ k + for all xX ∈ W + , where, recall, theoperator T is defined by the expression (3.6).3.3. Invariant Ricci-flat K ¨ahler metrics on G / H × R + . Let K ( G / H × R + ) = { ( g , ω , J Kc ) } (resp. R ( G / H × R + ) = { ( g , ω , J Kc ) } ) be the set of all G -invariant K¨ahler (resp. Ricci-flatK¨ahler) structures on G / H × R + , identified also with the set K ( T + ( G / K )) (resp. R ( T + ( G / K )) )of all G -invariant K¨ahler (resp. Ricci-flat K¨ahler) structures on the open dense subset T + ( G / K ) of T ( G / K ) , associated with J Kc , via the G -equivariant diffeomorphism φ ◦ f + : G / H × R + → T + ( G / K ) ( R + ∼ = W + ).Put { T , . . . , T n } = { Z X } ∪ { Z ξ j λ , λ ∈ Σ + , j = , . . . , m λ } . The following theorem is Theorem 4.8 from [8] (adapted to the rank one case) whichdescribes the spaces K ( G / H × R + ) and R ( G / H × R + ) in terms of invariant forms on thespace G × R + : Theorem 3.2. [8]
Let K ( G × R + ) = { e ω } be the set of all -forms e ω on G × R + such that ( ) the form e ω is closed; ( ) the form e ω is left G-invariant and right H-invariant; ( ) the kernel of e ω coincides with the subbundle H ⊂ T ( G × R + ) in (3.19); ( ) e ω ( T j , T k ) = , j , k = , . . . , n ; ( ) i e ω ( T , T ) > for each T = ∑ nj = c j T j , where ( c , . . . , c n ) ∈ C n \ { } . Let R ( G × R + ) = { e ω } be the subset of the set K ( G × R + ) = { e ω } consisting of all elements e ω such that the following condition holds ( in addition ): ( ) det (cid:0) e ω ( T j , T k ) (cid:1) = const on G × R + . Then (i)
For any -form e ω ∈ K ( G × R + ) there exists a unique -form ω on G / H × R + ∼ = T + ( G / K ) such that ( π H × id ) ∗ ω = e ω . The map e ω ω is a one-to-one map from K ( G × R + ) onto K ( G / H × R + ) ∼ = K ( T + ( G / K )) . (ii) If the group G is semisimple then the restriction of this map to R ( G × R + ) is a one-to-one map from R ( G × R + ) onto R ( G / H × R + ) ∼ = R ( T + ( G / K )) . Remark 3.3.
Note that condition ( ) of the previous theorem is equivalent to the followingcondition: the Hermitian matrix-function w ( x ) on R + with entries w jk ( x ) = i e ω ( T j , T k )( e , x ) ,j , k = , . . . , n, is positive-definite. To prove that a K¨ahler structure on T + ( G / K ) admits a K¨ahler extension to the whole T ( G / K ) we will use Corollary 4.10 from [8] (adapted to the rank one case): Corollary 3.4. [8]
Let ω ∈ K ( G / H × R + ) and e ω = ( π H × id ) ∗ ω . Then ω = (( φ ◦ f + ) − ) ∗ ω ∈ K ( T + ( G / K )) . Suppose that there exists a smooth form (extension) ω on the whole tangentbundle T ( G / K ) such that ω = ω on T + ( G / K ) . Then the form ω determines a G-invariantK¨ahler structure on T ( G / K ) (associated to the canonical complex structure J Kc ) if and onlyif for some sequence x m ∈ R + , m ∈ N , such that lim m → ∞ x m = , the Hermitian matrix w ( ) with entries w jk ( ) = lim m → ∞ w jk ( x m ) = lim m → ∞ i e ω ( T j , T k )( e , x m ) , j , k = , . . . , n, is positive-definite. General description of the space R ( G × R + ) . For any vector a ∈ g , denote by θ a theleft G -invariant 1-form on the group G such that θ a ( ξ l ) = h a , ξ i . Since r ∗ g θ a = θ Ad g a , where g ∈ G , the form θ a is right H -invariant if and only if Ad h a = a for all h ∈ H ⊂ G . Becaused θ a ( ξ l , η l ) = − θ a ([ ξ l , η l ]) = −h a , [ ξ , η ] i , the G -invariant form ω a on G , ω a ( ξ l , η l ) def = h a , [ ξ , η ] i , ξ , η ∈ g , is a closed 2-form on G .Let pr : G × R + → G and pr : G × R + → R + be the natural projections. Choosing someorthonormal basis { e , . . . , e N } of the Lie algebra g , where e = X , put e θ e k def = pr ∗ ( θ e k ) and e ω e k def = pr ∗ ( ω e k ) . For any vector-function a : R + → g , a ( x ) = ∑ Nk = a k ( x ) e k , denote by e θ a (resp. e ω a ) the G -invariant 1-form ∑ Nk = a k · e θ e k (resp. 2-form ∑ Nk = a k · e ω e k ).The following theorem [8, Theorem 5.1] (adapted to the rank one case) describes thespaces K ( G × R + ) and R ( G × R + ) in terms of some R + -parameter family of exact 1-formson the Lie group G : Theorem 3.5. [8]
Let e ω be a -form belonging to K ( G × R + ) , where the compact Liegroup G is semisimple. Then there exists a unique (up to a real constant) smooth functionf : R + → R , x f ( x ) , and a unique smooth vector-function a : R + → g H given by a ( x ) = a a ( x ) + z h + a k ( x ) + a m ( x ) , where a a ( x ) = f ′ ( x ) X , z h ∈ z ( h ) , a k ( x ) = ∑ λ ∈ Σ H ∩ Σ + c k λ cosh λ ′ ( xX ) ζ λ ∈ k + H , a m ( x ) = ∑ λ ∈ Σ H ∩ Σ + c m λ sinh λ ′ ( xX ) ξ λ ∈ m + H , (3.21) c m λ , c k λ ∈ R , such that e ω is the exact form expressed in terms of a as (3.22) e ω = d e θ a = d x ∧ e θ a ′ − e ω a , where a ′ = ∂ a ∂ x . ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 13
Moreover, for all points x ∈ R + , the following conditions ( ) − ( ) hold: ( ) the components a k ( x ) + z h and a m ( x ) of the vector-function a ( x ) in (3.21) satisfythe commutation relations (cid:0) R x · ad a k ( x ) · R x + S x · ad a k ( x ) · S x + ( R x + S x ) ad z h (cid:1) ( m + ) = , (cid:0) R x · ad a m ( x ) · S x − S x · ad a m ( x ) · R x (cid:1) ( m + ) = moreover, if G / K is an irreducible Riemannian symmetric space and a k ( x ) ≡ , thenz h = ; ( ) the Hermitian p × p-matrix-function w H ( x ) = (cid:0) w k | j ( x ) (cid:1) , p = dim m H = + card ( Σ H ∩ Σ + ) , with indices k , j ∈ { } ∪ { λ , λ ∈ Σ H ∩ Σ + } and entriesw | ( x ) = f ′′ ( x ) , w | λ ( x ) = λ ′ ( X ) (cid:16) i c k λ cosh λ ′ x − c m λ sinh λ ′ x (cid:17) , λ ∈ Σ H ∩ Σ + , w λ | µ ( x ) , λ , µ ∈ Σ H ∩ Σ + , determined by ( . ) , is positive-definite; ( ) if m + ∗ = then the Hermitian s × s-matrix w ∗ ( x ) = ( w j λ | k µ )( x ) , where s = dim m + ∗ = Σ λ ∈ Σ + \ Σ H m λ , with indices j λ , k µ ∈ { j λ , λ ∈ Σ + \ Σ H , j = , . . . , m λ } and entriesw j λ | k µ ( x ) = − λ ′ x sinh µ ′ x h ( ad a k ( x )+ z h ) ζ j λ , ζ k µ i (3.24) − λ ′ x sinh µ ′ x h ( ad a a ( x )+ a m ( x ) ) ξ j λ , ζ k µ i is positive-definite.If in addition ( ) either det w H ( x ) · det w ∗ ( x ) = const when m + ∗ = or det w H ( x ) ≡ const otherwise,then e ω ∈ R ( G × R + ) .Conversely, any -form as in (3.22) determined by a vector-function a : R + → g H asin (3.21) for which conditions ( ) − ( ) hold, belongs to K ( G × R + ) and if in addition ( ) holds, it belongs to R ( G × R + ) . Also Theorem 3.2 immediately implies
