INJECTIVITY RADIUS AND DIAMETER OF THE MANIFOLDS OF FLAGS IN THE PROJECTIVE PLANES

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1 INJECTIVITY RADIUS AND DIAMETER OF THE MANIFOLDS OF FLAGS IN THE PROJECTIVE PLANES THOMAS PÜTTMANN Abstract. The manifolds of flags in the rojective lanes RP, CP, HP, and OP are among the very few comact manifolds that are known to admit metrics with ositive sectional curvature. They also arise as isoarametric hyersurfaces in sheres. We show how their aearance in these two fields is related and study the global geodesic geometry of these saces in detail. 1. Introduction The manifolds F R, F C, F H, F O of flags in the rojective lanes RP, CP, HP, OP serve as imortant examles in two classical fields in Riemannian geometry: They are among the very few manifolds that are known to admit metrics with ositive sectional curvature and they arise as isoarametric hyersurfaces in the sheres S 4, S 7, S 13, and S 5 (see e.g. [Ka]. Their aearance in these two fields seems to be rather unrelated since the standard metrics on the sheres S 4, S 7, S 13, and S 5 induce metrics with some negative sectional curvatures on all the hyersurfaces in the isoarametric foliations. The isoarametric foliations of S 4, S 7, S 13, and S 5 are given by the isotroy reresentations of the rank symmetric saces (1 SU(3/SO(3, SU(3 SU(3/ SU(3, SU(6/S(3, E 6 /F 4, resectively. In this aer we study the geometry of the flag manifolds F R, F C, F H, F O via the orbital geometry of the isotroy actions of the symmetric saces in (1 rather than just via the infinitesimal icture rovided by the isotroy reresentations. This aroach in articular exhibits the following relation between the isoarametric foliations of the sheres and the metrics of ositive sectional curvature on the flag manifolds: Theorem 1.1. Any of the flag manifolds F R, F C, F H, F O with any homogeneous metric of ositive (or even nonnegative sectional curvature is isometric to a rincial orbit of the isotroy action of the corresonding symmetric sace in (1. In other words, when we consider sufficiently large distance sheres to a oint in one of the symmetric saces instead of the euclidian sheres in the tangent sace then some of the hyersurfaces in the isoarametric foliation of the distance sheres inherit metrics with ositive sectional curvature. All ossible homogeneous metrics with ositive sectional curvature on the hyersurfaces arise this way. This fact gives an answer in a articular case to one of the current general roblems in the field of ositive sectional curvature, namely, to the question of how one can construct ositively curved metrics on saces without using Riemannian submersions. The isotroy actions of the symmetric saces in (1 also allow us to determine the diameter of the normal homogeneous metrics on the flag manifolds: 000 Mathematics Subject Classification. 53C0, 53C30. 1

2 THOMAS PÜTTMANN Theorem 1.. Let O be a rincial orbit of the isotroy action of one of the symmetric saces in (1 such that O is isometric to a flag manifold F R, F C, F H, or F O with a normal homogeneous metric. Then O asses through two oints of maximal distance in the symmetric sace and the intrinsic diameter of O is equal to the diameter of the symmetric sace. Note that the diameter of a symmetric sace is equal to the diameter of its maximal torus and can thus be determined in a rather simle way. For the normal homogeneous metrics on the flag manifolds we know recisely which airs of oints have maximal distance. This knowledge enables us to give lower diameter bounds for all other homogeneous metrics with nonnegative curvature (these form a two dimensional family. It is of course interesting to comare these lower diameter bounds with the injectivity radius. Surrisingly, the injectivity radius can be determined recisely for all homogeneous metrics with nonnegative sectional curvature on the flag manifolds in the following way: For any ositively curved homogeneous metric on the three flag manifolds F C, F H, and F O the maximum of the sectional curvature was comuted by Valiev [Va] in By Klingenberg s classical injectivity radius estimate the injectivity radius of any simly connected, even dimensional manifold with sectional curvature 0 < sec Λ is bounded from below by π Λ. Insecting the length of certain closed geodesics on the flag manifolds we observe: Lemma 1.3. Klingenberg s injectivity radius estimate is shar for all homogeneous metrics with ositive sectional curvature on the manifolds F C, F H, and F O. Klingenberg