PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS OF ISOPARAMETRIC HYPERSURFACES IN SPHERES

REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 56, No. 2, 2015, Pges 119 133 Published online: September 30, 2015 PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS OF ISOPARAMETRIC HYPERSURFACES IN SPHERES Abstrct. The present pper contins some results bout the lgebric sets of plnr norml sections ssocited to the focl mnifolds of homogeneous isoprmetric hypersurfces in spheres. With the usul identifiction of the tngent spces to the focl mnifold with subspces of the tngent spces to the isoprmetric hypersurfce, it is proven tht the lgebric set of plnr norml sections of the focl mnifold is contined in tht of the originl hypersurfce. 1. Introduction In the present pper we study the rel lgebric set of unit tngent vectors generting plnr norml sections of the focl mnifolds of isoprmetric hypersurfces of the spheres. It is well known tht isoprmetric hypersurfces of the spheres cn be considered isoprmetric submnifolds of codimension two in ordinry Eucliden spces. By definition, norml sections re the curves cut out of submnifold M n of R n+k by the ffine subspce generted by unit tngent vector nd the norml spce, t given point p of M n. A norml section γ t ny point p of M n p = γ 0)) is clled plnr t p if its first three derivtives γ 0), γ 0), γ 0) re linerly dependent. The unit tngent vectors defining plnr norml sections t p M n, form rel lgebric set Xp M n see the next section for proper definition) which is of interest in the study of the geometry of submnifolds of Eucliden spces 2, 11). We restrict our considertions to homogeneous isoprmetric hypersurfces nd their focl mnifolds. The collection of homogeneous isoprmetric hypersurfces of the spheres hs finite number of members in opposition to tht of nonhomogeneous ones 10, 3). The reson for this restriction is twofold; on the one hnd our lgebric methods re intimtely relted to the R-spces see Section 4), nd on the other is the fct tht for R-spces the lgebric set X p M n cn in fct be considered independent of the point p M n. Recll tht the considered 2010 Mthemtics Subject Clssifiction. 53C30, 53C42. Prtil support from CONICET, SECYT-UNC nd SECYT-UNSL from Argentin is grtefully cknowledged. 119

120 submnifold M n is extrinsic homogeneous if for ny two points p, q M n there is n isometry g of the mbient spce R n+k such tht g M n ) = M n nd gp) = q. The objective of the present pper is to prove the following result. Theorem 1. Let M n S n+1 R n+2 be full homogenous isoprmetric hypersurfce nd M +) nd M ) its focl mnifolds. If p is point on M n nd p 1 is focl point of p in M +) or M ) then X p1 M±) Xp M n. The method to obtin Theorem 1 depends strongly on the fct tht the codimension of M n in R n+2 is 2, nd it seems cler tht the proof presented here cn not be extended to higher codimension. Theorem 1 indictes n importnt connection between the sets of plnr norml sections of two different mnifolds nd t the sme time shows the presence of interesting lgebric subsets of X p M n. For n isoprmetric submnifold M n of R n+k with codimension k 3 similr inclusion cn be proved 4, Corollry 13) but it requires tht the focl mnifold be symmetric R-spce. It is importnt to remrk the fct tht not ll focl mnifolds of homogeneous isoprmetric hypersurfces in spheres re symmetric R-spces. In fct, interesting exmples re the focl mnifolds M +) nd M ) of M = G 2 /T 2 which is the essentilly unique 7) isoprmetric