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• Rt = 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 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
• 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)
• 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
• 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)
• 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)
• Prop. ----- and Tarrío, Monatsh. Math. 154 (2008).- • For the manifold B7, ’2 = 1:
• i) Rt2s) = (-1)s-1 Rt
2)
• ii) Rt2s+1) = (-1)s Rt
1)
• Possibility of obtain an approximate solution of the Jacobi equation
• Prop. Macías, ----- and Tarrio, C. R. Acad. Sci. París, Ser. I, 346 (2008) 67- 70 For the manifold W7, we have:
• Rt1) + (5/2)Rt
3) = 0, Rt2) + (5/2)Rt
4) = 0,
• 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)Rt
3) + Rt5) = 0,
• (1/16)Rt2) + (5/8)Rt
4) + Rt6) = 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)Rt1) + (5/4)Rt
3) + Rt5) = 0,
• (1/4)Rt2) + (5/4)Rt
4) + Rt6) = 0,
• Compare with the result for F6
• (1/16)Rt1) + (5/8)Rt
3) + Rt5) = 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 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?
• 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), 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”.
• 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: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).
• (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).
• (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)).
• (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(1t)3 / (16 3t)(4 + 16t 11t2) 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 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.
• 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 m1,g = k m1 m2, g = h m2 areReductive 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
• 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);
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);
• 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
• 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 Ru
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):
4
1
4
1
• 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 Zare 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)
• (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
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*).
• 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
• (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: 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).