s estimate does not aly directly to the odd dimensional and non simly connected flag manifold F R = SO(3/(Z Z. It turns out, however, that the injectivity radius of F R is determined by the totally geodesic inclusion F R F C for all ositively curved homogeneous metrics on F C. Theorem 1.4. For the normal homogeneous metrics on the four flag manifolds the diameter is 4 3 times the injectivity radius. Moreover, we have diam inj 3 π arctan( for all homogeneous metrics with nonnegative sectional curvature. Note that the normal homogeneous metric on the real flag manifold has constant ositive sectional curvature while the normal homogeneous metrics on the other flag manifolds only have a nonnegative sectional curvature function, but are surrounded by homogeneous metrics with ositive sectional curvature. From our lower diameter bound for all ositively curved homogeneous metrics and the recise lower sectional curvature bound given by Valiev we deduce the following diameter inching estimate: Theorem 1.5. The manifolds F C, F H, and F O admit metrics such that (min sec ( diam π For the normal homogeneous metric on the flag manifold F R we have (min sec ( diam π = 1 9. This result should be seen in comlement to the diameter shere theorem of Grove and Shiohama [GS] and the related rigidity theorem of Gromoll and Grove (see [GG] and [Wi1] for the result in its final form: Every Riemannian manifold with ( (min sec ( diam π > 1 4

3 INJECTIVITY RADIUS AND DIAMETER 3 is homeomorhic to the standard shere, and if equality holds in ( then the manifold is homeomorhic to the standard shere or u to scaling isometric to one of the rojective saces CP k, HP k, or OP. The author would like to thank Uwe Abresch, A. Rigas, J.-H. Eschenburg, and Wolfgang Ziller for helful comments and discussions.. The manifolds of flags in the rojective lanes A flag in a rojective lane is a air (, l consisting of a oint and a line that are incident, i.e., the line asses through the oint. The set of flags in RP, CP, HP, and OP will be denoted by F R, F C, F H, and F O, resectively. In each case the isometry grou of the rojective lane acts transitively on the set of flags. One therefore gets the following homogeneous reresentations (3 F R = SO(3/(Z Z, F C = SU(3/T, F H = S(3/S(1 3, F O = F 4 /Sin(8, where in the first three cases the isotroy grous can be chosen to be the subgrous of all diagonal matrices. The dimensions of the flag manifolds are 3, 6, 1, and 4. There are three natural rojections from each of the flag manifolds to the corresonding rojective lane: First, we can forget about the line in the air. In this case the fiber over a oint consists of all lines assing through. This set can be identified with the cut locus of the oint, i.e., with its dual rojective line. Second, we can ma each air (, l to the oint l dual to the line l. Here, the fiber obviously consists of all oints in l. Finally, we can ma (, l to the oint l dual to the unique line l that also asses through and is erendicular to l. In the homogeneous descrition of F C, for examle, these three fibrations are given as follows: (4 S j = U( j /T F C = SU(3/T SU(3/U( j = CP, where U( 1, U(, U( 3 denote the centralizers of the matrices (5 I 1 =, I =, I 3 = ( ( ( in SU(3. It can easily be seen from the isotroy reresentation of each of the flag manifolds that the three fibers S m 1, S m, and S m 3 (where m is 1,, 4, or 8 according to whether we are in the real, comlex, quaternionic, or octonionic case are mutually erendicular for all homogeneous metrics on the flag manifold and that any homogeneous metric is determined by the size of the three fibers at one oint, i.e., by three ositive arameters. The fibers are connected comonents of fixed oint sets of isometries given by the matrices I j and hence totally geodesic. None of the three rojection mas from a flag manifold to the corresonding rojective lane is a Riemannian submersion until at least two of the fibers have the same size. Only in this case there is a free and isometric Z -action on the flag manifold. The metrics for which all three fibers have the same size are recisely the normal homogeneous metrics, i.e., the metrics that are induced from the biinvariant metrics on the Lie grous SO(3, SU(3, S(3, and F 4, resectively. Only for the normal homogeneous metrics there is a free and isometric action of the symmetric grou S 3 on three letters on the flag manifold. This free and isometric action is the action of the Weyl grou on the flag manifold from the right (see Section 5.