hypersurfce in the sphere of type 6, 2). Here M +) nd M ) re orbits of G 2 of the form G 2 /U 2) = Q 5 the complex qudrtic). They re not symmetric since G 2 hs only one symmetric spce, nmely G 2 /SO 4), which is n inner symmetric spce of quternionic type, hence it is not n R-spce. The structure of the pper is the following. In the next Section we indicte the bsic definitions to be used long the pper nd in Section 3 we recll bsic results from 11 bout the plnr norml sections of compct sphericl submnifolds nd the definition of the lgebric set X p M. In Section 4 we recll the definitions of R-spces nd in Section 5 we indicte other nottion ssocited to R-spces. Prticulrly in Subsection 5.2 we compute the covrint derivtive of the second fundmentl form of generl R-spce nd tht of the principl orbits. This bsic computtion is lso contined in 4 but it seems convenient to include it here. Section 6 records bsic fcts bout generl isoprmetric hypersurfces in spheres nd their focl mp. Detils could be obtined in 1; we just present wht seems strictly necessry. Section 7 contins the essentil steps to get the proof of Theorem 1, which is finlly contined in Section 8. 2. Bsic definitions We strt reclling some bsic definitions. Let M be compct connected n- dimensionl Riemnnin mnifold nd I : M R n+k n isometric embedding into the Eucliden spce R n+k. We identify M with its imge by I. A submnifold of Eucliden spce R n+k is usully clled full if it is not included in ny ffine hyperplne. Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS 121 Let, denote the inner product in R n+k. Let E be the Eucliden covrint derivtive in R n+k nd the Levi-Civit connection in M ssocited to the induced metric. We shll sy tht the submnifold M is sphericl if it is contined in sphere of rdius r in R n+k which we my think centered t the origin. Let α denote the second fundmentl form of the embedding in R n+k. We denote by T p M) nd T p M) the tngent nd norml spces to M t p, in R n+k, respectively. Let p be point in M nd consider unit vector Y in the tngent spce T p M). We my define in R n+k the ffine subspce { S p, Y ) = p + Spn Y, T p M) }. If U is smll enough neighborhood of p in M, then the intersection U S p, Y ) cn be considered the imge of C regulr curve γ s), prmetrized by rclength, such tht γ 0) = p, γ 0) = Y. This curve is clled norml section of M t p in the direction of Y. We sy tht the norml section γ of M t p in the direction of Y is plnr t p if its first three derivtives γ 0), γ 0) nd γ 0) re linerly dependent. We re interested in studying the set of those unit tngent vectors which generte plnr norml sections to M t the point p M. 3. Compct sphericl submnifolds We recll some known fcts nd definitions from 11 which re needed here. Lemm 2. Let M be sphericl. The norml section γ of M t p in the direction of X is plnr t p if nd only if the covrint derivtive of the second fundmentl form vnishes on the vector X = γ 0). Tht is, X stisfies the eqution X α ) X, X) = 0. Given point p in the submnifold M we shll denote, s in 11, X p M = { Y T p M) : Y = 1, Y α ) Y, Y ) = 0 }. By doing this for ech point in M we obtin subset of the unit tngent bundle of M which we shll denote by Ξ M). Let M be, s bove, compct connected n-dimensionl Riemnnin submnifold nd lso ssume tht M is sphericl. In these conditions, we my tke the imge X p M of Xp M in the rel projective spce RP T p M)). Then X p M is rel lgebric set of RP T p M)) nd its nturl complexifiction Xp C M is complex lgebric set of CP n 1. The submnifold M is sid to be extrinsiclly homogeneous in R n+k 1, p. 35 if for ny two points p nd q in M n there is n isometry g of R n+k such tht g M) = M nd gp) = q. If this is the cse, X p M cn be considered independent of the point p M. In fct ) ) g px) α g p X, g p X = g p X α ) X, X), nd we clerly hve tht X q M = X ) gp) M = g p Xp M. Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