Corollary 3.6. [8]
Let G / K be a rank-one Riemannian symmetric space of compact type.Each G-invariant K¨ahler metric g , associated with the canonical complex structure J Kc onG / H × R + ∼ = T + ( G / K ) ( T + ( G / K ) is an open dense subset of T ( G / K )) , is uniquely deter-mined by the K¨ahler form ω ( · , · ) = g ( − J Kc · , · ) on G / H × R + given by ( π H × id ) ∗ ω = d e θ a , where a is the unique smooth vector-function a : R + → g H in (3.21) satisfying conditions ( ) − ( ) of Theorem 3.5 . If, in addition, condition ( ) of Theorem 3.5 holds, this metric g isRicci-flat. Corollary 3.7. [8]
Let ω be a G-invariant symplectic form on G / H × R + such that ( π H × id ) ∗ ω = d ˜ θ a , where a : R + → a , a ( x ) = f ′ ( x ) X , for some function f ∈ C ∞ ( R + , R ) . Then thepair ( ω , J Kc ) is a K¨ahler structure on G / H × R + (equivalently ( π H × id ) ∗ ω ∈ K ( G × R + )) if and only if f ′ ( x ) > and f ′′ ( x ) > for all x ∈ R + . In this case, the G-invariant functionQ : G / H × R + → R , Q ( gH , x ) = f ( x ) , is a potential function of the K¨ahler structure ( ω , J Kc ) on G / H × R + .The K¨ahler structure ( ω , J Kc ) with G-invariant potential function Q is Ricci-flat K¨ahler(equivalently ( π H × id ) ∗ ω ∈ R ( G × R + )) if and only iff ′′ · ∏ λ ∈ Σ + (cid:18) λ ′ ( a ) sinh 2 λ ′ ( a ) (cid:19) m λ = f ′′ · (cid:18) f ′ sinh ( f ′ ) (cid:19) m ε · (cid:18) f ′ sinh ( f ′ ) (cid:19) m ε / ≡ const .
4. C
OMPLETE INVARIANT R ICCI - FLAT
K ¨
AHLER METRICS ON TANGENT BUNDLES OFRANK - ONE R IEMANNIAN SYMMETRIC SPACES OF COMPACT TYPE
Let g be a compact Lie algebra and let σ , k , m , a , X ∈ a , Σ , etc. be as in Section 3. Wecontinue with the previous notations but in this section it is assumed in addition that thesubgroup K is connected.In this Section using Theorem 3.5 we describe all invariant Ricci-flat K¨ahler structureson the tangent bundles of the spaces under study, in terms of explicit expressions of thecorresponding vector-valued functions a .To this end we give with more detail the facts concerning the case G / K = C P n ( n > ) .These spaces are Hermitian symmetric spaces and therefore we will review a few facts aboutthem [9, Ch. VIII, §§ k of the semisimple Lie algebra g = su ( n + ) is the direct sum k = z ⊕ [ k , k ] of the one-dimensional center z and the semisimpleideal [ k , k ] ∼ = su ( n ) . The subalgebra k coincides with the centralizer of z in g . Here su ( n + ) denotes the space of traceless skew-Hermitian ( n + ) × ( n + ) complex matrices and k = { ( b jk ) ∈ su ( n + ) : b j = b j = , j = , . . . , n + } . Fix on g = su ( n + ) the invarianttrace-form given by h B , B i = − B B , B , B ∈ su ( n + ) . There exists a unique (up to asign) element Z ∈ z ( k ) such that the endomorphism I = ad Z | m : m → m satisfies I = − Id m .Choose Z as(4.1) Z = diag ( ib , i ( b − ) , . . . , i ( b − )) , b = n / ( n + ) . By the invariance of the form h· , ·i on g , the form h· , ·i| m is I -invariant. Moreover, by theJacobi identity,(4.2) [ I ξ , I η ] = [ ξ , η ] , I [ ζ , η ] = [ ζ , I η ] for all ξ , η ∈ m , ζ ∈ k . Denote by E jk the elementary ( n + ) × ( n + ) matrix whose entries are 0 except for 1 atthe entry in the j th row and k th column. Choose as basis vector X ∈ a the matrix X = E − E ∈ m ⊂ su ( n + ) . We will show below (using direct matrix calculations) that this ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 15 choice is consistent with the notation of the previous sections, i.e. in this case h X , X i = Σ of ( g , k , a ) coincides with the set {± ε } if n = {± ε , ± ε } if n >
2. The center z ( h ) of the centralizer h = g X ∩ k of X ∈ a in k is trivial for n = n > z ( h ) = R Z , where(4.3) Z = diag (cid:0) ib , ib , i ( b − ) , . . . , i ( b − ) (cid:1) , b = ( n − ) / ( n + ) , n > . Note that Z = n = Theorem 4.1.
Let G / K be a rank-one Riemannian symmetric space of compact type withK connected. A -form ω on the punctured tangent bundle T + ( G / K ) of G / K determinesa G-invariant K¨ahler structure, associated to the canonical complex structure J Kc , and thecorresponding metric g = ω ( J Kc · , · ) is Ricci-flat, if and only if the -form e ω = (cid:0) ( φ ◦ f + ) ◦ ( π H × id ) (cid:1) ∗ ω on G × R + may be expressed as e ω = d e θ a , where ( ) for G / K ∈ { S n ( n > ) , H P n ( n > ) , C a P } the vector-function a ( x ) = f ′ ( x ) X , where (4.4) (cid:0) f ′ ( x ) (cid:1) m ε + m ε / + = C · Z x ( sinh 2 t ) m ε ( sinh t ) m ε / d t + C , C , C ∈ R , C > , C > ( ) for G / K ∈ { C P n ( n > ) } the vector-function is a ( x ) = f ′ ( x ) X + c Z cosh x [ IX , X ] − c Z Z , where c Z is an arbitrary real number and (4.5) f ′ ( x ) = q ( C n sinh n x + C ) / n + c Z sinh x cosh − x , C , C ∈ R , C > , C > .The corresponding G-invariant Ricci-flat K¨ahler metric g = g ( C , C , c Z ) on T + ( G / K ) isuniquely extendable to a smooth complete metric on the whole tangent bundle T ( G / K ) ifand only if C = (that is, lim x → f ′ ( x ) = ) .Proof. By Theorem 3.5 we have to describe all vector-functions a : R + → g H satisfyingconditions ( ) − ( ) of that theorem. Then the 2-form e ω = d e θ a belongs to the space R ( G × R + ) . We consider the following two cases: (1) G / K ∈ (cid:8) S n ( n > ) , H P n ( n > ) , C a P (cid:9) . In this case by (3.15) g H = g h = a . Onegets that m H = a and k H =