4 4 THOMAS PÜTTMANN (6 There is the natural chain of inclusions F R F C F H F O which identifies the cones of homogeneous metrics on the flag manifolds in the natural way, i.e., a homogeneous metric on F O induces metrics on F R, F C, and F H given by the same three arameters. Each of the inclusions in (6 is totally geodesic (it is easy to see that the submanifolds are fixed oint sets of certain isometries. It is also not hard to see that given a oint F C and a tangent vector v T F O there is an element of the isotroy grou of F O that moves v into T F C (see [Wi]. This shows that the diameters of corresonding metrics on F C, F H, and F O are the same, since a geodesic from to a oint of maximal distance to in F O can be moved into the totally geodesic submanifold F C. The flag manifolds arise as isoarametric hyersurfaces of the sheres S 4, S 7, S 13, and S 5. The isoarametric foliations of these sheres are given by the isotroy reresentations of the rank symmetric saces in (1. This means that the isotroy grous of the symmetric saces act on the euclidian sheres in the tangent sace with hyersurface orbits that are diffeomorhic to the flag manifolds. As ointed out in the introduction we consider the isotroy actions of the symmetric saces on themselves rather than the isotroy reresentations. Like the flag manifolds themselves the rank symmetric saces in (1 are nested totally geodesically. Using this fact it is easy to see that it suffices to investigate just one of the isotroy actions. Note that the symmetric sace SU(3 SU(3/ SU(3 is the Lie grou SU(3. In this case the isotroy action is just the action of SU(3 on itself by conjugation. We investigate this action and the geometry of its rincial orbits in Section Sectional curvature and injectivity radius As exlained in the revious section the homogeneous metrics on the flag manifolds can be arametrized by three ositive arameters s 1, s, s 3 such that π sj is the diameter of the fiber S m j in the fibration (4 or its real, quaternionic, or octonionic analogues. Any such metric will be denoted by, s. For our uroses, it is convenient to use the elementary symmetric olynomials σ 1 = s 1 + s + s 3, σ = s s 3 + s 3 s 1 + s 1 s, σ 3 = s 1 s s 3 and additionally the symmetric olynomial σ := 4σ σ 1 = s s 3 + s 3 s 1 + s 1 s s 1 s s 3. The three critical values of the sectional curvature of the metric, s on the real flag manifold F R = SO(3/Z Z are given by the three numbers d 1 = σ 4σ 3 + s1+s+s3 s s 3, d = σ 4σ 3 + s1 s+s3 s 3s 1, d 3 = σ 4σ 3 + s1+s s3 s 1s and the scalar curvature is given by (d 1 + d + d 3 = σ σ 3. Wallach [Wa] discovered in 197 that the other three flag manifolds F C, F H, and F O admit homogeneous metrics with ositive sectional curvature as well. Since the real flag manifold F R is a totally geodesic submanifold of the other flag manifolds a metric, s on any of the four flag manifolds can only have ositive sectional curvature if d 1, d, d 3 and hence also σ are ositive. Theorem 3.1 (Valiev, see [Va] or [Pü]. For any homogeneous metric, s on the flag manifolds F C, F H, or F O we have max sec = 4 s 1 and min sec = min { } 3σ 1σ 3 σ σ σ σ 3, d 3 rovided that s 1 s s 3 and d 3 0.

