Apr 28, 2011 - geodesics or geodesic orbit manifold (shortly, GO-manifold) if any geodesic Î³ of M ... L. Vanhecke classified all GO-spaces of dimensi...

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arXiv:1104.2664v2 [math.DG] 28 Apr 2011

YU.G. NIKONOROV Abstract. The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian geodesic orbit manifold (M, g) and the structure of its full isometry group. The Lie algebra of the full isometry group of (M, g) is identified with the Lie algebra of Killing fields g on (M, g). We prove the following result: If a is an abelian ideal of g, then every Killing field X ∈ a has constant length. On the ground of this assertion we give a new proof of one result of C. Gordon: Every Riemannian geodesic orbit manifold of nonpositive Ricci curvature is a symmetric space. 2000 Mathematical Subject Classification: 53C20, 53C25, 53C35. Key words and phrases: Killing fields of constant length, homogeneous Riemannian manifolds, geodesic orbit spaces, symmetric spaces, Ricci curvature.

1. Introduction, notation and useful facts All manifolds in this paper are supposed to be connected. At first, we recall and discuss important definitions. Definition 1. A Riemannian manifold (M, g) is called a manifold with homogeneous geodesics or geodesic orbit manifold (shortly, GO-manifold) if any geodesic γ of M is an orbit of 1-parameter subgroup of the full isometry group of (M, g). Definition 2. A Riemannian manifold (M = G/H, g), where H is a compact subgroup of a Lie group G and g is a G-invariant Riemannian metric, is called a space with homogeneous geodesics or geodesic orbit space (shortly, GO-space) if any geodesic γ of M is an orbit of 1-parameter subgroup of the group G. This terminology was introduced in [13] by O. Kowalski and L. Vanhecke, who initiated a systematic study of such spaces. In the same paper, O. Kowalski and L. Vanhecke classified all GO-spaces of dimension ≤ 6. Many interesting results about GO-manifolds and its subclasses one can find in [1, 2, 3, 4, 5, 7, 9, 14, 15, 17], and in the references therein. In [10], C. Gordon obtained some structure results on GO-spaces, in particular, the following one: Every Riemannian GO-manifold of nonpositive Ricci curvature is a symmetric space. The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian GO-manifold (M, g) and the structure of its full isometry group. The Lie algebra of the full (connected) isometry group of (M, g) is identified naturally with the Lie algebra of Killing fields g on (M, g). We prove the following result: If a is an abelian ideal of g, then every Killing field X ∈ a has constant length (Theorem 1). On the ground of this theorem we give a new proof of the above mentioned result of C. Gordon on GO-manifold with nonpositive Ricci curvature (Theorem 2). 1

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Let (M, g) be a GO-manifolds and G is its connected full isometry group. Obviously, (M, g) is homogeneous and M = G/H, where H is the isotropy subgroup at a point o ∈ M . Since H is compact, there is an Ad(H)-invariant decomposition (1)

g = h ⊕ m,

where g = Lie(G) and h = Lie(H). The Riemannian metric g is G-invariant and is determined by an Ad(H)-invariant Euclidean metric g = (·, ·) on the space m which is identified with the tangent space To M at the initial point o = eH. In what follows we identify elements of g with corresponded Killing vector fields on (M, g). Now, we recall some well known formulas for a homogeneous Riemannian manifold (M, g = (·, ·)) [8]. Let us choose some g-orthonormal basis (Xi ) in m. Consider also a vector Z ∈ m defined by the condition (Z, X) = trace(adX ) for every X ∈ m. Therefore, Z = 0 iff the Lie algebra g (and the Lie group G) is unimodular. The following formulae (that is more simple for the unimodular case) is useful for the Ricci curvature calculations: 1X 1 |[X, Xi ]m |2 + Ric(X, X) = − Bg (X, X) − 2 2 i

(2)

