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Historical remarks

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.

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Historical remarks

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

  2. 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 • Rt = R(’,.) ’

  3. ------- 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.

  4. Some notations: • g TeG, k TeK, m = g / k • Evidently, [k, k] k • Reductive homogeneous space, [k, m] m • Naturally reductive homogeneous space • [k, m] m and <w, [u, v]m> = <[w, u] m, v> • Normal Riemannian homogeneous space • Riemannian connection: uv = (1/2)[u, v]m

  5. 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 B7 = Sp(2)/ SU(2) • The manifold B13 = SU (5)/ (Sp (2)xS1)

  6. 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).

  7. The Wilking’s manifold • W7 = (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) / S1SO(3),  :U(2) SU(3), • (A) = Diag (A, - Tr A)

  8. One result of Tsukada ,Kodai Math. J. 19 (1996), about the “constant osculating rank” of a curve in the Euclidean space • Prop.- Rt = R0 + i ai (t)R0i)

  9. Prop.----- and Tarrío, Monatsh. Math. 154 (2008).- • For the manifold B7, ’2 = 1: • i) Rt2s) = (-1)s-1 Rt2) • ii) Rt2s+1) = (-1)s Rt1) • Possibility of obtain an approximate solution of the Jacobi equation

  10. Prop. Macías, ----- and Tarrio, C. R. Acad. Sci. París, Ser. I, 346 (2008) 67- 70 For the manifold W7, we have: • Rt1) + (5/2)Rt3) = 0, Rt2) + (5/2)Rt4) = 0,

  11. Well known classification of the 3-symmetric spaces, • Gray, J. Diff. Geom. 7 (1972). • Example most studied in the literature: • F6 = SU(3) / S(U(1) x U(1)x U(1)) • Prop. Arias, Archiv. Mathematicum (Brno), 45 (2009).- For the manifold F6, we have: • (1/16)Rt1) + (5/8)Rt3) + Rt5) = 0, • (1/16)Rt2) + (5/8)Rt4) + Rt6) = 0,

  12. 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.

  13. ------- and Arias-Marco in Publ. Math. Debrecen74 (2009) we prove that the Kaplan’s example satisfies: • (1/4)Rt1) + (5/4)Rt3) + Rt5) = 0, • (1/4)Rt2) + (5/4)Rt4) + Rt6) = 0, • Compare with the result for F6 • (1/16)Rt1) + (5/8)Rt3) + Rt5) = 0, • Open problems: Determine the osculating rang in other examples and families of 3-symmetric and g. o. spaces

  14. The solution of the Jacobi equation is very easy for the symmetric spaces. • One result of González-Dávila and Salazar, Publ. Math. Debrecen66 (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”

  15. On B7, Chavel Bull. Amer. Math. Soc. 73 (1976), • On B13, Chavel Comment. Math. Helv . 42 (1967). • He use the “canonical connection” c. • Why is interesting work with the canonical connection? • Because (i), cg = cTc = cRc = 0  •  Jacobi eq. has const. Coef. • (ii)  and c have the same geodesics • What happens with W7?

  16. 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.

  17. The main results • The notion of “variationally complete action” is of Bott and Samelson, Amer. J. Math. 80 (1958), 964-1029. 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”.

  18. 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:

  19. 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:Sn,( = 1);CPn, HPn, CaP2,(=1/4); • (ii) the complex projective space CPn = Sp(m+1)/(Sp(m) x U(1)), n = 2m + 1, equipped with a standard Sp(m+1)homogeneous metric(=1/16).

  20. (iii) the Berger spheres (S2m+1 = SU(m+1/SU(m), gs), 0 < s  1 • ((s) = {s(m+1)/(8m  3s(m+1)} • (iv) (S4m+3 = Sp(m+1)/Sp(m), gs), 0 < s  1, • ((s) = {s/(8  3s)}, if s  2/3, and • (s) = s2/4, if s < 2/3).

  21. (v) B7 = SU(5) / (SU(2) equipped with a standard Sp(2)  homogeneous metric ( = 1/27). • (vi) B13 = Sp(2) / (Sp(2) x S1) equipped with a standard SU(5)  homogeneous metric ( = 1/ (29x27)).

  22. (vii) W7 = {(SU(3) x SO(3) / U(2), gs) s > 0, • ((s) = t2/4, 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  11t2) if 2/5  t < 1, where t = t(s) = 2s / (2s + 3)

  23. Eliasson, Math. Ann. 164 (1966), and Heintze, Invent. Math. 13 (1971) compute the pinching constants 1/37 and 16/(29x37) for B7 and B13 respectively. • Püttmann, gives the optimal pinching constant 1/37 for any invariant metric on B13 and W7. • 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.

  24. 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

  25. Some properties:  h = k m1,g = k m1m2, g = hm2 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 = m1 and H = m2. F is totally geodesic submanifold

  26. Homogeneous fibrations on rank one normal homogeneous spaces  S1 ( S2m+1 = U(m+1) / U(m), gk,s = (1/k)gs)  CPm(k);  S2 ( CP2m+1 = Sp(m+1) / (Sp(m) x U(1)), gk = (1/k)g)  HPm(k);  S3 ( S4m+3 = Sp(m+1) / (Sp(m) x U(1)), gk,s = (1/k)gs)  HPm(k);

  27.  RP3 ( W7 = (S0(3) x SU(3) / U(2), gk,s = (1/2k)gs)  CP2(2k);  RP5 ( B13 = SU(5) / (Sp(2) x S1), gk = (1/2k)g)  CP4(k);

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

  29. Normal homogeneous spaces and isotropic Jacobi fields • Rc 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 Ruc). • 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):

  30. 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

  31. 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)  • (S2m+1, gk,s), s  1 2ks(m+1)m 2k(2m  s(m+1)) • ( CP2m+1, gk) 2k 2k • ( S4m+3, gk,s), s  1 2ks 2k(2-s) • ( W7, gk,s), 2ks/(1+s) 2k/(1+s) • ( B13, gk), 2k 2k

  32. Horizontal geodesics If u(t0) is a G-isotropic conjugate point along a horizontal geodesic u(t) = (exp tu)0 then u*(to) is G-isotropic conjugate to 0* = (0), whereu* = (u) on (M*, g*).

  33. Theorem On the normal homogeneous spaces (S2m+1, gk,s), ( CP2m+1, gk), ( S4m+3, gk,s), ( W7, gk,s) and ( B13, gk) 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

  34. (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.

  35. 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: S2n, ( = 1); CPn, HPn/2(n even), CaP2, ( = 1/4); or to the complex projective space CPn= Sp(m+1)/(Sp(m) x U(1)), n = 2m + 1, equipped with the standard Sp(m+1)-homogeneous Riemannian metric ( = 1/16).

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