5 INJECTIVITY RADIUS AND DIAMETER 5 It is imortant to note that s 1 s s 3 and d 3 0 imlies that 3σ1σ3 σσ σ σ 3 0, i.e., the domains of homogeneous metrics with nonnegative sectional curvature on all four flag manifolds are identical. The domains of homogeneous metrics with ositive sectional curvature differ only by the normal homogeneous metrics, which have ositive sectional curvature in case of the real flag manifold F R and nonnegative sectional curvature in case of the other flag manifolds F C, F H, and F O. We will now see that Klingenberg s injectivity radius estimate is shar for all homogeneous metrics with ositive sectional curvature on the flag manifolds F C, F H, and F O : Corollary 3.. The injectivity radius of a metric, s with nonnegative sectional curvature on any of the four flag manifolds is equal to π min{ s 1, s, s 3 }, i.e., to the minimal diameter of the totally geodesic fibers S m 1, S m, S m 3 in (4. Proof. The flag manifolds F C, F H, and F O are simly connected and even dimensional. Klingenberg s estimate together with the revious theorem yields that π s1 is a lower bound for the injectivity radius if s 1 s s 3 and d 3 > 0. This continues to hold if we weaken d 3 > 0 to d 3 0. By definition π s1 is the diameter of the totally geodesic fiber S 1 and hence an uer bound for the injectivity radius of the flag manifold. Concerning the real flag manifold F R note that the totally geodesic inclusion F R F C imlies that the injectivity radius of F R is not less than the injectivity radius of F C. We would like to oint out that Valiev did not consider the arameters s 1, s, s 3 as the size of fibers, so the corollary is not obvious from his aer. One actually has to check that his definition of the arameters is identical to our geometric one. This can be done by a straightforward comutation. (7 4. Metrics on the comlex flag manifold induced by the adjoint action of SU(3 We will now be concerned with the fundamental action SU(3 SU(3 SU(3, (A, B ABA 1 of SU(3 on itself by conjugation. There are obviously three orbit tyes: If the matrix B is a scalar multile of the identity (i.e., in the center of SU(3 then B is a fixed oint of the action. If only two eigenvalues of B are equal then its centralizer is isomorhic to U( and the orbit is diffeomorhic to a comlex rojective lane. If all three eigenvalues are distinct then the centralizer is a maximal torus of SU(3 and the orbit is diffeomorhic to the manifold F C = SU(3/T of flags in the comlex rojective lane. We characterize the metrics that the latter orbits inherit from SU(3 in an intrinsic way using the symmetric olynomials from the revious section. Lemma 4.1. The metrics on the comlex flag manifold F C induced by conjugation on the Lie grou SU(3 equied with the biinvariant metric κ tr(xy are recisely all metrics, s for which κσ = σ 3. The metrics on the flag manifold F C induced by the adjoint action of SU(3 on its Lie algebra are recisely all metrics for which σ = 0. Note that any metric with σ > 0 can be rescaled such that κσ = σ 3. Hence, we see from the revious section that all homogeneous metrics with ositive sectional curvature on the flag manifold F C are induced by the action of SU(3 on itself by conjugation. As one exects the condition σ = 0 for the metrics that are induced