+

1X ([Xi , Xj ]m , X)2 − ([Z, X]m , X), 4 i,j

where Bg is the Killing form of the Lie algebra g, X ∈ m, and Vm means the m-part of a vector V ∈ g. Lemma 1 ([13]). A homogeneous Riemannian manifold (M = G/H, g) with the reductive decomposition (1) is GO-space if and only if for any X ∈ m there is HX ∈ h such that ([HX + X, Y ]m , X) = 0 for all Y ∈ m. This lemma shows that the property to be GO-space depends only on the reductive decomposition (1) and the Euclidean metric g on m. In other words, if (M = G/H, g) is a GO-space, then any locally isomorphic homogeneous Riemannian manifold (M ′ = G′ /H ′ , g ′ ) is a GO-space. Also a direct product of Riemannian manifolds is a manifold with homogeneous geodesics if and only if each factor is a manifold with homogeneous geodesics. For any subspace l ⊂ g and any U ∈ g we use the symbol adlU for a restriction of the operator adU to l, i.e. adlU : l → l, adlU (X) = [U, X]l . Lemma 2. Suppose that (M = G/H, g) is a GO-space. Let m1 and m2 be Ad(H)invariant subspaces of m such that (m1 , m2 ) = 0 and m = m1 ⊕ m2 . Then for any 2 U ∈ m1 the operator adm U is skew-symmetric. If, in addition, [h, m1 ] = 0, then the operator adm U is skew-symmetric. Proof. For any X ∈ m2 there is HX ∈ h such that ([HX + X, Y ]m , X) = 0 for all Y ∈ m (see Lemma 1). Therefore, 0 = ([HX + X, U ]m , X) = ([HX , U ]m , X) + ([X, U ]m , X) = ([X, U ]m , X), because [HX , U ] ∈ m1 and X ∈ m2 . If [h, m1 ] = 0, then the same is true for any X ∈ m. This proves the lemma.

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Lemma 3 ([10]). Let (M = G/H, g) be a GO-space, then the group G is unimodular. Proof. Here we give a more direct proof, than the original one in [10]. Suppose that the Lie algebra g = Lie(G) is not unimodular and consider its proper subspace u = {X ∈ g | trace(adX ) = 0}. Obviously, h ⊂ u. Since ad[X,Y ] = [adX , adY ] and trace(ad[X,Y ] ) = trace([adX , adY ]) = 0, then [u, g] ⊂ [g, g] ⊂ u, hence, u is an ideal of g. Consider m1 = m ∩ u and let m2 be a (non-trivial) g-orthogonal complement to m1 in m. Since u is an ideal of g and g is ad(h)-invariant, then m1 and m2 are ad(h)-invariant. On the other hand, [h, m2 ] ⊂ [g, g] ⊂ u. Therefore, [h, m2 ] = 0. Now, consider any non-trivial Y ∈ m2 . By our construction, trace(adY ) 6= 0. On the other hand, by Lemma 1 for every X ∈ m there is HX ∈ h such that ([HX + X, Y ]m , X) = 0. Since [h, m2 ] = 0, we get ([X, Y ]m , X) = 0 (see also Lemma 2), that implies trace(adY ) = 0, a contradiction. Lemma 4. Let (M, g) be a Riemannian manifold, X a Killing field on (M, g). Consider any point x ∈ M such that X(x) 6= 0. Then the integral curve of X through the point x is a geodesic if and only if x is a critical point of the function y ∈ M 7→ gy (X, X). Proof. In fact, this is proved in Proposition 5.7 of Chapter VI in [12]. We use the symbol Mx for the tangent space of a manifold M at a point x ∈ M . Lemma 5. Let (M, g) be a Riemannian manifold, g its Lie algebra of Killing field. Then (M, g) is a GO-manifold if and only if for any x ∈ M and any v ∈ Mx there is X ∈ g such that X(x) = v and x is a critical point of the function y ∈ M 7→ gy (X, X). If (M, g) is homogeneous, then the latter condition is equivalent to the following one: for any Y ∈ g the equality gx ([Y, X], X) = 0 holds. Proof. By Lemma 4 an integral curve of X through the point x ∈ M is geodesic if and only if x is a critical point of the function y ∈ M 7→ gy (X, X). If (M, g) is homogeneous, then it is equivalent to the condition Y · g(X, X)|x = 2gx ([Y, X], X) = 0 for every Y ∈ g. In what follows we need the following Proposition 1 (Theorem 2.10 in [18]). If a Killing vector field X on a compact Riemannian manifold M satisfies the condition Ric(X, X) ≤ 0, then X is parallel on M and Ric(X, X) ≡ 0. Corollary 1. If a compact homogeneous Rimannian manifold (M, g) has nonpositive Ricci curvature, then it is an Euclidean torus. In particular, its full connected isometry group is abelian. 2. Main results At first, we get the following remarkable result. Theorem 1. Let (M, g) be a GO-manifold, g is its Lie algebra of Killing fields. Suppose that a is an abelian ideal of g. Then any X ∈ a has constant length on (M, g).