122 Then X q M nd X p M re isomorphic for ech pir p, q nd we my free ourselves from the point p. 4. R-spces Let G/K be compct simply connected irreducible Riemnnin symmetric spce where G is compct connected Lie group nd K is closed subgroup. K is the isotropy subgroup of the bsic point o = K. Let g nd k denote the Lie lgebrs of G nd K respectively. The structure of G/K genertes n involutive utomorphism of θ of g such tht k is the fixed point set of θ nd setting p = {X g : θ X) = X} we get direct sum decomposition with the properties: g = k p, k, k k, k, p p, p, p k. We cn identify T o G/K), the tngent spce to G/K t o = K, with the subspce p. We my ssume tht the invrint Riemnnin metric on G/K is such tht t T o G/K) the inner product coincides with the restriction to p p of the opposite of the Killing form B of g. Tht is, we hve for X, Y in p X, Y = B g X, Y ). 1) The group Ad K) cts on p effectively, s compct group of liner isometries. The orbit of non-zero element of p by this group is submnifold of p clled n R-spce. Let S p be the unit sphere. Since the orbit of n element λa λ 0) is homothetic to tht of A it is customry to consider only the orbits of elements in S. Such orbits re, of course, contined in S. Let p be mximl belin subspce of p which we shll mintin fixed. It is furthermore known tht the orbit of ny A S meets orthogonlly polrity, 1, p. 41-46), so the relevnt orbits re those of vectors in the unit sphere of, tht is S. We refer the reder to 1, p. 37, p. 46 3.2.10) for description of orbit types. The relevnt comment to mke here is tht, in our setting, for the ction of Ad K) on p we hve two clsses of orbits, nmely the principl orbits generted by the points in n open dense subset U of S) nd the singulr ones generted by the points in S) U)). In the following lines we shll cll regulr elements to those in the set U. The objective of the present prt is to study the set of plnr norml sections of the orbit of generl element in S. 5. The geometry of n R-spce We re going to study the geometry of the orbit for ny F S which my not be regulr) by the group Ad K). We strt by fixing some nottion tht will simplify things. Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS 123 5.1. Nottion. We hve M = Ad K) F p. It is convenient to define for λ the dul spce of ) the subspces nd observe tht obviously k λ = {X k : d H)) 2 X = λ 2 H) X, p λ = {X p : d H)) 2 X = λ 2 H) X, k λ = k λ, p λ = p λ, p 0 =, H }, H }, nd k 0 is the centrlizer of in k. If λ is λ 0 nd p λ {0} then λ is clled restricted root of g with respect to. We denote by R = R g, ) the set of restricted roots of g with respect to. It is importnt to notice tht the set of restricted roots R {0}) my be non-reduced. Now consider the subsets of restricted roots ssocited to F : R0 F ) = {λ R : λ F ) = 0} R, R+ F ) = {λ R : λ F ) > 0} R. With them we my define the following four subspces: k F = k 0 k λ, p F = p λ, λ R0 λ R0 k + F ) = k λ, p + F ) = p λ, λ R+ λ R+ where ll sums re direct, nd of course g = k p = k F k + ) p F p + ). A more complete nottion my be required to indicte the dependence of k + nd