0. Then a ( x ) = f ′ ( x ) X , x ∈ R + . Let us describe the Hermitianmatrix-functions w H ( x ) and w ∗ ( x ) from Theorem 3.5. As it is easily seen, the first matrix w H ( x ) contains a unique element w | ( x ) = f ′′ ( x ) and the second one, w ∗ ( x ) , is diagonalwith elements(4.6) w j ε | j ε ( x ) = f ′ ( x ) cosh x · sinh x , w k ε / | k ε / ( x ) = f ′ ( x ) cosh x · sinh x , where j = , . . . , m ε and k = , . . . , m ε / . These matrices are positive definite if and onlyif f ′ ( x ) > f ′′ ( x ) > x ∈ R + . Hence the vector-function a : R + → a satisfiesconditions ( ) − ( ) of Theorem 3.5 (see also Corollary 3.7) if and only if f ′ ( x ) > , f ′′ ( x ) > , f ′′ ( x ) · (cid:16) f ′ ( x ) cosh x sinh x (cid:17) m ε (cid:16) f ′ ( x ) cosh x sinh x (cid:17) m ε / ≡ const , x ∈ R + . It is clear that the unique possible solution of these equations is of form (4.4). (2) G / K = C P n ( n > ) . Theorem 3.5 was shown for G / K = C P ∼ = S in our paper [8,Theorem 6.1]. Therefore in this proof we will suppose that n >
2. Since we have chosen thematrix X = E − E ∈ m ⊂ su ( n + ) as the basis vector X ∈ a , it follows that Y def = IX = [ Z , X ] = i E + i E ∈ m Z def = [ IX , X ] = − i E + i E ∈ k . (4.7)It is easy to verify that the set { X , Y , Z } is an orthonormal system of vectors in g and(4.8) [ X , Y ] = − Z , [ X , Z ] = Y , [ Z , Y ] = X , i.e. the vectors { X , Y , Z } form a canonical basis of the Lie algebra isomorphic to su ( ) .By (4.8), ad X ( IX ) = − IX and as it is easy to verify, ad X ( ξ ) = − ξ for any vector ξ fromthe set of vectors(4.9) ξ j − ε / = E ( + j ) − E ( + j ) , ξ j ε / = i E ( + j ) + i E ( + j ) , j = , . . . , n − . Defining the restricted root ε ∈ ( a C ) ∗ by the relation ε ′ ( X ) = ε ( X ) = i ), we obtain that m ε = R ( IX ) and that the set (4.9) is an orthonormal basis of the space m ε / of dimension2 n − a ⊕ m ε in m ). Moreover, I ξ j − ε / = ξ j ε / for each j = , . . . , n −
1. Therefore Σ + = { ε , ε } ( n > g h and g H of g determined by relations (3.8) and (3.9).By (3.10) the space a ⊕ z ( h ) , where a = R X and z ( h ) = R Z , is a Cartan subalgebra of thealgebra g h . Since z ( h ) belongs to the center of g h we see that rank [ g h , g h ] dim a =
1, thatis, g h ∼ = su ( ) ⊕ z ( h ) if the algebra g h is not commutative.Since by definition [ X , h ] = h ⊂ k , by (4.2) one gets that [ IX , h ] =
0, that is, IX ∈ g h .By (4.8), the subalgebra in g generated by the vectors X and IX = Y is not commutative.Thus g h is not commutative and, consequently, g h ∼ = su ( ) ⊕ z ( h ) . The vectors { X , Y , Z , Z } form an orthonormal basis of g h . Therefore Σ h = {± ε } .Let us find now the algebra g H . The finite group D a defined by relation (3.13) is given by D a = { exp tX : exp tX = exp ( − tX ) } ∩ K = { exp 4 π X , exp 2 π X } = { diag ( , . . . , ) , diag ( − , − , , . . . , ) } . It is clear that the group Ad ( D a ) acts trivially on the space generated by the vectors X , Y , Z and Z . Therefore by (3.14) we have that g H = g h and, consequently, Σ H = Σ h = {± ε } . Notethat D a ⊂ H ∼ = U ( ) × SU ( n − ) , U ( ) ∼ = { exp tZ , t ∈ R } , i.e. the subgroup H is connected. ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 17
Using properties (4.2) of the automorphism I , we can describe the actions of the operatorsad Y and ad Z on m ⊕ k in terms of the operators I and ad X . Specifically, for any vectors ξ ∈ m , ζ ∈ k , we have [ Y , ξ ] = [ IX , ξ ] ( . ) = [ − X , I ξ ] = − ad X I ξ , (4.10) [ Y , ζ ] = [ IX , ζ ] ( . ) = I [ X , ζ ] = I ad X ζ . (4.11)Similarly, for Z = [ Y , X ] , using the Jacobi identity and relations (4.10), (4.11) we obtain that [ Z , ξ ] = [ Y , [ X , ξ ]] − [ X , [ Y , ξ ]] = I ad X ξ + ad X I ξ , (4.12) [ Z , ζ ] = [ Y , [ X , ζ ]] − [ X , [ Y , ζ ]] = − X I ad X ζ . (4.13)From the definitions of Z and Z in (4.1) and (4.3), respectively, it follows that ad Z | m ε / = ad Z | m ε / . Moreover, since b − = ( b − ) , then from (4.1) and (4.7) we obtain that Z − Z = Z . In other words,(4.14) ad Z | m ε / = I | m ε / and Z − Z = − Z . The operator-functions in (3.20) are given here by R x | m ε ⊕ k ε = x Id m ε ⊕ k ε , S x | m ε ⊕ k ε = x ad X | m ε ⊕ k ε , (4.15) R x | m ε / ⊕ k ε / = x Id m ε / ⊕ k ε / , S x | m ε / ⊕ k ε / = x ad X | m ε / ⊕ k ε / . (4.16)Put ξ ε = Y ∈ m ε . With the notation of the previous subsection, ζ ε = Z ∈ k ε . Now we haveto verify conditions ( ) − ( ) of Theorem 3.5 for the vector-function(4.17) a ( x ) = a a ( x ) + a k ( x ) + a m ( x ) + z h = f ′ ( x ) X + c Z ϕ ( x ) Z + c Y ψ ( x ) Y + c Z , where f ∈ C ∞ ( R + , R ) , ϕ ( x ) = x , ψ ( x ) = x , c Y , c Z , c ∈ R . Consider now the first condition in (3.23). We have the splitting m + = m ε ⊕ m ε / . Takinginto account that by its definition [ z ( h ) , g h ] = m ε = R Y ⊂ g h , using relations (4.15),we can rewrite the first condition in (3.23) for the vector Y = ξ ε as(4.18) x · R x (cid:2) c Z ϕ ( x ) Z , Y (cid:3) + x · S x (cid:2) c Z ϕ ( x ) Z , ad X Y (cid:3) = . The first term in (4.18) vanishes because [ Z , Y ] = X ∈ a and R x ( a ) =