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Proof. Let x be any point in M . We will prove that x is a critical point of the function y ∈ M 7→ gy (X, X). Since (M, g) is homogeneous, then (by Lemma 5) it suffices to prove that gx ([Y, X], X) = 0 for every Y ∈ g. Consider any Y ∈ a, then Y · g(X, X) = 2g([Y, X], X) = 0 on M , since a is abelian. Now, consider Y ∈ g such that gx (Y, U ) = 0 for every U ∈ a. We will prove that gx ([Y, X], X) = 0. By Lemma 5, for the vector X(x) ∈ Mx there is a Killing field Z ∈ g such that Z(x) = X(x) and gx ([V, Z], Z) = 0 for any V ∈ g. In particular, gx ([Y, Z], Z) = 0. Now, W = X − Z vanishes at x and we get gx ([Y, X], X) = gx ([Y, Z + W ], Z + W ) = gx ([Y, Z + W ], Z) = gx ([Y, Z], Z) + gx ([Y, W ], Z) = gx ([Y, W ], Z). Note that gx ([Y, W ], Z) = −gx ([W, Y ], Z) = gx (Y, [W, Z]) = 0 because W (x) = 0 (0 = W ·g(Y, Z)|x = gx ([W, Y ], Z)+ gx (Y, [W, Z])) and [W, Z] = [X, Z] ∈ a. Therefore, gx ([Y, X], X) = 0. Hence, x is a critical point of the function y ∈ M 7→ gy (X, X). Since every x ∈ M is a critical point of the function y ∈ M 7→ gy (X, X), then X has constant length on (M, g). Remark 1. This result can be easily generalized to some cases when g is a subalgebra of the Lie algebra of the full connected isometry group of (M, g). It suffices that a connected subgroup G (with the Lie algebra g) of the full isometry group of (M, g) is such that (M = G/H, g) is a GO-space. In the rest of the paper we reprove the following Theorem 2 (C. Gordon [10]). Every Riemannian GO-manifold of nonpositive Ricci curvature is symmetric. Remark 2. It should be noted that the original proof of this theorem (Theorem 5.1 in [10]) has an error in the claim “Since U ∗ /L∗ is a compact homogeneous space, its Ricci curvature Ric∗ is nonnegative”. Nevertheless, this error could be corrected, and the proof in [10] requires only a little modification. But here we present a more simple proof, in which some constructions from [10] are essentially used. It suffices to prove Theorem 2 for simply connected manifolds. Indeed, if a Rimannian homogeneous manifold M has a Riemannian symmetric space of nonpositive Ricci curvature (equivalently, nonpositive sectional curvature) as a universal covering, then it is symmetric too [16, 11]. Let (M, g) be a simply connected GO-manifolds with nonpositive Ricci curvature, an let G be its full connected isometry group. We know that G is unimodular (Lemma 3), and the isotropy subgroup H must be connected. At first, we reduce the problem to the case when G is semisimple. Proposition 2. Let (M, g) be a simply connected GO-manifold with nonpositive Ricci curvature. Then it is a direct metric product of an Euclidean space Em and a simply connected GO-manifold (M1 , g1 ) (with nonpositive Ricci curvature) with semisimple full isometry groups. Proof. In the notation of Theorem 1, any Killing field X ∈ a has constant length on (M, g). Since the Ricci curvature is nonpositive, then by Theorem 4 in [6] we get that Ric(X, X) = 0, moreover, the Killing field X is parallel on (M, g), and the Riemannian manifold (M, g) is a direct metric product of two Riemannian manifolds, one of which is a one-dimensional manifold tangent to Killing field X and another one

