historical remarks

36
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

Upload: hilde

Post on 14-Jan-2016

44 views

Category:

Documents


0 download

DESCRIPTION

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. - PowerPoint PPT Presentation

TRANSCRIPT

Page 1: 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

Page 2: Historical remarks

• 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(’,.) ’

Page 3: Historical remarks

• ------- 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.

Page 4: Historical remarks

• 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

Page 5: Historical remarks

• 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)

Page 6: Historical remarks

• 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).

Page 7: Historical remarks

• 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)

Page 8: Historical remarks

• 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)

Page 9: Historical remarks

• 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

Page 10: Historical remarks

• 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,

Page 11: Historical remarks

• 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,

Page 12: Historical remarks

• 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.

Page 13: Historical remarks

• ------- 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

Page 14: Historical remarks

• 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”

Page 15: Historical remarks

• 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?

Page 16: Historical remarks

• 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.

Page 17: Historical remarks

• 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”.

Page 18: Historical remarks

• 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:

Page 19: Historical remarks

• 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).

Page 20: Historical remarks

• (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).

Page 21: Historical remarks

• (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)).

Page 22: Historical remarks

• (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)

Page 23: Historical remarks

• 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.

Page 24: Historical remarks

• 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

Page 25: Historical remarks

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

Page 26: Historical remarks

• 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);

Page 27: Historical remarks

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);

Page 28: Historical remarks

• Theorem,

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

Page 29: Historical remarks

• 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

Page 30: Historical remarks

• 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

Page 31: Historical remarks
Page 32: Historical remarks

• 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

Page 33: Historical remarks

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*).

Page 34: Historical remarks

• 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

Page 35: Historical remarks

• (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.

Page 36: Historical remarks

• 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).