0; the second termvanishes because ad X Y = − Z .Since in our case m ∗ = m ε / and k ∗ = k ε / , then by Remark 3.1 [ g H ⊕ [ h , h ] , m ε / ⊕ k ε / ] ⊂ m ε / ⊕ k ε / . Let now ξ ∈ m ε / . Using relations (4.12), (4.13) and (4.16), expression (4.14) and the fact that I ξ ∈ m ε / , we can rewrite the first condition in (3.23) as0 = c Z ϕ ( x ) cosh x · R x (cid:2) Z , ξ (cid:3) + c Z ϕ ( x ) sinh x · S x (cid:2) Z , ad X ξ (cid:3) + (cid:16) x − x (cid:17) · [ c Z , ξ ]= c Z ϕ ( x ) cosh x · ( I ad X ξ + ad X I ξ ) + c Z ϕ ( x ) sinh x · ad X ( − X I ad X ξ ) − c cosh x sinh x · I ξ . Then0 = (cid:18) x + x (cid:19) − c Z x I ξ − c cosh x sinh x I ξ = x sinh x ( − c Z − c ) I ξ , because ad X | m ε / = − Id m ε / . Thus c = − c Z .We can also rewrite the second condition in (3.23) for ξ ∈ m ε / as x · R x (cid:2) c Y ψ ( x ) Y , ad X ξ (cid:3) − x · S x (cid:2) c Y ψ ( x ) Y , ξ (cid:3) = . Taking into account the relations (4.10), (4.11) and (4.16) we obtain that c Y ψ ( x ) sinh x cosh x · ( I ad X ξ + ad X I ξ ) = − c Y sinh x cosh x sinh x · I ξ = . Thus c Y = a m ( x ) of a ( x ) vanishes. The second conditionin (3.23) holds.Summarizing the results proved above, we obtain that for the vector-function (4.17), con-dition ( ) of Theorem 3.5 for G / K = C P n ( n > ) is equivalent to the conditions { c Z ∈ R , c = − c Z , c Y = } .Let us describe the 2 × w H ( x ) , 2 = dim a + dim m ε , accordingto condition ( ) of Theorem 3.5.It is clear that w | ( x ) = f ′′ ( x ) . The function w | ε ( x ) is determined by the relation (for λ = ε ): w | ε ( x ) = c Z cosh x − c Y sinh x .The function w ε | ε ( x ) is determined by the relation (3.24) for ζ ε = Z and ξ ε = Y . Byrelations (4.8) and the invariance of the form h· , ·i , w ε | ε ( x ) = − x (cid:10)(cid:2) c Z ϕ ( x ) Z + c Z , Z (cid:3) , Z (cid:11) − x sinh x (cid:10)(cid:2) f ′ ( x ) X , Y (cid:3) , Z (cid:11) = f ′ ( x ) x sinh x . Hence, we conclude that the entries of w H ( x ) are(4.19) w ( x ) = f ′′ ( x ) , w | ε ( x ) = c Z cosh x , w ε | ε ( x ) = f ′ ( x ) cosh x sinh x . Let us describe the Hermitian s × s -matrix w ∗ ( x ) = (cid:0) w jk ( x ) (cid:1) , s = dim m ε / = n −
2, withentries w jk ( x ) = w j ε / | k ε / ( x ) , j , k = , . . . , n −
2, determined by relations (3.24): w jk ( x ) = − x (cid:10)(cid:2) c Z ϕ ( x ) Z + c Z , − X ξ j (cid:3) , − X ξ k (cid:11) − x sinh x (cid:10)(cid:2) f ′ ( x ) X , ξ j (cid:3) , − X ξ k (cid:11) , ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 19 where we put ξ j = ξ j ε / to simplify notation. Taking into account relations (4.13), (4.10) andthe commutation relation [ ad Z , ad X ] =
0, we obtain that w jk ( x ) = − x (cid:0) − c Z ϕ ( x ) h ad X I ad X ξ j , ad X ξ k i + c h ad X I ξ j , ad X ξ k i (cid:1) − x sinh x ( − ) f ′ ( x ) h ad X ξ j , ad X ξ k i = − isinh x c Z (cid:16) x − (cid:17) h I ξ j , ξ k i + x sinh x f ′ ( x ) h ξ j , ξ k i . But the orthonormal basis { ξ j ε / } n − j = is chosen in such a way that ξ j ε / = I ξ j − ε / . Thusfrom the relations above it follows that the Hermitian matrix w ∗ ( x ) is a block-diagonal ma-trix, where each block is an Hermitian 2 × (cid:0) ξ j − ε / , ξ j ε / (cid:1) , j = , . . . , n −
1, and it is a 2 × w ( j − )( j − ) ( x ) = w ( j )( j ) ( x ) = f ′ ( x ) cosh x sinh x , w ( j − )( j ) ( x ) = i c Z sinh x (cid:16) − x (cid:17) . (4.20)It is easily checked (calculating determinants of order 2) that vector-function (4.17) sat-isfies conditions ( ) , ( ) and ( ) of Theorem 3.5 if and only if all the mentioned Hermitian2 × w H ( x ) · det w ∗ ( x ) = n − · C n , C >
0, i.e. for all x ∈ R + the following relations hold: ( a ) f ′′ ( x ) > , ( b ) f ′′ ( x ) f ′ ( x ) cosh x sinh x − c Z cosh x > , ( c ) f ′ ( x ) > , ( d ) f ′ ( x ) f ′ ( x ) cosh x sinh x − c Z sinh x (cid:16) − x (cid:17) > , (4.21)and(4.22) (cid:16) f ′′ ( x ) f ′ ( x ) cosh x sinh x − c Z cosh x (cid:17)(cid:18) f ′ ( x ) f ′ ( x ) cosh x sinh x − c Z sinh x (cid:16) − x (cid:17) (cid:19) n − = n − C n . However, there exists an exact general solution of equation (4.22). Indeed, taking into ac-count some well-known identities for the functions cosh x and sinh x , and using the substitu-tion g ( x ) = ( f ′ ( x )) one can rewrite (4.22) as g ′ ( x ) = c Z sinh x cosh x + n − C n cosh x sinh x cosh x sinh x g ( x ) cosh x − c Z sinh x ! n − . Next, using the substitution g ( x ) = g ( x ) cosh x − c Z sinh x we obtain the Bernoulli equa-tion g ′ ( x ) = x cosh x · g ( x ) + C n cosh x sinh x (cid:18) cosh x sinh xg ( x ) (cid:19) n −
10 P. M. GADEA, J. C. GONZ ´ALEZ-D ´AVILA, AND I. V. MYKYTYUK with solutions g ( x ) = cosh x (cid:0) C n sinh n x + C (cid:1) / n , C >
0, on the whole semi-axis, i.e. weobtain that(4.23) f ′ ( x ) = q ( C n sinh n x + C ) / n + c Z sinh x cosh − x . and therefore f ′′ ( x ) = ( f ′ ( x )) − (cid:0) C n · ( C n sinh n x + C ) − nn · sinh n − x cosh x + c Z ( tanh x − tanh x ) (cid:1) . (4.24)For these functions on the whole semi-axis relations (4.21c) and (4.21a) hold since sinh x > x > tanh x on this set (0 < tanh x < x − tanh x = sinh x cosh − x ; and (4.21d) hold, as sinh x cosh − x = cosh x sinh x (cid:16) − x (cid:17) .The form e ω = d e θ a on G × R + determines a unique form ω on G / H × R + = G / H × W + such that e ω = ( π H × id ) ∗ ω (see Corollary 3.6). Let us study when the form ω on G / H × W + ∼ = T + ( G / K ) admits an smooth extension to the whole tangent space T ( G / K ) . To thisend we will find the expression of the form ω R = (( f + ) − ) ∗ ω on the space G × K m R ∼ = T + ( G / K ) , where, recall, f + : G / H × R + → G × K m R is a G -equivariant diffeomorphism.However, by the commutativity of diagram (3.16) there exists a unique form e ω R on G × m R such that(4.25) e ω R = π ∗ ω R and e ω = id ∗ e ω R . Thus it is sufficient to calculate the form e ω R on the space G × m R , because the form ω R on the space G × K m R ∼ = T + ( G / K ) may be extended (in a unique way if the extension doesexist) to the whole tangent space T ( G / K ) if and only if the form e ω R is extendable (admitsan extension to the whole space G × m ).By the second expression in (4.25),(4.26) e ω R ( g , xX ) (cid:0)(cid:0) ξ l ( g ) , t X (cid:1) , (cid:0) ξ l ( g ) , t X (cid:1)(cid:1) = e ω ( g , x ) (cid:0)(cid:0) ξ l ( g ) , t ∂∂ x (cid:1) , (cid:0) ξ l ( g ) , t ∂∂ x (cid:1)(cid:1) . To describe e ω R we consider again the two cases (1) and (2): (1) G / K ∈ { S n ( n > ) , H P n ( n > ) , C a P } . Since a ( x ) = f ′ ( x ) X , by (3.22) at the point ( g , xX ) ∈ G × W + and from (4.26) we have that e ω R ( g , xX ) (cid:0) ( ξ l ( g ) , t X ) , ( ξ l ( g ) , t X ) (cid:1) = − (cid:10) f ′ ( x ) X , [ ξ , ξ ] (cid:11) + f ′′ ( x ) (cid:0) t (cid:10) X , ξ (cid:11) − t (cid:10) X , ξ (cid:11)(cid:1) , where ξ , ξ ∈ g = T e G and t , t ∈ R . Consider on the whole tangent space T ( g , w ) ( G × m R ) ( w ∈ m R = m \ { } ) , the bilinear form ∆ given by ∆ ( g , w ) (cid:0) ( ξ l ( g ) , u ) , ( ξ l ( g ) , u ) (cid:1) = − (cid:10) f ′ ( r ) r w , [ ξ , ξ ] (cid:11) + f ′ ( r ) r (cid:0) h u , ξ i − h u , ξ i (cid:1) + r (cid:16) f ′ ( r ) r (cid:17) ′ (cid:0) h u , w ih w , ξ i − h u , w ih w , ξ i (cid:1) , (4.27)where ξ , ξ ∈ g = T e G , u , u ∈ m = T w m R . Here r = r ( w ) , r ( w ) def = h w , w i ( r ( xX ) = x ). Itis clear that this form is skew-symmetric. ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 21