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is a GO-manifold with nonpositive Ricci curvature. This procedure could be repeated unless the obtained second manifold has a semisimple full isometry group. In what follows we suppose that the group G is semisimple. Now we consider a reductive decomposition (see (1)) g = h ⊕ m, where g = Lie(G), h = Lie(H), and m is an orthogonal complement to h in h with respect to the Killing form Bg of the (semisimple) Lie algebra g. The Riemannian metric g is G-invariant and is determined by an Ad(H)-invariant Euclidean metric g = (·, ·) on the space m. Now we consider a maximal compactly embedded subalgebra k ⊂ g such that h ⊂ k. If h = k, then the manifold under consideration is a symmetric space [16, 11]. Suppose now, that h 6= k. Then there are Ad(H)-invariant subspaces m1 , m2 ⊂ m such that (m1 , m2 ) = 0, m = m1 ⊕ m2 , and k = h ⊕ m1 . Let K ∗ be a compact Lie group with the Lie algebra k and H ∗ be its subgroup corresponded to subalgebra h ⊂ k. We have k = h⊕m1 , therefore m1 could be identified with the tangent space at the point eH ∗ of a compact homogeneous manifold M ∗ = K ∗ /H ∗ . We consider K ∗ -invariant Riemannian metric g∗ on M ∗ that is generated with the inner product (·, ·)|m1 . Note that K ∗ may not act effectively on M ∗ = K ∗ /H ∗ , but this is not important for calculation of the Ricci curvature of (M ∗ , g∗ ). We choose a (·, ·)-orthonormal basis X1 , X2 , . . . , Xr , r = dim(m1 ), in m1 , and (·, ·)orthonormal basis Y1 , Y2 , . . . , Ys , s = dim(m2 ), in m2 . Denote the Ricci curvature of (M ∗ , g∗ ) by Ric∗ and the Killing forms of k and g by Bk and Bg respectively. Using (2) and the fact that g is unimodular (Lemma 3 is not necessary, because every semisimple Lie algebra is unimodular), we get 1X 1X 1 |[X, Xi ]m |2 − |[X, Yi ]m |2 + Ric(X, X) = − Bg (X, X) − 2 2 2 i

+

1X 1X 1X ([Xi , Xj ]m , X)2 + ([Yi , Yj ]m , X)2 + ([Xi , Yj ]m , X)2 4 4 2 i,j

and

i

i,j

i,j

1X 1 1X |[X, Xi ]m1 |2 + ([Xi , Xj ]m1 , X)2 Ric∗ (X, X) = − Bk (X, X) − 2 2 4 i

i,j

for any X ∈ m1 . By Lemma 2 X X Bg (X, X) = Bk (X, X) + ([X, [X, Yi ]]m , Yi ) = Bk (X, X) − |[X, Yi ]m2 |2 , i

i

then, using Lemma P 2 again (([Xi , Yj ]m , X)2 = ([Yj , Xi ]m1 , X)2 = ([Yj , X]m1 , Xi )2 and P 2 ([Xi , Yj ]m , X) = |[X, Yi ]m1 |2 ), we get i,j

i

Proposition 3 ([10]). For any X ∈ m1 the equality 1 X Ric∗ (X, X) = Ric(X, X) − ([Yi , Yj ]m1 , X)2 2 1≤i

holds.