p + from F ; in tht cse we shll use k + F ) nd p + F ), but if there is no risk of confusion we shll keep the simpler nottion. If X k λ λ R+ ) we hve X, F p nd furthermore k λ, F p λ, nd similrly tking X p λ we obtin p λ, F k λ. But in fct, the function d F ) which is d F ) : k λ p λ, d F ) : p λ k λ is n isomorphism for ech λ R+ becuse in ny of these two spces d F ) 2 = λ F ) 2 Id. Then we hve T F M) = k, F = k +, F = p +. Now we notice tht p + nd p F re orthogonl, p +, p F = 0; in fct, since ny Y p + my be written s Y = X, F for some X k +, by tking U p F we see tht U, Y = U, X, F = X, F, U = X, F, U = 0 Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

124 becuse U, F = 0. Then we hve T F M) = p F ; we must observe tht if we tke insted of F ) n element E in S which is regulr then T E M) =. 5.2. Computtion of α. We hve the Eucliden covrint derivtive E in p,, ) nd the Levi-Civit connection ssocited to the induced metric on M. Also on M we re going to consider the cnonicl connection determined by the decomposition k = k F k +, which we shll denote by c 1, p. 203. We hve the second fundmentl form of M on p nd we hve the Guss formul E U W = U W + α U, W ). We need to compute E U W. Let us tke, for some X k, the curve in M of the form γ t) = Ad exp tx)) F ). 2) Its tngent vector t F is γ 0) = X, F nd if we tke t 1 > 0 then we my compute the derivtive γ t 1 ) by γ t 1 ) = d dt t=0 Ad exp t 1 + t) X)) F ) = d dt t=0 Ad exp t 1 ) X)) Ad exp t) X)) F ) 3) = Ad exp t 1 ) X)) d dt t=0 Ad exp t) X)) F = Ad exp t 1 ) X)) X, F, nd so this gives the tngent field long γ t). It is well known tht the curves like γ in 2) re c -geodesics nd tht the c -prllel trnsltion long these geodesics is precisely given by 3) 1, p. 203. Let us tke tngent vector t F nd extend it to field long γ by Y, F T F M) = k, F = k +, F Y, F = Ad exp tx)) Y, F. 4) Then compute, for X, Y k, E X,F Y, F = d dt Ad exp tx)) Y, F ) = X, Y, F p, t=0 noticing tht, in fct, we my tke X, Y k +. Now writing the Guss formul for X, F nd Y, F we hve then clerly E X,F Y, F = X,F Y, F + α X, F, Y, F ) ; X,F Y, F = X, Y, F ) p+ α X, F, Y, F ) = X, Y, F ) pf. 5) Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS 125 Let us now compute the covrint derivtive of α. By definition this is X,F α ) Y, F, Z, F ) = X,F α Y, F, Z, F ) α X,F Y, F, Z, F ) α Y, F, X,F Z, F ). Now in 8 see lso 1, p. 212) the cnonicl covrint derivtive of the second fundmentl form α ws introduced s ) c X,F α Y, F, Z, F ) = X,F α Y, F, Z, F ) α c X,F Y, F, Z, F ) αy, F, c X,F Z, F ), nd n importnt result of 8 is tht the condition c X,F α ) Y, F, Z, F ) = 0 6) in fct chrcterizes R-spces. Since M is R-spce we hve: ) 0 = X,F α Y, F, Z, F ) α c X,F Y, F, Z, F ) α Y, F, c X,F Z, F, 7) nd subtrcting 7) from 6) we get X,F α ) Y, F, Z, F ) = α D X, F, Y, F ), Z, F ) where α Y, F, D X, F, Z, F )), D = c is the difference tensor of the two connections, nd so we hve n importnt equivlence: X,F α ) X, F, X, F ) = 0 α D X, F, X, F ), X, F ) = 0. 8) So we need to compute the difference tensor D X, F, Y, F ). To tht end we use the fct tht the tngent field Y, F 4) is prllel with respect to the cnonicl connection c long γ. We hve D X, F, Y, F ) = X,F Y, F c X,F Y, F = X,F Y, F, nd going bck to 5) we hve D X, F, Y, F ) = X,F Y, F = T X, Y, F ) = X, Y, F ) p+. Then we hve tht the indicted equivlence 8) becomes here: X,F α ) ) X, F, X, F ) = 0 X, X, X, F ) p+f ) = 0. 9) p F Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