Since r (cid:16) f ′ ( r ) r (cid:17) ′ = r f ′′ ( r ) − f ′ ( r ) r , it is easy to verify that ∆ ( g , xX ) (cid:0) ( ξ l ( g ) , t X ) , ( ξ l ( g ) , t X ) (cid:1) = e ω R ( g , xX ) (cid:0) ( ξ l ( g ) , t X ) , ( ξ l ( g ) , t X ) (cid:1) , i.e. the restrictions of e ω R and ∆ to G × W + ⊂ G × m R coincide. Now to prove that thedifferential forms e ω R and ∆ coincide on the whole tangent bundle T ( G × m R ) it is sufficientto show that the form ∆ is left G -invariant, right K -invariant and its kernel contains (andtherefore coincides with) the subbundle K = ker π ∗ .Since for each k ∈ K the scalar product h· , ·i is Ad k -invariant and Ad k is an automorphismof g , the following relations hold: ∆ ( g , w ) (( ξ l ( g ) , u ) , ( ξ l ( g ) , u )) = ∆ ( e , w ) (( ξ , u ) , ( ξ , u ))= ∆ ( e , Ad k w ) (( Ad k ξ , Ad k u ) , ( Ad k ξ , Ad k u )) . (4.28)Hence, ∆ is left G -invariant and right K -invariant.The kernel K ⊂ T ( G × m ) of the tangent map π ∗ : T ( G × m ) → T ( G × K m ) is generatedby the (left) G -invariant vector fields ζ L , ζ ∈ k (3.17) on G × m . Then, since m ⊥ k and h w , [ w , ζ ] i =
0, we obtain ∆ ( g , w ) (cid:0) ( ξ l ( g ) , u ) , ζ L ( g , w ) (cid:1) = − (cid:10) f ′ ( r ) r w , [ ξ , ζ ] (cid:11) + f ′ ( r ) r (cid:0) h u , ζ i − h [ w , ζ ] , ξ i (cid:1) + r (cid:16) f ′ ( r ) r (cid:17) ′ (cid:0) h u , w ih w , ζ i − h [ w , ζ ] , w ih w , ξ i (cid:1) = − (cid:10) f ′ ( r ) r w , [ ξ , ζ ] (cid:11) − (cid:10) f ′ ( r ) r [ w , ζ ] , ξ (cid:11) = . This means that K ⊂ ker ∆ . Thus e ω R = ∆ on G × m R ( K = ker ∆ because the form ω isnondegenerate).Expression (4.27) determines a smooth 2-form on the whole tangent bundle T ( G × m ) if and only if lim x → f ′ ( x ) =
0, that is, C =
0. Indeed, if C > x → f ′ ( x ) = C / m and lim x → f ′′ ( x ) =
0, where m = dim m = m ε + m ε / +
1. Therefore,by (4.27), lim x → ∆ ( e , xX ) (cid:0) ( ξ , u ) , ( ξ , u ) (cid:1) = ∞ for some vectors ξ , ξ ∈ g , u , u ∈ m suchthat lim x → f ′ ( x ) x (cid:0) h u , ξ i − h u , ξ i − h u , X ih X , ξ i + h u , X ih X , ξ i (cid:1) = ∞ . Let C =
0. Since sinh xx > x >
0, there exists an even real analytic function on the wholeaxis, ψ ( x ) , such that(4.29) f ′ ( x ) = x (cid:16) m ε Cm + x ψ ( x ) (cid:17) / m , m ε Cm + x ψ ( x ) > , ∀ x > . In this case expression (4.27) determines a smooth 2-form on the whole space G × m .We will denote this form (extension) on G × m by e ω R . There exists a unique 2-form ω R on G × K m ∼ = T ( G / K ) such that e ω R = π ∗ ω R . The forms ω R and ω R coincide, by construction,on the open submanifold G × K m R ∼ = T + ( G / K ) , that is, ω R is a smooth extension of ω R . Now we will prove, applying Corollary 3.4, that this extension is the K¨ahler form of themetric g on the whole tangent bundle T ( G / K ) . Indeed, by (4.6) and (4.29) for C = x → w | ( x ) = (cid:18) m ε Cm (cid:19) / m , lim x → w j ε | j ε ( x ) = lim x → w k ε / | k ε / ( x ) = (cid:18) m ε Cm (cid:19) / m , that is, the corresponding limit diagonal Hermitian matrices lim x → w H ( x ) and lim x → w ∗ ( x ) are positive-definite. Thus by Corollary 3.4, ω R is the K¨ahler form of the metric g (theextension of g ) on G × K m ∼ = T ( G / K ) . (2) G / K = C P n ( n ≥ ) . In this case the vector-function a takes the form a ( x ) = f ′ ( x ) X + c Z ϕ ( x ) Z + c Z = f ′ ( x ) X + c Z (cid:0) ϕ ( x ) − (cid:1) Z − c Z Z , because Z = ( Z + Z ) and c = − c Z . Here f ′ ( x ) is given in (4.5), ϕ ( x ) = x and c Z isan arbitrary real number. Then, from (3.22), we have e ω R ( g , xX ) (cid:0) ( ξ l ( g ) , t X ) , ( ξ l ( g ) , t X ) (cid:1) = − (cid:10) f ′ ( x ) X + c Z (cid:0) ϕ ( x ) − (cid:1) Z − c Z Z , [ ξ , ξ ] (cid:11) + f ′′ ( x ) (cid:0) t (cid:10) X , ξ (cid:11) − t (cid:10) X , ξ (cid:11)(cid:1) + c Z ϕ ′ ( x ) (cid:0) t (cid:10) Z , ξ (cid:11) − t (cid:10) Z , ξ (cid:11)(cid:1) , (4.30)where ξ , ξ ∈ g = T e G and t , t ∈ R .Consider on the whole tangent space T ( g , w ) ( G × m R ) ( w = ∆ : ∆ ( g , w ) (cid:0) ( ξ l ( g ) , u ) , ( ξ l ( g ) , u ) (cid:1) = − (cid:10) f ′ ( r ) r w + c Z r (cid:0) ϕ ( r ) − (cid:1) [ Iw , w ] − c Z Z , [ ξ , ξ ] (cid:11) + f ′ ( r ) r (cid:0) h u , ξ i − h u , ξ i (cid:1) + r (cid:16) f ′ ( r ) r (cid:17) ′ (cid:0) h u , w ih w , ξ i − h u , w ih w , ξ i (cid:1) + c Z (cid:16) r ϕ ′ ( r ) − ( ϕ ( r ) − ) r (cid:17)(cid:0) h u , w ih [ Iw , w ] , ξ i − h u , w ih [ Iw , w ] , ξ i (cid:1) + c Z (cid:16) r ϕ ′ ( r ) − ( ϕ ( r ) − ) r (cid:17)(cid:0) h u , Iw ih w , u i − h u , Iw ih w , u i (cid:1) + c Z ( ϕ ( r ) − ) r (cid:16) h [ Iu , w ] , ξ i − h [ Iu , w ] , ξ i + h u , Iu i (cid:17) , (4.31)where ξ , ξ ∈ g = T e G , u , u ∈ m = T w m R . It is clear that this form is skew-symmetric.From the expression of e ω R at the point ( g , xX ) ∈ G × W + given in (4.30) and taking intoaccount that [ IX , X ] = Z and h X , IX i =