6 6 THOMAS PÜTTMANN by the adjoint action of SU(3 on its Lie algebra su(3 arises when taking the limit κ. It is easy to see that at least one of sectional curvatures d j is negative in the case σ = 0. Hence, if we consider the flag manifolds as isoarametric submanifolds of the euclidean sheres S 4, S 7, S 13, S 5 then they always inherit metrics with some negative sectional curvatures. In order to ut metrics on the sheres S 4, S 7, S 13, S 5 that induce metrics with ositive sectional curvature on the flag manifolds one can ass to sufficiently large distance sheres to any base oint in the rank symmetric saces given in (1. Such a distance shere is the union of adjoint orbits that ass through a circle around the origin in E that intersects the domain F in Figure. Proof of Lemma 4.1. We will only rove the first art of the lemma. The roof of the second art is similar and simler. Let T be the fixed maximal torus of SU(3 that consists of all diagonal matrices. This torus meets all orbits and is erendicular to all of them. We consider the universal covering ma E = { α R 3 α 1 + α + α 3 = 0} T, α diag(e iα1, e iα, e iα3. The maximal torus T corresonds to the lattice in E generated by the vectors v 1 = π( and v = π( 1 while the elements in the center of SU(3 corresond to the lattice in E generated by the vectors ( 1 ( 1 w 1 = π 3 and w = π These two lattices are illustrated in Figure 1. The solid lines reresent all singular matrices in the torus T, i.e., all matrices whose centralizer is larger than T. Let E be the lane E without the solid lines, so that the orbits through matrices corresonding to oints in E are diffeomorhic to F C = SU(3/T. We arametrize the lane E by ( β1+β β 1 π w β π w 1 = 1 β 3 1 β β 1 β with β 1, β R. A straightforward comutation shows that the orbit through the matrix determined by β 1 and β inherits the metric given by the arameters (8 s 1 = κ(1 cos β 1, s = κ(1 cos β, s 3 = κ(1 cos β 3 where β 3 = (β 1 + β. From these exressions for s 1, s, s 3 we get immediately σ 3 κσ = 4κ 3 ( 1 + cos β 1 + cos β + cos β 3 cos β 1 cos β cos β 3. In order to see that this exression is actually zero we substitute the right hand side of the identity cos β 1 = cos(β + β 3 = cos β cos β 3 sin β sin β 3 for one of the two factors cos β 1 in the square cos β 1 and do the same with the cos β and cos β 3. This yields σ 3 κσ = 4κ 3 ( 1 + cos β 1 cos β cos β 3 cos β 1 sin β sin β 3 sin β 1 cos β sin β 3 sin β 1 sin β cos β 3 = 4κ 3 ( 1 + cos(β 1 + β + β 3 = 0. We will now investigate which oints in E will rovide isometric metrics. If a matrix C is in the center of SU(3 then the orbit through BC is evidently an isometric coy of the orbit through B. Moreover, the reflection in E that fixes R w 1 mas the arameter triel (s 1, s, s 3 to the triel (s 1, s 3, s, and the reflection in 1

7 INJECTIVITY RADIUS AND DIAMETER 7 v 1 w 1 w F v Figure 1. The lattices in E that corresond to the maximal tori in SU(3 and PSU(3. β 1 = β s 1 = s β 1 = 0 s 1 = 0 s = s 3 F β = π β1 s = s 3 Figure. The fundamental domain F in the maximal torus of SU(3 and the subdomain F that corresonds to homogeneous metrics of nonnegative sectional curvature on the flag manifolds.

