EXISTENCE OF NON-ISOTROPIC CONJUGATE POINTS ON RANK ONE NORMAL HOMOGENEOUS SPACES C. González-Dávila(U. La Laguna) and A. M. Naveira, U. Valencia, Spain

Historical remarks The Jacobi equation for a Riemannian manifold with respect to a connection with torsion. One talk with Prof. K. Nomizu, Lyon, 1985 The Jacobi equation for the Levi-Civita connection The Jacobi operator R t = R(  ’,.)  ’

and Tarrío, A. Monatsh. Math. 154 (2008) Theorem., Warner, Scott Foresman (1970) Let G be a Lie group, H  G a closed subgroup, then M = G/H has a unique structure of differentiable manifold making the natural projection a submersion.

Some notations: g  T e G,k  T e K,m = g / k Evidently, [k, k]  k Reductive homogeneous space, [k, m]  m Naturally reductive homogeneous space [k, m]  m and = Normal Riemannian homogeneous space Riemannian connection:  u v = (1/2)[u, v] m

The classification of M. Berger, Ann. Scuola Norm. Sup. Pisa 15 (1961), of G/K which admit a normal G- invariant Riemannian metric with strictly positive sectional curvature: Rank one symmetric spaces The manifold B 7 = Sp(2)/ SU(2) The manifold B 13 = SU (5)/ (Sp (2)xS 1 )

One remark of Berard-Bergery, J. Math. P. and Appl. 55 (1961) The family of 7-manifolds of Aloff, and Wallach, Bull. Amer. Math. Soc. 81 (1975).

The Wilking’s manifold W 7 = (SU(3)x SO(3)/ U  (2) U  (2) is the image of U(2) under the embedding ( ,  ): U(2)  SO(3) x SU(3) /  : U (2)  U (2) / S 1  SO(3),  :U(2)  SU(3),  (A) = Diag (A, - Tr A)

One result of Tsukada,Kodai Math. J. 19 (1996), about the “constant osculating rank” of a curve in the Euclidean space Prop.- R t = R 0 +  i a i (t)R 0 i)

Prop and Tarrío, Monatsh. Math. 154 (2008).- For the manifold B 7,  ’  2 = 1: i)R t 2s) = (-1) s-1 R t 2) ii)R t 2s+1) = (-1) s R t 1) Possibility of obtain an approximate solution of the Jacobi equation

Prop. Macías, and Tarrio, C. R. Acad. Sci. París, Ser. I, 346 (2008) For the manifold W 7, we have: R t 1) + (5/2)R t 3) = 0, R t 2) + (5/2)R t 4) = 0,

Well known classification of the 3-symmetric spaces, Gray, J. Diff. Geom. 7 (1972). Example most studied in the literature: F 6 = SU(3) / S(U(1) x U(1)x U(1)) Prop. Arias, Archiv. Mathematicum (Brno), 45 (2009).- For the manifold F 6, we have: (1/16)R t 1) + (5/8)R t 3) + R t 5) = 0, (1/16)R t 2) + (5/8)R t 4) + R t 6) = 0,

One geometric property: Def. Riemannian homogeneous spaces verifying that each geodesic of (G/K, g) is an orbit of a one parameter group of isometries {exp tZ}, Z  g, are called g. o. spaces, studied firstly by Kaplan, Bull. London Math. Soc. 15(1983). Kaplan gives the first example of one g. o. space which is not naturally reductive: one generalized Heisenberg group. There exist a rich literature about the geometry of g. o. spaces.

and Arias-Marco in Publ. Math. Debrecen 74 (2009) we prove that the Kaplan’s example satisfies: (1/4)R t 1) + (5/4)R t 3) + R t 5) = 0, (1/4)R t 2) + (5/4)R t 4) + R t 6) = 0, Compare with the result for F 6 (1/16)R t 1) + (5/8)R t 3) + R t 5) = 0, Open problems: Determine the osculating rang in other examples and families of 3-symmetric and g. o. spaces

The solution of the Jacobi equation is very easy for the symmetric spaces. One result of González-Dávila and Salazar, Publ. Math. Debrecen 66 (2005): “Every Jacobi field vanishing at two points is the restriction of a Killing vector field along the geodesic. One very interesting paper: “Isotropic Jacobi vector field” along one geodesic, Ziller, Comment. Math. Helv. 52 (1977). “Anisotropic Jacobi vector field”

On B 7, Chavel Bull. Amer. Math. Soc. 73 (1976), On B 13, Chavel Comment. Math. Helv. 42 (1967). He use the “canonical connection”  c. Why is interesting work with the canonical connection? Because (i),  c g =  c T c =  c R c = 0   Jacobi eq. has const. Coef. (ii)  and  c have the same geodesics What happens with W 7 ?