0, it is easy to verify that ∆ ( g , xX ) (cid:0) ( ξ l ( g ) , t X ) , ( ξ l ( g ) , t X ) (cid:1) = e ω R ( g , xX ) (cid:0) ( ξ l ( g ) , t X ) , ( ξ l ( g ) , t X ) (cid:1) . Since for each k ∈ K the scalar product h· , ·i is Ad k -invariant, Ad k is an automorphismof g and Ad k ( Z ) = Z , Ad k I = I Ad k , relations (4.28) hold now for this ∆ , that is, ∆ isleft G -invariant and right K -invariant. We now prove that ker ∆ ⊃ K . By (3.17), since theform ∆ is left G -invariant, right K -invariant and Ad ( K )( R X ) = m , it is sufficient to showthat the vectors ( ζ , x [ X , ζ ]) , ζ ∈ k , belong to the kernel of ∆ ( e , xX ) . Indeed, using the fact that ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 23 [ Z , k ] = k ⊥ m and h· , ·i is Ad ( G ) -invariant, we have that ∆ ( e , xX ) (cid:0) ( ξ , u ) , ( ζ , x [ X , ζ ]) (cid:1) = c Z (cid:0) ϕ ( x ) − (cid:1)(cid:10) − [ IX , X ] , [ ξ , ζ ] (cid:11) + x c Z (cid:0) ϕ ( x ) − (cid:1) · (cid:16) h [ Iu , X ] , ζ i − h [ I [ xX , ζ ] , X ] , ξ i + h u , I [ X , ζ ] i (cid:17) . This expression vanishes because the endomorphism I on m is skew-symmetric and − h [ IX , X ] , [ ξ , ζ ] i − h [ I [ X , ζ ] , X ] , ξ i ( . ) = h [ Z , ζ ] , ξ i + h [ X , I [ X , ζ ]] , ξ i ( . ) = h− X I ad X ζ , ξ i + h [ X , I [ X , ζ ]] , ξ i = . Thus the differential forms e ω R and ∆ coincide on the whole tangent bundle T ( G × m R ) .Our expression (4.31) of ∆ determines a smooth 2-form on the whole tangent bundle T ( G × m ) if and only if lim x → f ′ ( x ) =
0, that is, C =
0. Indeed, if C > x → f ′ ( x ) = C / n and lim x → f ′′ ( x ) =
0. Therefore by (4.31), lim x → ∆ ( e , xX ) (cid:0) ( ξ , u ) , ( ξ , u ) (cid:1) = ∞ for some vectors ξ , ξ ∈ g , u , u ∈ m such thatlim x → f ′ ( x ) x (cid:0) h u , ξ i − h u , ξ i − h u , X ih X , ξ i + h u , X ih X , ξ i (cid:1) = ∞ . Let C =
0. In this case, the expression for the function f ′ ( x ) in (4.5) is independent of n andthere exists an even real analytic function on the whole axis, ϕ ( x ) , such that(4.32) f ′ ( x ) = x (cid:0) C + c Z + x ϕ ( x ) (cid:1) / , C + c Z + x ϕ ( x ) > , ∀ x > . Hence by (4.5) the functions f ′ ( x ) x and x (cid:16) f ′ ( x ) x (cid:17) ′ are even real analytic functions on the wholeaxis. Also taking into account that ϕ ( x ) = x and ϕ ′ ( x ) = − ϕ ( x ) tanh x and tanh ′ x = ϕ ( x ) we obtain that ϕ ( x ) = − x + x + ϕ ( x ) x , where ϕ ( x ) is an even real analyticfunction on the whole axis R . Therefore the functions ϕ ( x ) − x and x ϕ ′ ( x ) − ( ϕ ( x ) − ) x = + ϕ ( x ) x + ϕ ′ ( x ) x are even real analytic functions defined on the whole axis. Therefore theexpression (4.31) determines a smooth 2-form on the whole tangent space T ( G / K ) . We willdenote, as in the previous cases, this form (extension) on T ( G × m ) by e ω R .By continuity the form e ω R is closed, left G -invariant, right K -invariant and K ⊂ ker e ω R .It is clear that there exists a unique (closed) 2-form ω R on T ( G / K ) such that e ω R = π ∗ ω R .Now we will prove, applying Corollary 3.4, that this extension is the K¨ahler form of themetric g on the whole tangent bundle T ( G / K ) . Indeed, the entries of w H ( x ) are determinedby expressions (4.19) and therefore by (4.32),lim x → w | ( x ) = q C + c Z , lim x → w | ε ( x ) = c Z , lim x → w ε | ε ( x ) = q C + c Z . Also from relations (4.20) it follows that for the block-diagonal Hermitian matrix w ∗ ( x ) foreach its 2 × x → w ( j − )( j − ) ( x ) = lim x → w ( j )( j ) ( x ) = q C + c Z , lim x → w ( j − )( j ) ( x ) = c Z . It is easy to check that the corresponding limit diagonal Hermitian matrices lim x → w H ( x ) and lim x → w ∗ ( x ) are positive-definite. Thus by Corollary 3.4, ω R is the K¨ahler form of themetric g (the extension of g ) on G × K m ∼ = T ( G / K ) .Let us prove that the metric g determined by the form ω R on the whole tangent bundle T ( G / K ) ∼ = G × K m is complete.First of all suppose that ω R is determined by the vector-function a ( x ) = f ′ ( x ) X . By Corol-lary 3.7, such a metric admits a G -invariant potential function 2 f ( r ) on T ( G / K ) \ G / K ,where r is the norm function determined by a G -invariant metric on G / K . Since in our cases f ( x ) is the restriction of an even smooth function on the whole axis R , there exist a smoothextension of 2 f ( r ) to the whole tangent bundle T ( G / K ) . By continuity, this extension isa potential function on T ( G / K ) . Now, Stenzel described all G -invariant K¨ahler structures ( ω , J Kc ) on T ( G / K ) , where G / K is a compact symmetric space of rank one admitting a G -invariant potential function [18]. Thus the set of metrics c g , c >
0, coincides with Stenzel’sset of metrics. The completeness of these metrics is proved in Stenzel’s paper [18] (see alsoanother proof of this fact in Mykytyuk [14]).Let us prove that the metric g determined by the form ω R on the whole tangent bundle T ( G / K ) ∼ = G × K m is complete if G / K = C P n , n >
2. To this end, consider again its descrip-tion (4.30) on the space G / H × R + ( G = SU ( n + ) , H ∼ = U ( ) × SU ( n − ) ). For our aim itis sufficient to calculate the distance dist ( b , c ) between the compact subsets G / H × { b } and G / H × { c } , where dist ( b , c ) = inf { d ( p b , p c ) , p b ∈ G / H × { b } , p c ∈ G / H × { c }} . Since thesets G / H × { x } are compact, it is clear that the metric g is complete if and only if for some b > c → ∞ dist ( b , c ) = ∞ .To calculate the function dist ( b , c ) note that the tangent bundle T ( G / K ) ∼ = G × K m is acohomogeneity-one manifold, i.e. the orbits of the action of the Lie group G have codimen-sion one. We will use only one fundamental fact on the structure of these manifolds [1]: Aunit smooth vector field U on a G -invariant domain D ⊂ T ( G / K ) which is g -orthogonal toeach G -orbit in D is a geodesic vector field, i.e. its integral curves are geodesics of the metric g .We now describe such a vector field U on the domain G × R + ∼ = T ( G / K ) \ G / K . Put(4.33) f U ( x ) = (cid:18) f ′ ( x ) f ′ ( x ) f ′′ ( x ) + c Z ϕ ( x ) ϕ ′ ( x ) (cid:19) / , x ∈ R + , where, recall, ϕ ( x ) = x . ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 25
Lemma 4.2.
