EXISTENCE OF NON-ISOTROPIC CONJUGATE POINTS ON RANK ONE NORMAL HOMOGENEOUS SPACES C. Gonz

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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
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• 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(?,.) ?

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• ------- 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.

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• 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 ltw, u, vmgt ltw, u m, vgt
• Normal Riemannian homogeneous space
• Riemannian connection ?uv (1/2)u, vm

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• 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)

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• 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).

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• The Wilkings 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)

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• 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)

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• Prop. ----- and Tarrío, Monatsh. Math. 154
  (2008).-
• For the manifold B7, ???2 1
• i) Rt2s) (-1)s-1 Rt2)
• ii) Rt2s1) (-1)s Rt1)
• Possibility of obtain an approximate solution of
  the Jacobi equation

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• 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,

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• 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,

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• 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.

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• ------- and Arias-Marco in Publ. Math. Debrecen
  74 (2009) we prove that the Kaplans 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

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• 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

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• 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?

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• 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.

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• 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.

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• Bergers 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

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• 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 metricsSn,(? 1)CPn, HPn,
  CaP2,(?1/4)
• (ii) the complex projective space
• CPn Sp(m1)/(Sp(m) x U(1)), n 2m 1,
• equipped with a standard Sp(m1)homogeneous
  metric(?1/16).

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• (iii) the Berger spheres
• (S2m1 SU(m1/SU(m), gs), 0 lt s ? 1
• (?(s) s(m1)/(8m ? 3s(m1)
• (iv) (S4m3 Sp(m1)/Sp(m), gs), 0 lt s ? 1,
• (?(s) s/(8 ? 3s), if s ? 2/3, and
• ?(s) s2/4, if s lt 2/3).

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• (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)).

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• (vii)
• W7 (SU(3) x SO(3) / U?(2), gs) s gt 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 lt 1, where t t(s) 2s / (2s 3)

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• 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.

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• Homogeneous fibrations
• (M G/K, g) normal homogeneous space, lt ?, ?gt
  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

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• 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 lt ?, ?gtm x
  m is a
• Riemannian submersion. Put V m1 and H m2.
• ? F is totally geodesic submanifold

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• Homogeneous fibrations on rank one normal
  homogeneous spaces
• ? S1 ? ( S2m1 U(m1) / U(m),
• gk,s (1/k)gs) ? CPm(k)
• ? S2 ? ( CP2m1 Sp(m1) / (Sp(m) x U(1)),
• gk (1/k)g) ? HPm(k)
• ? S3 ? ( S4m3 Sp(m1) / (Sp(m) x U(1)),
• gk,s (1/k)gs) ? HPm(k)

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• ? 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)

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

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• 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)

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• 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, vk, 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

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• Conjugate points ?u(s/(? ?)1/2)), ?, ? gt 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) ? ?
• (S2m1, gk,s), s ? 1 2ks(m1)m 2k(2m ? s(m1))
• ( CP2m1, gk) 2k 2k
• ( S4m3, gk,s), s ? 1 2ks 2k(2-s)
• ( W7, gk,s), 2ks/(1s) 2k/(1s)
• ( B13, gk), 2k 2k

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• 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).

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• Theorem
• On the normal homogeneous spaces (S2m1, gk,s),
  ( CP2m1, gk), ( S4m3, 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

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• (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 Chavels conjecture follows now
  immediately from the Fundamental Theorem.

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• 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(m1)/(Sp(m) x U(1)), n
  2m 1, equipped with the standard
  Sp(m1)-homogeneous Riemannian metric (? 1/16).