8 8 THOMAS PÜTTMANN E that fixes R v mas the triel (s 1, s, s 3 to the triel (s 3, s, s 1. These two reflections generate the dihedral grou D 6. A fundamental domain F for the action on E generated by the two reflections and the translations by central elements is given by 0 < β 1 π 3 and β 1 β π β1 and deicted in Figure 1 by the shaded triangle (see also Figure. It is easy to see that the arameters s 1, s, s 3 induced by oints in F satisfy s 1 s s 3. It can be shown by a sequence of elementary arguments that 0 < s 1 s s 3 and κσ = σ 3 imlies that s 3 4κ and that s 3 is uniquely determined by s 1 and s by the formula s 3 = s 1 + s s1s κ + (s 1 + s s1s κ (s 1 s. From this fact it follows that the assignment (8 mas the fundamental domain F injectively into the set S of arameters with 0 < s 1 s s 3 and κσ = σ 3. It is not difficult to see that this assignment mas F also surjectively onto the connected set S (first insect the assignment along the boundary and then aly a connectedness argument. Remark 4.. The rincial orbits of the adjoint action of SU(3 that inherit normal homogeneous metrics corresond to the lower left corner of the domain F in Figure 1, i.e., to the midoints of the equilateral triangles bounded by the solid lines. If one considers the adjoint action of PSU(3 instead of the adjoint action of SU(3 then these orbits become isolated excetional orbits diffeomorhic to F C /Z The diameter of the flag manifolds In order to describe the oints of maximal distance in the flag manifolds it is useful to consider the Weyl grous of the flag manifolds first. The Weyl grou of a homogeneous sace G/H is by definition the quotient N(H/H where N(H denotes the normalizer of H in G. The Weyl grou is obviously a homogeneous subsace of G/H, namely, the fixed oint set of the isotroy grou H. (The Weyl grou also acts freely from the right on G/H by diffeomorhisms. This latter action induces an action on the cone of left invariant metrics on G/H. In our case of the flag manifolds the three arameters of a left invariant metric are ermuted. For the homogeneous reresentations of the flag manifolds given in (3 the Weyl grous consist of the following six oints: ( 0 0 ( 0 0 ( 0 0 id = 0 0, = 0 0, = 0 0, (9 3 =, 31 =, 1 =, ( ( ( where the asterisks can be filled in any way that is consistent with the grou G. This makes immediately sense for the matrix grous SO(3, SU(3, and S(3 in the case of the flag manifolds F R, F C, and F H, and it is not difficult to give some meaning to it in the case of the octonionic flag manifold F O = F 4 /Sin(8. The Weyl grous are, as indicated by the notation, isomorhic to the symmetric grou on three letters. The three oints in the first line of (9 form the alternating grou on three letters, i.e., the cyclic grou of order three. The oints in the second line form the other coset of this grou. At each oint of the Weyl grou there are the three fibers S m of the fibrations to the rojective sace. The antiodes of in these three fibers are recisely the oints in the coset of the alternating grou to which the oint does not belong. Hence, there are in total nine sheres that meet erendicularly at the oints of the Weyl grou. The entire configuration

9 INJECTIVITY RADIUS AND DIAMETER 9 3 S m 1 c S m S m 3 id 31 1 Figure 3. The nine sheres that join the six oints of the Weyl grou. is illustrated in Figure 3. The Normalizer N(H acts transitively on the set of unordered airs of oints in N(H/H which belong to a common coset of the alternating grou: (10 { id, }, {, }, {, id }, { 1, 31 }, { 31, 3 }, { 3, 1 }. Therefore, the distance between the two oints in one of these airs is indeendent of the air. It will turn out that the diameter of the flag manifolds is given by this distance (at least for the normal homogeneous metrics. Lemma 5.1. Let, s be a metric on one of the flag manifolds that corresonds to a diagonal matrix D in the fundamental domain F of the revious section given by the arameters β 1 and β. Then the distance between the two oints id and with resect to the metric, s is bounded from below as follows: dist( id, β1 + β + β 1β. Proof. The distances between the two oints id and are the same in the three flag manifolds F C, F H, and F O since any geodesic in F O between the two oints can be moved isometrically into the totally geodesic submanifold F C. In F R the distance between the two oints cannot be smaller than the distance in F C. Hence, it suffices to consider F C. The orbit of the diagonal matrix D with resect to the action by conjugation on SU(3 is isometric to F C with the metric, s. The isometry is given by the embedding SU(3/T SU(3, A T ADA 1. The oint id is maed to the oint diag(e iα1, e iα, e iα3 in T SU(3, and the oint is maed to the oint diag(e iα3, e iα1, e iα. It is easy to comute that the distance between these two oints with resect to the biinvariant metric 1 tr(xy on SU(3 is given by β1 + β + β 1β.