Such a unit vector field U on G / H × R + is G-invariant and at the point ( o , x ) ,o = { H } , x ∈ R + , is determined by the expression (4.34) U ( o , x ) = f U ( x ) · (cid:18) − c Z ϕ ′ ( x ) f ′ ( x ) Y , ∂∂ x (cid:19) . For the coordinate function x on G / H × R + the following inequality holds (4.35) (cid:12)(cid:12) d x ( o , x ) (cid:0) ξ , t ∂∂ x (cid:1)(cid:12)(cid:12) f U ( x ) · (cid:13)(cid:13) ( ξ , t ∂∂ x ) (cid:13)(cid:13) ( o , x ) , where (cid:0) ξ , t ∂∂ x (cid:1) ∈ T ( o , x ) ( G / H × R + ) = ( m ⊕ k + ) × R and k · k is the norm given by the met-ric g .Proof (of Lemma) Since the vector field U is unique (up to sign), it is sufficient to verifythat each vector U ( o , x ) in (4.34) is g -orthogonal to the G -orbit through ( o , x ) , i.e. to thesubspace V ( o , x ) ⊂ T ( o , x ) ( G / H × R + ) generated by the vectors ( ξ , ) , ξ ∈ m ⊕ k + , and that k U ( o , x ) k = e ω R we obtain the following expression for the form ω at ( o , x ) : ω ( o , x ) (cid:16)(cid:0) ξ , t ∂∂ x (cid:1) , (cid:0) ξ , t ∂∂ x (cid:1)(cid:17) = − (cid:10) f ′ ( x ) X + c Z (cid:0) ϕ ( x ) − (cid:1) Z − c Z Z , [ ξ , ξ ] (cid:11) + f ′′ ( x ) (cid:0) t (cid:10) X , ξ (cid:11) − t (cid:10) X , ξ (cid:11)(cid:1) + c Z ϕ ′ ( x ) (cid:0) t (cid:10) Z , ξ (cid:11) − t (cid:10) Z , ξ (cid:11)(cid:1) , (4.36)where ξ , ξ ∈ m ⊕ k + , t , t ∈ R .Fix a point ( o , x ) ∈ G / H × R + and consider a tangent vector e Y = (cid:16) bY , ∂∂ x (cid:17) , b ∈ R , at ( o , x ) . This vector is g -orthogonal to V ( o , x ) if and only if this vector is ω -orthogonal tothe subspace J Kc ( V ( o , x )) generated by the vectors ( ξ , t ∂∂ x ) , ξ ∈ m + ⊕ k + (cid:0) h ξ , X i = (cid:1) , t ∈ R , because by (3.18),(4.37) J Kc ( o , x )( X , ) = (cid:0) , ∂∂ x (cid:1) , J Kc ( o , x )( Y , ) = (cid:0) − cosh x sinh x Z , (cid:1) and J Kc ( o , x )( ξ j ε / , ) = ( − cosh x sinh x ζ j ε / , ) , j = , . . . , ( n − ) . By (4.36) for any ξ ∈ m + ⊕ k + , t ∈ R we obtain ω ( o , x ) (cid:0)(cid:0) ξ , t ∂∂ x (cid:1) , (cid:0) bY , ∂∂ x (cid:1)(cid:1) = b (cid:10) [ f ′ ( x ) X + c Z (cid:0) ϕ ( x ) − (cid:1) Z − c Z Z , Y ] , ξ (cid:11) + f ′′ ( x ) (cid:0) t (cid:10) X , bY (cid:11) − (cid:10) X , ξ (cid:11)(cid:1) + c Z ϕ ′ ( x ) (cid:0) t (cid:10) Z , bY (cid:11) − (cid:10) Z , ξ (cid:11)(cid:1) = − ( b f ′ ( x ) + c Z ϕ ′ ( x )) h Z , ξ (cid:11) , (4.38)because ξ ⊥ X and [ Z , Y ] = IY = − X (see also relations (4.8)). By (4.38) the vector U ( o , x ) = f U ( x ) e Y with b = − c Z ϕ ′ ( x ) / f ′ ( x ) is g -orthogonal to the subspace V ( o , x ) .But by (4.37), J Kc ( o , x ) (cid:16) − c Z ϕ ′ ( x ) f ′ ( x ) Y , ∂∂ x (cid:17) = (cid:16) − c Z ϕ ( x ) f ′ ( x ) Z − X , (cid:17) because ϕ ′ ( x ) = − ϕ ( x ) tanh x .Taking into account relations (4.36), (4.31), (4.8) and the fact that h Z , Z i = − ω ( J Kc ( U ) , U ) = f U (cid:16) f ′′ + c Z ϕ ′ ϕ f ′ (cid:17) ≡
1, i.e. k U k ≡ To prove the inequality in the statement it is sufficient to find the Hamiltonian vector field H x of the function x . This vector field is G -invariant as so are the form ω and the function x .Let us show that H x ( o , x ) = (cid:0) a ( x ) X + c ( x ) Z , (cid:1) , where a , c are some functions of x . Indeed,using relations (4.36), (4.7), (4.8), [ Z , Z ] = h· , ·i , we obtainthe following expression at the point ( o , x ) for any ξ ∈ m ⊕ k + , t ∈ R : ω (cid:16)(cid:0) ξ , t ∂∂ x (cid:1) , (cid:0) aX + cZ , (cid:1)(cid:17) = − (cid:10) f ′ X + c Z (cid:0) ϕ − (cid:1) Z − c Z Z , [ ξ , aX + cZ ] (cid:11) + f ′′ t (cid:10) X , aX + cZ (cid:11) + c Z ϕ ′ t (cid:10) Z , aX + cZ (cid:11) = ( c f ′ − c Z a ϕ ) (cid:10) Y , ξ (cid:11) + ( a f ′′ + c Z c ϕ ′ ) t , (4.39)Now it is easy to see that ω (cid:0) ( ξ , t ∂∂ x ) , H x (cid:1) def = d x (cid:0) ξ , t ∂∂ x (cid:1) = t at the point ( o , x ) for arbitrary t ∈ R , ξ ∈ m ⊕ k + if and only if(4.40) a = c · f ′ c Z ϕ and c = c Z ϕ f ′′ f ′ + c Z ϕ ′ ϕ . Since J Kc ( H x )( o , x ) = (cid:0) c sinh x cosh x Y , a ∂∂ x (cid:1) and a = f U we obtain at the point ( o , x ) k H x k = ω (cid:0) ( c sinh x cosh x Y , a ∂∂ x ) , ( aX + cZ , ) (cid:1) = d x (cid:0) c sinh x cosh x Y , a ∂∂ x (cid:1) = a = f U . Now, by the Cauchy-Schwarz inequality for metrics one has at the point ( o , x ) (cid:12)(cid:12) d x ( ξ , t ∂∂ x ) (cid:12)(cid:12) = (cid:12)(cid:12) ω (( ξ , t ∂∂ x ) , H x ) (cid:12)(cid:12) = (cid:12)(cid:12) g (( ξ , t ∂∂ x ) , J Kc ( H x )) (cid:12)(cid:12) (cid:13)(cid:13) J Kc ( H x ) (cid:13)(cid:13) · (cid:13)(cid:13) ( ξ , t ∂∂ x ) (cid:13)(cid:13) = k H x k · (cid:13)(cid:13) ( ξ , t ∂∂ x ) (cid:13)(cid:13) , that is, we obtain (4.35). (cid:3) Using now the vector field U we shall calculate the distance between the level sets G / H ×{ b } and G / H × { c } in G / H × R + with respect to the metric g . Let γ ( t ) = ( b g ( t ) H , b x ( t )) , t ∈ [ , T ] , be the integral curve of the vector field U with initial point p b in G / H × { b } , thatis, b x ( ) = b . There exists a function h on R + such that the function h ( b x ( t )) is linear in t . It iseasy to verify that h ( x ) = Z xb d sf U ( s ) , because by (4.34)dd t h (cid:0)b x ( t ) (cid:1) = h ′ (cid:0)b x ( t ) (cid:1) · d x (cid:0) γ ′ ( t ) (cid:1) = h ′ (cid:0)b x ( t ) (cid:1) · b x ′ ( t ) = h ′ (cid:0)b x ( t ) (cid:1) · (cid:0) f U ( b x ( t )) (cid:1) = . Suppose that p c ∈ G / H × { c } , where p c = γ ( t c ) , t c ∈ [ , T ] . Since the curve γ is a geodesic,the length of the curve γ ( t ) , t ∈ [ , t c ] , from p b to p c is t c = h ( x ( p c )) − h (cid:0) x ( p b ) (cid:1) = h ( c ) − h ( b ) .Thus dist ( b , c ) > h ( c ) − h ( b ) .For any other curve γ ( t ) = (cid:0)b g ( t ) H , b x ( t ) (cid:1) , with k γ ′ ( t ) k =
1, starting at the point p b , andending at a point p c ∈ G / H × { c } , p c = γ ( t c ) (of length t c ), we obtain by Lemma 4.2dd t h (cid:0)b x ( t ) (cid:1) = h ′ (cid:0)b x ( t ) (cid:1) · d x (cid:0) γ ′ ( t ) (cid:1) f U (cid:0)b x ( t ) (cid:1) · f U ( b x ( t )) (cid:1) · k γ ′ ( t ) k = . ICCI-FLAT K ¨AHLER METRICS ON TANGENT BUNDLES OF RANK-ONE SYMMETRIC SPACES 27
Thus h ( c ) − h ( b ) t c and the length t c of the curve γ from p b to p c is not less than thelength of the curve γ ( t ) , t ∈ [ , t c ] . Thus, the distance between the level surfaces G / H × { b } and G / H × { c } is | h ( c ) − h ( b ) | .Now, since by (4.23) and (4.24) for C = f ′ ( x ) = q C sinh x + c Z sinh x cosh − x , f ′′ ( x ) = (cid:0) C sinh x cosh x + c Z sinh x cosh − x − c Z sinh x cosh − x (cid:1) / f ′ ( x ) , we obtain that f ′ ( x ) ∼ √ C sinh x , f ′′ ( x ) ∼ √ C sinh x and, by (4.33), f U ( x ) ∼ (cid:0) √ C sinh x (cid:1) / as x → ∞ . Therefore lim x → ∞ h ( x ) = ∞ . Hence the metric g = g ( C , c Z , ) (that is, for C = T ( G / K ) is complete for any C > c Z ∈ R . (cid:3) It is well known that R P n ∼ = S n / Z as R P n = SO ( n + ) / O ( n ) and S n = SO ( n + ) / SO ( n )( n > ) . Hence each SO ( n + ) -invariant Ricci-flat K¨ahler structure on T R P n is uniquelydetermined by a Z -invariant Ricci flat K¨ahler structure on T S n . Corollary 4.3.