Studing conjugate points on odd-dimensional Berger spheres, Chavel in J. Diff. Geom. 4 (1970), proposed the following conjecture: “If every conjugate point of a simply-connected normal homogeneous Riemannian manifold G/K of rank one is isotropic, then G/K is isometric to a Riemannian symmetric space of rank one.” With González-Dávila, we think we have the solution to this conjecture.

The main results The notion of “variationally complete action” is of Bott and Samelson, Amer. J. Math. 80 (1958), Correction in: Amer. J. Math. 83 (1961). One result of González-Dávila, J. Diff. Geom. 83 (2009): “If the isotropy action of K on G/K is variationally complete then all Jacobi field vanishing at two points are G - isotropic” Then, Chavel conjecture  “If the isotropy action on a simple-connected rank one normal homogeneous space is variationally complete then it is a compact rank one symmetric space”.

Berger’s classification is under diffeomorphisms and not under isometries. Using results of Wallach, Ann. of Math. 96 (1972) and Ziller, Comment. Math. Helv. 52 (1977), Math. Ann. 259 (1982) and denoting by  the corresponding pinching constant, we can prove:

Th.- A simply-connected, normal homogeneous space of positive curvature is isometric to one of the following Riemannian spaces: (i)compact rank one symmetric spaces with their standard metrics:S n,(  = 1);CP n, HP n, CaP 2,(  =1/4); (ii)the complex projective space CP n = Sp(m+1)/(Sp(m) x U(1)), n = 2m + 1, equipped with a standard Sp(m+1)homogeneous metric(  =1/16).

(iii) the Berger spheres (S 2m+1 = SU(m+1/SU(m), g s ),0 < s  1 (  (s) = {s(m+1)/(8m  3s(m+1)} (iv)(S 4m+3 = Sp(m+1)/Sp(m), g s ), 0 < s  1, (  (s) = {s/(8  3s)}, if s  2/3, and  (s) = s 2 /4, if s < 2/3).

(v)B 7 = SU(5) / (SU(2) equipped with a standard Sp(2)  homogeneous metric (  = 1/27). (vi)B 13 = Sp(2) / (Sp(2) x S 1 ) equipped with a standard SU(5)  homogeneous metric (  = 1/ (29x27)).

(vii) W 7 = {(SU(3) x SO(3) / U  (2), g s ) s > 0, (  (s) = t 2 /4, if t  (8  2 /3 ;  (s) = t / (16  3t) if (8  2 /3)  t  2/5 and  (s) = 16(1  t) 3 / (16  3t)(4 + 16t  11t 2 ) if 2/5  t < 1, where t = t(s) = 2s / (2s + 3)

Eliasson, Math. Ann. 164 (1966), and Heintze, Invent. Math. 13 (1971) compute the pinching constants 1/37 and 16/(29x37) for B 7 and B 13 respectively. Püttmann, gives the optimal pinching constant 1/37 for any invariant metric on B 13 and W 7. Using results of Sagle, Nagoya Math. J. 91 (1968), adapting the Lie triple systems to the NRHS, we obtain some results about totally geodesic submanifolds used after.