126 We wnt to mention the importnt cse in which the orbit M is principl. To distinguish this cse we use E insted of F, so here E is regulr element in S). Formul 9) becomes then X,E α ) ) X, E, X, E) = 0 X, X, X, E) p+e) = 0. Observe tht when E is regulr, p = p + E), nd in ll these formule X k = k F k + F ), but we my tke in fct X k + F ). 6. Isoprmetric hypersurfces in spheres We need to indicte in this section some nottion for the cse of isoprmetric hypersurfces in spheres or equivlently isoprmetric submnifolds of rnk two in Eucliden spces. The literture bout this type of hypersurfces nd their mny interesting properties is quite vst, nd we refer the reder to the book 1 nd its references for rther complete tretment of this prticulr subject within the frmework of submnifold geometry. In fct we shll only indicte here the things we need to develop the present work nd refer to this source whenever it is necessry. Let us recll tht compct connected hypersurfce M S n+1 is clled isoprmetric if its principl curvtures λ 1 p),..., λ g p) re constnt functions of p M. It is usul 1, p. 152), for the constnts λ 1,..., λ g, to dopt the nottion λ j = cot θ j ), 0 < θ 1 < < θ g < π, nd one hs the result of Münzner 1, Th. 5.2.16, p. 152: θ k = θ 1 + k 1 π, k = 1,..., g. g The multiplicity of the principl curvture λ j is denoted by m j. These stisfy m j = m j+2 modulo g indexing). Any pir of shpe opertors corresponding to pir of norml vectors in R n+2 ) commute: A ξ, A γ = 0. It follows from the Ricci eqution 1, p. 11 tht the norml curvture tensor R vnishes on M nd ll the shpe opertors cn be simultneously digonlized on ech tngent spce. We hve then in T p M) common eigendistributions D j p) j = 1,..., g) clled the curvture distributions of M, nd we my write T p M) s direct sum g T p M) = D j p). j=1 Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS 127 6.1. The focl mp. For ech point p M the norml spce in R n+2 is the two dimensionl rel vector spce T p M). We my tke the norml vector H p) such tht {p, H p)} is n orthonorml bsis of T p M) notice tht H p) is tngent to the sphere S n+1 t ech p M). Let us consider the unit speed geodesic in S n+1 norml to M strting t p, moving in the direction of H p). We my write it s Φ s p) = cos s) p + sin s) H p), s 0, π. 10) We hve then for ech fixed s) function Φ s : M S n+1. The derivtive of Φ s t p M is 1, p. 152: nd we observe tht for Φ s ) p = cos s) Id sin t) A Hp), 11) s 0, π, s θ j j = 1,..., g) 12) the derivtive 11), t ech p M, is monomorphism in ech D j p). In fct, if X D j p), Φ s ) p X = sin θ j s) X sin θ j ) nd, by our condition on s, Φ s ) p is monomorphism on T p M) 1, p. 152. Remrk 3. It is importnt to notice tht the subspce D j p) is the eigenspce of eigenvlue λ j = cot θ j ) for the opertor A Hp). Ech of the points Φ s p), for s stisfying 12), belongs to nother isoprmetric hypersurfce of the fmily. On the other hnd if s = θ j we see tht Φ s ) p sends D j p) to zero nd therefore p = Φ θj p) is focl point of p long the norml geodesic, nd so p is in one of the two focl mnifolds. Let us cll M θj the focl mnifold reched t s = θ j of course this is either M +) or M ) but this is not relevnt for our purposes). The tngent spce to the corresponding focl mnifold M θj t p cn be identified with ) g T p Mθj = D k p). k=1,k j Let us notice tht the two dimensionl plne generted by {p, H p)} which coincides with T p M) is orthogonl to M θj t p. But lso the component D j p) is now contined in the norml spce of this focl mnifold t p. Furthermore we my identify T p Mθj ) = SpnR {p, H p)} D j p). Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