If n > , each G-invariant Ricci-flat K¨ahler structure ( g ( C , C ) , J Kc ) on thepunctured tangent bundle T + ( G / K ) = T + ( SO ( n + ) / SO ( n )) = T + S n determines an invari-ant Ricci-flat K¨ahler structure on T + R P n . If n = , the G-invariant Ricci-flat K¨ahler struc-ture ( g ( C , C , c Z ) , J Kc ) on T + ( G / K ) = T + ( SO ( ) / SO ( )) = T + S determines an invariantRicci-flat K¨ahler structure on T + R P if and only if c Z = . All these invariant Ricci-flatK¨ahler metrics on T + R P n are uniquely extendable to complete metrics on the whole tangentbundle T R P n , n > , if and only if C = .Proof. We will use the notations of the proof of Theorem 4.1. As it follows from its proof theK¨ahler structure ( g ( C , C ) , J Kc ) on T + ( G / K ) = T + ( SO ( n + ) / SO ( n )) ( n > ) is Z -invariantif and only if the form e ω R = ∆ (see (4.27)) on G × m is right K -invariant, where K = O ( n ) ( K ⊂ K ⊂ G ). The form e ω R is right K -invariant because Ad ( K )( m ) = m and Ad ( K ) is a subgroup of the group of inner automorphisms Ad ( G ) of g . Similarly, if n =
2, theform e ω R = ∆ (see (4.31)) is right K -invariant if and only if c Z = ( K ) Z = { Z } ( Z = Z if n = R P is not a homogeneous complex manifold). Now the last assertionof the corollary immediately follows from the last assertion of Theorem 4.1. (cid:3) R EFERENCES [1] Alekseevsky, A.V., Alekseevsky, D.V: Riemannian G -manifolds with one-dimensional orbit space. Ann.Global Anal. Geom. (3), 197–211 (1993)[2] Biquard, O., Gauduchon, P.: Hyperk¨ahler metrics on cotangent bundles of Hermitian symmetric spaces,in: J. E. Andersen, J. Dupont, H. Pedersen, A. Swann (Eds.), Geometry and Physics, Vol. 184, Lect. NotesPure Appl. Math., Marcel Dekker, pp. 287–298 (1996)[3] Biquard, O., Gauduchon, P.: G´eom´etrie hyperk¨ahl´erienne des espaces hermitiens sym´etriques complex-ifi´es. S´eminaire de Th´eorie spectrale et G´eom´etrie , 127–173 (1998)[4] Calabi, E.: M´etriques k¨ahl´eriennes et fibr´es holomorphes. Ann. Sci. ´Ecole Norm. Sup. (4) , 269–294(1979) [5] Cvetiˇc, M., Gibbons, G.W., L¨u, L., Pope, C.N.: Ricci-flat metrics, harmonic forms and brane resolutions.Comm. Math. Phys. , 457–500 (2003)[6] Dancer, A., Wang, M.Y.: K¨ahler-Einstein metrics of cohomogeneity one. Math. Ann. (3), 503–526(1998)[7] Eguchi, T., Hanson, A.J.: Asymptotically flat self-dual solutions to Euclidean gravity. Phys. Lett. B74 ,249–251 (1978)[8] Gadea, P.M., Gonz´alez-D´avila, J.C., Mykytyuk, I.V.: Invariant Ricci-flat K¨ahler metrics on tangent bun-dles of compact symmetric spaces. http://arxiv.org/abs/1903.00044 [9] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Academic Press, New York-SanFrancisco-London (1978)[10] Ionel, M., Ivey, T.A.: Austere submanifolds in C P n . Comm. Anal. Geom. (4), 821–841 (2016)[11] Lee, T.C.: Complete Ricci-flat K¨ahler metric on M nI , M nII , M nIII . Pacific J. Math. (2), 315–326 (1998)[12] Mostow, G.D.: Some new decomposition theorems for semisimple groups. Mem. Amer. Math. Soc. ,31–54 (1955)[13] Mostow, G.D.: On covariant fiberings of Klein spaces. Amer. J. Math. , 247–278 (1955)[14] Mykytyuk, I.V.: Invariant K¨ahler structures on the cotangent bundles of compact symmetric spaces.Nagoya Math. J. , 191–217 (2003)[15] Oliveira, G.: Calabi-Yau monopoles for the Stenzel metric. Comm. Math. Phys. (2), 699–728 (2016)[16] Onishchik, A.L., Vinberg, E.V.: Lie groups and algebraic groups. Springer Series in Soviet Mathematics.Springer-Verlag, Berlin (1990)[17] Patrizio, G., Wong, P.: Stein manifolds with compact symmetric center. Math. Ann. , 355–382 (1991)[18] Stenzel, M.: Ricci-flat metrics on the complexification of a compact rank one symmetric space.Manuscripta Math. , 151–163 (1993)I NSTITUTO DE F´ ISICA F UNDAMENTAL , CSIC, S
ERRANO
BIS , 28006-M
ADRID , S
PAIN . E-mail address : [email protected] D EPARTAMENTO DE M ATEM ´ ATICAS , E
STAD ´ ISTICA E I NVESTIGACI ´ ON O PERATIVA , U
NIVERSITY OF L A L AGUNA , 38200 L A L AGUNA , T
ENERIFE , S
PAIN . E-mail address : [email protected] I NSTITUTE OF M ATHEMATICS , C
RACOW U NIVERSITY OF T ECHNOLOGY , W
ARSZAWSKA
24, 31155,C
RACOW , P
OLAND .I NSTITUTE OF A PPLIED P ROBLEMS OF M ATHEMATICS AND M ECHANICS , N
AUKOVA S TR . 3 B , 79601,L VIV , U
KRAINE . E-mail address ::