10 10 THOMAS PÜTTMANN Theorem 5.. The diameter of each of the flag manifolds with the normal homogeneous metric, (3,3,3 is equal to the diameter of the Lie grou SU(3 with the biinvariant metric 1 π tr(xy, i.e., equal to 3. Proof. The metric induced on the orbit of the adjoint action on SU(3 through the oint in the fundamental domain F given by β 1 = β = π 3 is the metric, (3,3,3. Hence, the revious lemma shows that the diameter is π 3. In order to see the converse inequality for F C, F H, and F O note that the diameter of PSU(3 equied with the biinvariant metric 1 tr XY equals π 3 (see Figure 1 and that there is a Riemannian submersion from PSU(3 equied with the metric 3 tr XY to the flag manifold F C = SU(3/T equied with the metric, (3,3,3. Since Riemannian submersions are distance nonincreasing, the diameter of F C, F H, and F O is π 3. In the case of F R we refer to the comments on the cut locus of F R at the end of this section. in SO(3 with c(0 = 1l and c( π 3 = ( For the normal homogeneous metrics a minimal geodesic segment from id to is exlicitely given as follows: Consider the geodesic ( c(t = ( cos t 1 1 ( sin t The rojection of the segment c([0, π 3 ] to the real flag manifold F R = SO(3/Z Z is a bijection and joins the oints id and (see Figure 3. Theorem 5.3. On the four flag manifolds the inequality diam inj 3 π holds for all homogeneous metrics with σ 0 and the inequality diam inj 3 π arctan( holds for all homogeneous metrics with nonnegative sectional curvature. Proof. Every homogeneous metric with σ > 0 can be rescaled such that σ = σ 3. The latter metrics corresond to the fundamental domain F of the revious section that is deicted in Figure 1 and Figure. From Lemma 5.1 and Corollary 3. we see that f(β 1, β := β 1 +β +β1β (1 cos β 1 ( π diam. inj Since f β > 0 holds on the interior of the fundamental domain F, the infimum of f on F is aroched when 0 F is aroached along the left edge. Along this edge we have β 1 = β and 3β f(β, β = (1 cos β = 3 ( β/ sin(β/ which is increasing in β and tends to 3 if β 0. This roves the first art of the roosition. The subdomain F of F that corresonds to all metrics with nonnegative sectional curvature on the flag manifolds is in the notation of Section 3 given by d 3 0. This is equivalent to tan(β 1 / tan(β / (note that σ = σ 3. The subdomain F is given in Figure. The uer boundary curve is given by the function β (β 1 = arctan tan(β 1/. with β (0 = π. The derivative of this function is β (β 1 = tan(β 1/ 1.

11 INJECTIVITY RADIUS AND DIAMETER Figure 4. The identifications in the tangential cut locus of F R at id. Therefore β (β 1 π β1 and the oint given by β 1 and β (β 1 belongs to F until β (β 1 = β 1, i.e., if and only if 0 < β 1 arctan. The arguments that were used to determine the infimum of f on the whole fundamental domain F show that the minimum of the function f on F is attained along the uer boundary curve of F. Therefore, we consider the derivative of the function g(β 1 := f(β 1, β (β 1 : g (β 1 := (β1+ββ +β+β1β (1 cos β1 (β 1 +β +β1β sin β1 (1 cos β 1 If β 1 + β β + β + β 1 β 0 then evidently g (β 1 < 0 for 0 < β 1 arctan. Otherwise, the elementary inequalities 1 cos β β 1 and sin β 1 β1 3 for 0 < β 1 π 3 together with β 1 β and β 1 show that g (β 1 < 0 for 0 < β 1 arctan. Therefore, the minimum of f on F is attained for β 1 = β = arctan. Proof of Theorem 1.5. It follows from Lemma 4.1 with κ = 1 and Theorem 3.1 that the minimal sectional curvature of a metric corresonding to a oint in F is given by min { 3σ 1 σ 1, } (11 s1+s s3 s 1s 1 4 where the arameters s 1 s s 3 are determined by (8. Hence, Lemma 5.1 imlies (β1 + β + β 1 β min { 3σ 1 σ 1, s1+s s3 s 1s 4} 1 diam (min sec. It can be seen that the left hand side attains its maximum at the oint in F given by β 1 = arccos 1 8, β = arccos( 3 4, which corresonds to the metric, s with the arameters (1 s 1 = 7 4, s = s 3 = 