128 7. Plnr norml sections of M θj The fundmentl fct here is tht the mnifold M θj is sphericl submnifold of R n+2 nd so we my study its set of plnr norml sections t the point p s in Section 3. However we shll restrict our considertions to the cse in which M is homogeneous isoprmetric hypersurfce in the sphere S n+1. This shll llows us to use the bove fcts for R-spces which pply both to the cses of M nd M θj. We need to mke comptible the nottions used bove. We hve the Crtn decomposition g = k p. We hve p mximl belin subspce dim = 2), E S) regulr element, nd M = Ad K) E is principl orbit in p. This is the isoprmetric hypersurfce of the unit sphere S p) p under considertion. We hve the orthonorml bsis {E, H E)} of =T E M) nd consider the focl point in M θj ) E 1 = Φ θj E) = cos θ j ) E + sin θ j ) H E) given by 10) t s = θ j. We wnt to pply 9) to the sphericl submnifold M θj hence here we hve F = E 1, nd we need to recover some of the nottion from Section 5 in this specil cse. In prticulr, we hve to identify the subspce D j E) in the nottion of the R-spces. By definition Remrk 3), D j E) = eigenspce of cot θ j ) of A HE). It is on the other hnd known 1, p. 63 tht for the principl orbit M = Ad K) E) the eigenvlues of ny shpe opertor cn be computed in terms of the restricted roots λ R s A ξ X) = λ ξ) ) X, X p λ, ξ, 13) λ E) so the common eigenspces of the shpe opertors of M re Ω λ = p λ + p 2λ, where p 2λ = {0} if 2λ is not restricted root. By Remrk 3 we hve tht, on X D j, ) λ H E)) A HE) X) = cot θ j ) X = = cot θ j ). λ E) We need to identify which is the λ R such tht D j = Ω λ. To tht end, since E 1, we my use formul 13) to compute t the point E M) the shpe opertor A E1 X) for X p λ. We get A E1 X) = λ E ) 1) X, X p λ. λ E) Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS 129 On the other hnd, by the definition of E 1, we my compute gin t the point E M) on X D j, A E1 X) = cos θ j ) A E X) + sin θ j ) A HE) X = cos θ j ) X) + sin θ j ) cot θ j ) X = 0. Then we see tht X D j implies λ E 1 ) = 0. We conclude tht D j λ Ω λ such tht λe 1 ) = 0. 14) Then in our nottion for the R-spces we hve the two sets of restricted roots ssocited to the point E 1, tht is R0 E 1 ) = {λ R : λ E 1 ) = 0}, R+ E 1 ) = {λ R : λ E 1 ) > 0}. nd 14) yields D j = p λ. 15) λ R0 E 1) We lso hve k = k E k + E) k = k E1 k + E 1 ) k E1 = k 0 λ R0 k λ p E1 = λ R0 p λ k + E 1 ) = λ R+ k λ p + E 1 ) = λ R+ p λ. To determine the plnr norml sections of M θj hve the orthogonl direct sum p E1 = D j, we see tht X, X, X, E 1 ) p+e 1) p E1 = X, X, X, E 1 ) p+e 1) we consider now 9). Since we + X, X, X, E 1 ) p+e 1). 16) D j Therefore, if n X vnishes the left hnd side i.e. if the vector X, E 1 defines plnr norml section to M θj t E 1 ) then it hs to vnish ech term on the right hnd side becuse they belong to different orthogonl subspces). In these formuls X k = k E1 k + E 1 ), but we my tke in fct X k + E 1 ) nd k + E 1 ) k + E). So, we study the condition X, X, X, E 1 ) p+e 1) = 0. Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