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Since (M = G/H, g) has nonpositive Ricci curvature, then from Proposition 3 we get Ric∗ (X, X) ≤ 0 for any X ∈ m1 , but Ric∗ (X, X) = 0 implies Ric(X, X) = 0 and ([m2 , m2 ]m1 , X) = 0. Since (M ∗ = K ∗ /H ∗ , g∗ ) is a compact homogeneous Riemannian manifold with nonpositive Ricci curvature, then it is Euclidean torus by Corollary 1. This implies Ric∗ (X, X) = Ric(X, X) = 0 for all X ∈ m1 , m1 lies in the center of k, and [m2 , m2 ] ⊂ h ⊕ m2 . Let p be a Bg -orthogonal compliment to k in g. It is well known that [p, p] ⊂ k; if h1 := [p, p], then g1 := h1 ⊕ p is a maximal noncompact semisimple ideal in the Lie algebra g. Now we will prove that p = m2 . By Lemma 2 we get that for any U ∈ m1 the operator adm U is skew-symmetric. The same is true for any U ∈ h, and, therefore, for any U ∈ k = h ⊕ m1 . From the relations p ⊂ m, h1 ⊂ k, [k, m1 ] = 0, and [k, p] = p we get (m1 , p) = (m1 , [k, p]m ) = ([k, m1 ]m , p) + (m1 , [k, p]m ) = 0, since the operators adm U are skew-symmetric for all U ∈ k. This proves p = m2 . Therefore, h1 := [p, p] = [m2 , m2 ] ⊂ h ⊕ m2 , and we get h1 ⊂ h. Let g2 be a Bg -orthogonal compliment to g1 in g. Then g2 is a maximal compact semisimple ideal in the Lie algebra g. If h2 is a Bg -orthogonal compliment to h1 in h, then g2 = h2 ⊕ m1 . Recall that [h2 , m1 ] ⊂ [h, m1 ] = 0. Hence, h2 is an ideal in the Lie algebra g. Since the space G/H is effective, h2 is trivial. Since m1 = g2 is commutative (m1 lies in the center of k), then it is trivial too (otherwise, g2 is not semisimple). Hence, h = k, and (M = G/H, g) is a symmetric space. Therefore, Theorem 2 is completely proved. Acknowledgements. The project was supported in part by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (grant NSh6613.2010.1), by Federal Target Grant “Scientific and educational personnel of innovative Russia” for 2009-2013 (government contract No. 02.740.11.0457), and by RFBR (grant 10-01-9000-Bel-a). References [1] D.V. Alekseevsky, A. Arvanitoyeorgos, Riemannian flag manifolds with homogeneous geodesics, Trans. Amer. Math. Soc., 359 (2007), 3769–3789. [2] D.V. Alekseevsky, Yu.G. Nikonorov, Compact Riemannian manifolds with homogeneous geodesics, SIGMA 5 (2009), 093, 16 pages. [3] D.N. Akhiezer, E.B. Vinberg, Weakly symmetric spaces and spherical varieties, Transf. Groups, 4 (1999), 3–24. [4] V.N. Berestovskii, Yu.G. Nikonorov, On δ-homogeneous Riemannian manifolds, Differential Geometry and its Applications 26(5) (2008), 514–535. [5] V.N. Berestovskii, Yu.G. Nikonorov, On δ-homogeneous Riemannian manifolds, II, Siber. Math. J., 50(2) (2009), 214–222. [6] V.N. Berestovskii, Yu.G. Nikonorov, Killing vector fields of constant length on Riemannian manifolds, Siber. Math. J., 49 (3) (2008) 395–407. [7] V.N. Berestovskii, Yu.G. Nikonorov, Clifford-Wolf homogeneous Riemannian manifolds, J. Differ. Geom. 82(3), 2009, 467–500. [8] A.L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987. [9] Z. Du˘sek, O. Kowalski, S. Nik˘cevi´c, New examples of Riemannian g.o. manifolds in dimension 7. Diff. Geom. Appl., 21 (2004), 65–78.

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[10] C. Gordon, Homogeneous Riemannian manifolds whose geodesics are orbits, 155–174. In: Progress in Nonlinear Differential Equations. V. 20. Topics in geometry: in memory of Joseph D’Atri. Birkh¨ auser, 1996. [11] S. Helgason, Differential geometry and symmetric spaces, Academic Press Inc., New-York, 1962. [12] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Vol. I – A Wiley-Interscience Publication, New York, 1963; Vol. II – A Wiley-Interscience Publication, New York, 1969. [13] O. Kowalski, L. Vanhecke, Riemannian manifolds with homogeneous geodesics, Boll. Unione Mat. Ital. Ser. B. 5(1) (1991), 189-246. [14] H. Tamaru, Riemannin g.o. spaces fibered over irreducible symmetric spaces, Osaka J. Math. 15 (1998) 55–67. [15] H. Tamaru, Riemannin geodesic orbit space metrics on fiber bundles, Algebras Groups Geom. 36 (1999) 835–851. [16] J.A. Wolf, Spaces of constant curvature, Publish or Perish, Inc., Wilmington, Delaware (U.S.A.), 1984. [17] J.A. Wolf, Harmonic Analysis on Commutative Spaces, American Mathematical Society, 2007. [18] K. Yano, S. Bochner, Curvature and Betti numbers, Princeton, New Jersey, Princeton University Press, 1953. Nikonorov Yuri˘ı Gennadyevich South Mathematical Institute of VSC RAS, Vladikavkaz, Russia E-mail address: [email protected]