7. This metric rovides the lower bound for the diameter inching constant that is stated in Theorem 1.5 in the introduction. Note that the metric given by the arameters in (1 is also the metric of otimal sectional curvature inching A reason for this coincidence is that the minimum of the sectional curvature in (11 is given by the minimum of two functions, which are equal for these secific arameters. Finally, we describe the cut locus of the real flag manifold F R with the normal homogeneous metric, (1,1,1. Since F R can be regarded equivalently as the quotient of the shere S 3 with radius by the quaternion grou the cut locus can be determined by exlicit comutations, which we omit. The result is as follows:

12 1 THOMAS PÜTTMANN Given a unit initial vector v in the tangent sace at id the geodesic defined by v is minimal recisely on the interval from 0 to arccot( v. Here, the -norm is taken with resect to the coordiante system given by the three irreducible subsaces of the isotroy reresentation. It follows that the diameter of F R is π 3. The identifications in the tangential cut locus are illustrated in Figure 4. For the sake of simlicity we have rojected the tangential cut locus radially to a cube. Each edge is ointwise identified with two other edges as indicated. Each oint in the interior of a face is identified with just one other oint on the oosite face in a way that is determined by the identification on the edges. An imortant subset of the cut locus of a oint in a Riemannian manifold is the set of critical oints for the distance function to the oint in the sense of Grove and Shiohama (see [GS]. This is the set of oints q for which the initial vectors of minimal geodesic segments from q to are not contained in an oen halfsace. A oint of maximal distance to is necessarily a critical oint for the distance function to. For the normal homogeneous metric on F R one can see from Figure 4 that only the oints in the Weyl grou are critical oints for the distance function to the oint id. The oints in the Weyl grou are critical oints of the distance function to the oint id for all homogeneous metrics on all four flag manifolds. This follows immediately from the isotroy reresentations. A larger set of oints that is contained in the cut locus of id for all homogeneous metrics is given by the union of the six sheres in Figure 3 that do not contain id, since these sheres are connected comonents of the fixed oint sets of the isometries I 1, I, or I 3 given in (5 that all fix id as well. In case of the normal homogeneous metric on F R these six sheres can be found as the diagonals of the faces in Figure 4. References [CE] J. Cheeger, D. Ebin, Comarison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, New York, [GG] D. Gromoll, K. Grove, A generalization of Berger s rigidity theorem for ositively curved manifolds, Ann. Sci. Ecole Norm. Su. 0 (1987, [GS] K. Grove, K. Shiohama, A generalized shere theorem, Ann. Math. 106 (1977, [Ka] H. Karcher, A geometric classification of ositively curved symmetric saces and the isoarametric construction of the Cayley lane, Asterisque 163/164 (1988, [Pü] T. Püttmann, Otimal inching constants of odd-dimensional homogeneous saces, Invent. Math. 138 (1999, [Ra] C. Rakotoniaina, Cut locus of the B-sheres, Ann. Global Anal. Geom. 3 (1985, [Va] F. M. Valiev, Shar estimates of the sectional curvatures of homogeneous Riemannian metrics on Wallach saces, Sibirsk. Mat. Zh. 0 (1979, [Wa] N. R. Wallach, Comact homogeneous Riemannian manifolds with strictly ositive sectional curvature, Ann. Math. 96 (197, [Wi1] B. Wilking, Index arity of closed geodesics and rigidity of Hof fibrations, Invent. Math. 144 (001, [Wi] B. Wilking, Riemannsche Submersionen und ositive Krümmung, Seminar notes, University of Münster, Ruhr-Universität Bochum, Fakultät für Mathematik, D Bochum, Germany address: uttmann@math.ruhr-uni-bochum.de