Homogeneous fibrations: (M = G/K, g) normal homogeneous space, Ad(G) – invariant Inner product of g and H closed subgroup s. t. K  H  G. The homogeneous fibration: F = H/K  M = G/K  M* = G/H, gK  gH

Some properties:  h = k  m 1,g = k  m 1  m 2, g = h  m 2 are Reductive decompositions for F, M and M*, respectively  : (M, g)  (M*, g*), g* induced by m x m is a Riemannian submersion. Put V = m 1 and H = m 2.  F is totally geodesic submanifold

Homogeneous fibrations on rank one normal homogeneous spaces  S 1  ( S 2m+1 = U(m+1) / U(m), g k,s = (1/k)g s )  CP m (k);  S 2  ( CP 2m+1 = Sp(m+1) / (Sp(m) x U(1)), g k = (1/k)g)  HP m (k);  S 3  ( S 4m+3 = Sp(m+1) / (Sp(m) x U(1)), g k,s = (1/k)g s )  HP m (k);

 RP 3  ( W 7 = (S0(3) x SU(3) / U  (2), g k,s = (1/2k)g s )  CP 2 (2k);  RP 5  ( B 13 = SU(5) / (Sp(2) x S1), g k = (1/2k)g)  CP 4 (k);

Theorem, On all these spaces, there exist conjugate points to the origin along any geodesic starting at this point which are not isotropic

Normal homogeneous spaces and isotropic Jacobi fields R c represents the curvature of the canonical connection Lemma, González-Dávila, J. Diff. Geom. 83 (2009).- A Jacobi field V along one geodesic  u(t) is G-isotropic if and only if V’(0)  (Ker R u c ) . Key result for this article is the following result which is a more complete version of results of González-Dávila in J. Diff. Geom. 83 (2009):

Conjugate points in normal homogeneous spaces Lemma Let (M = G/K, g) be a normal homogeneous space and u, v orthonormal vectors in m s. t. [u, v]  m \ {0}. If there exist positive numbers and  satisfying [[u, v], u] m = v,[u, [u, v]] k, u] =  [u, v], Then  u (s/( +  ) 1/2 ), where 1.s is a solution of the equation tan (s/2) =  s/ 2, or 2.s = 2p , p  Z are conjugate points to the origin along  u (t) = (exp tu)0. In 1. they are not strictely G-isotropic and in 2., they are G-isotropic

Conjugate points  u (s/( +  ) 1/2 )),,  > 0 Any pair of unit vectors (u, v)  H x V satisfy the hypothesis of the lemma, the scalars and  are the same for any (u, v) and they are given by (M, g) (S 2m+1, g k,s ), s  12ks(m+1)m2k(2m  s(m+1)) ( CP 2m+1, g k )2k2k ( S 4m+3, g k,s ), s  12ks2k(2-s) ( W 7, g k,s ),2ks/(1+s)2k/(1+s) ( B 13, g k ),2k2k

Horizontal geodesics  If  u (t 0 ) is a G-isotropic conjugate point along a horizontal geodesic  u (t) = (exp tu)0 then  u *(t o ) is G-isotropic conjugate to 0* =  (0), where  u * =  (  u ) on (M*, g*).

Theorem On the normal homogeneous spaces (S 2m+1, g k,s ), ( CP 2m+1, g k ), ( S 4m+3, g k,s ), ( W 7, g k,s ) and ( B 13, g k ) the points  u *(t/2) of any horizontal geodesic  u, where 1.t is a solution of the equation tan (t/s) =  t/ 2. Or 2.t= 2p , p  Z are conjugate points to the origin along  u (t) = (exp tu)0. In 1. they are not isotropic and in 2., they are isotropic

(ii)-(iv) and (vi)-(vii) in the Fundamental Theorem follows now from the above results. (v) is a result of Chavel, Bull. Amer. Math. Soc. 73 (1967. For (i) we have the compact rank one symmetric spaces with their standard metric. The Proof of the Chavel’s conjecture follows now immediately from the Fundamental Theorem.

Normal homogeneous metrics of positive curvature on symmetric spaces Even-dimensional case. Normal homogeneous metrics on symmetric spaces with positive curvature, Wallach, Ann. of Math. 96 (1972). Prop..- A simple connected, 2n-dimensional, normal homogeneous space of positive sectional curvatura is isometric to a compact rank one symmetric space: S 2n, (  = 1); CP n, HP n/2 (n even), CaP 2, (  = 1/4); or to the complex projective space CP n = Sp(m+1)/(Sp(m) x U(1)), n = 2m + 1, equipped with the standard Sp(m+1)- homogeneous Riemannian metric (  = 1/16).