130 It is lso cler, by definition of E 1, tht we hve writing H insted of H E) to simplify nottion): X, X, X, E 1 ) p+e 1) = cos θ j ) X, X, X, E) p+e 1) + sin θ j ) X, X, X, H) p+e 1). 17) Now we study the two terms in 17) seprtely, X, X, X, E) p+e 1) nd X, X, X, H) p+e 1). 7.1. First term X, X, X, E) p+e 1). Since p + E) = p + E 1 ) D j, it is cler tht we hve X, X, X, E) p+e) = X, X, X, E) p+e 1) + X, X, X, E) Dj. Now we study X, X, X, E) Dj Since, by definition, X, X, E) Dj D j nd X k + E 1 ) k + E) we hve X, X, X, E) Dj k + E 1 ), D j, so we study k + E 1 ), D j in the following lemm. Lemm 4. k + E 1 ), D j = 0. Proof. By 15) we hve to consider here k + E 1 ), D j = k λ, It is well known 5 tht λ R+ E 1) µ R0 E 1) p µ = k λ, p µ p λ+µ + p λ µ,. λ R+ E 1) µ R0 E 1) k λ, p µ. nd since λ R+ E 1 ) nd µ R0 E 1 ) we see tht λ + µ nd λ µ belong to R+ E 1 ). Then we hve k + E 1 ), D j p + E 1 ) p + E). But p + E) is orthogonl to nd therefore the lemm follows. We see then tht X, X, X, E) Dj nd hence we hve the equlity X, X, X, E) p+e) = X, X, X, E) p+e 1). 18) = 0 Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS 131 7.2. Second term in 17), tht is X, X, X, H) p+e 1). Let us consider now the second term X, X, X, H) p+e 1). 19) Since H = Φ s E) for s = π 2 ) we hve two possibilities: either H is lso regulr if θ k π 2, k = 1,..., g) or it is focl point of E which requires θ k = π 2 for some k {1,..., g}). In ny cse we study 19) in the sme wy s before. We hve gin the sme decomposition X, X, X, H) p+e) = X, X, X, H) p+e 1) + X, X, X, H) Dj, nd since X k + E 1 ) k + E) we hve here X, X, X, H) Dj k + E 1 ), D j, but lso in this cse Lemm 4 yields X, X, X, H) Dj = 0, so we hve gin version of 18) but here with H insted of E. Tht is X, X, X, H) p+e) = X, X, X, H) p+e 1). 20) Then in view of 18) nd 20) we my chnge the two terms in 17) nd write X, X, X, E 1 ) p+e 1) = cos θ 1 ) X, X, X, E) p+e) + sin θ 1 ) X, X, X, H) p+e). 21) Now we hve the following proposition. Proposition 5. For ny X k + E 1 ) we hve tht X, X, X, E) p+e) is orthogonl to E nd X, X, X, H) p+e) is orthogonl to H. Proof. In fct we prove tht for ny nonzero vector V in nd E regulr we hve: X, X, X, V ) p+e) is orthogonl to V for ny X k. We hve to compute X, X, X, V ) p+e), V = X, X, X, V ) p+e), V. Recll tht the inner product tht we hve in p = p + E) is 1). Then we hve to compute: ) ) B g X, X, X, V ) p+e), V = B g X, X, V ), X p+e), V ) = B g X, X, V ), X, V p+e) = B g X, X, V, X, V ) Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

132 becuse X, V p + E). Now B g X, X, V, X, V ) = B g X, V, X, X, V ) But the equlity clerly implies nd in turn = B g X, V, X, X, V ) = B g X, X, V, X, V ) by symmetry of B g ). B g X, X, V, X, V ) = B g X, X, V, X, V ) B g X, X, V, X, V ) = 0, B g X, X, X, V ) p+e) ), V = 0. Now setting V = E nd V = H we hve both ssertions in Proposition 5. Corollry 6. For ny X k + E 1 ), if X, X, X, E) p+e) nd X, X, X, H) p+e) re both non-zero then they re linerly independent in. 8. Proof of Theorem 1 Proof. As we hve observed bove, if n X vnishes the left hnd side of 16) then it hs to vnish ech term on the right hnd side. Therefore X, X, X, E 1 ) p+e 1) = 0 = X, X, X, E 1 ) p+e p 1) = 0; F on the other hnd 21) nd Corollry 6 show tht 0 = X, X, X, E 1 ) p+e 1) X, X, X, E) p+e) = 0 nd X, X, X, H) p+e) = 0. In fct, if one of the two terms on the right hnd side of 21) vnishes then so does the other one. On the other hnd if both terms re non-zero then by Corollry 6 they must be linerly independent nd this leds to contrdiction becuse cos θ 1 nd sin θ 1 re not zero. Since by definition { } X M = X k : X, X, X, E) p+e) = 0 nd we get the inclusion X M θj = {X k + : } X, X, X, E 1 ) p+e 1) = 0, p F X M θj X M. Rev. Un. Mt. Argentin, Vol. 56, No. 2 2015)

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