Math Phys Anal Geom (2008) 11:1–9 DOI 10.1007/s11040-008-9037-8 On the Flag Curvature of Invariant Randers Metrics Hamid Reza Salimi Moghaddam Received: 10 December 2007 / Accepted: 6 February 2008 / Published online: 21 March 2008 © Springer Science + Business Media B.V. 2008 Abstract In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field. Keywords Invariant metric · Flag curvature · Randers space · Homogeneous space · Lie group Mathematics Subject Classifications (2000) 22E60 · 53C60 · 53C30 1 Introduction The geometry of invariant structures on homogeneous spaces is one of the interesting subjects in differential geometry. Invariant metrics are of these invariant structures. K. Nomizu studied many interesting properties of in- variant Riemannian metrics and the existence and properties of invariant affine connections on reductive homogeneous spaces (see [14, 16]). Also some H. R. S. Moghaddam (B) Department of Mathematics, Shahrood University of Technology, Shahrood, Iran e-mail: 2 H.R.S. Moghaddam curvature properties of invariant Riemannian metrics on Lie groups has studied by J. Milnor [15]. So it is important to study invariant Finsler metrics which are a generalization of invariant Riemannian metrics. S. Deng and Z. Hou studied invariant Finsler metrics on reductive homo- geneous spaces and gave an algebraic description of these metrics [12, 13]. Also, in [10, 11], we have studied the existence of invariant Finsler metrics on quotient groups and the flag curvature of invariant Randers metrics on naturally reductive homogeneous spaces. In this paper we study the flag cur- vature of invariant Randers metrics on homogeneous spaces and Lie groups. Flag curvature, which is a generalization of the concept of sectional curvature in Riemannian geometry, is one of the fundamental quantities which associate with a Finsler space. In general, the computation of the flag curvature of Finsler metrics is very difficult, therefore it is important to find an explicit and applica- ble formula for the flag curvature. One of important Finsler metrics which have found many applications in physics are Randers metrics (see [2, 3]). In this article, by using Püttmann’s formula [17], we give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and Lie groups. Then the Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups are studied. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field. 2 Flag Curvature of Invariant Randers Metrics on Homogeneous Spaces The aim of this section is to give an explicit formula for the flag curvature of invariant Randers metrics of Berwald type, arising from invariant Riemannian metrics, on homogeneous spaces. For this purpose we need the Püttmann’s formula for the curvature tensor of invariant Riemannian metrics on homoge- neous spaces (see [17]). Let G be a compact Lie group, H a closed subgroup, and g0 a bi-invariant Riemannian metric on G. Assume that g and h are the Lie algebras of G and H respectively. The tangent space of the homogeneous space G/H is given by the orthogonal compliment m of h in g with respect to g0. Each invariant metric g on G/H is determined by its restriction to m. The arising AdH-invariant inner product from g on m can extend to an AdH-invariant inner product on g by taking g0 for the components in h. In this way the invariant metric g on G/H determines a unique left invariant metric on G that we also denote by g. The values of g0 and g at the identity are inner products on g which we denote as < ., . >0 and < ., . >. The inner product < ., . > determines a positive definite endomorphism φ of g such that < X, Y >=< φX, Y >0 for all X, Y ∈ g. Now we give the following lemma which was proved by T. Püttmann (see [17]). On the flag curvature of invariant Randers metrics 3 Lemma 1 The curvature tensor of the invariant metric < ., . > on the compact homogeneous space G/H is given by ( ) <R(X, Y)Z, W >=1/2 < B−(X, Y), [Z, W] >0 +< [X, Y], B−(Z, W) >0 + ( + 1/4 < [X, W], [Y, Z]m >−< [X, Z], [Y, W]m > − ) − 2 < [X, Y], [Z, W]m > + ( −1 + < B+(X, W), φ B+(Y, Z) >0 − ) −1 − < B+(X, Z), φ B+(Y, W) >0 , (1) where the symmetric resp. skew symmetric bilinear maps B+ and B− are defined by ( ) B+(X, Y) = 1/2 [X, φY] + [Y, φX] , ( ) B−(X, Y) = 1/2 [φX, Y] + [X, φY] , and [., .]m is the projection of [., .] to m. ˜ Let X be an invariant vector field on the homogeneous space G/H such that √ ˜ ˜ ˜ ‖ X ‖= g(X, X) < 1. A case happen when G/H is reductive with g = m ⊕ h ˜ and X is the corresponding left invariant vector field to a vector X ∈ m such that < X, X >< 1 and Ad(h)X = X for all h ∈ H (see [13] and [10]). By using ˜ X we can construct an invariant Randers metric on the homogeneous space G/H in the following way: √ ( ) ˜ F(xH, Y) = g(xH)(Y, Y) + g(xH) Xx, Y ∀Y ∈ TxH(G/H). (2) Now we give an explicit formula for the flag curvature of these invariant Randers metrics. Theorem 1 Let G be a compact Lie group, H a closed subgroup, g0 a bi- invariant metric on G, and g and h the Lie algebras of G and H respectively. Also let g be any invariant Riemannian metric on the homogeneous space ˜ G/H such that < Y, Z >=< φY, Z >0 for all Y, Z ∈ g. Assume that X is an ˜ ˜ invariant vector field on G/H which is parallel with respect to g and g(X, X) < ˜ ˜ 1 and XH = X. Suppose that F is the Randers metric arising from g and X, and (P, Y) is a flag in TH(G/H) such that {Y, U} is an orthonormal basis of P with respect to < ., . >. Then the flag curvature of the flag (P, Y) in TH(G/H) is given by A K(P, Y) = , (3) 2 (1+ < X, Y >) (1− < X, Y >) 4 H.R.S. Moghaddam where A = α. < X, U > +γ (1+ < X, Y), and for A we have: ( ) α = 1/4 < [φU, Y] + [U, φY], [Y, X] >0 + < [U, Y], [φY, X] + [Y, φX] >0 + ( ) −1 + 3/4 < [Y, U], [Y, X]m >+1/2 < [U, φX] + [X, φU], φ [Y, φY] >0 − ( ) −1 − 1/4 < [U, φY] + [Y, φU], φ [Y, φX] + [X, φY] >0, (4) and γ = 1/2 < [φU, Y] + [U, φY], [Y, X] >0 + −1 + 3/4 < [Y, U], [Y, U]m > + < [U, φU], φ ([Y, φY]) >0 − ( ) −1 − 1/4 < [U, φY] + [Y, φU], φ [Y, φU] + [U, φY] >0 . (5) ˜ Proof X is parallel with respect to g, therefore F is of Berwald type and the Chern connection of F and the Riemannian connection of g coincide (see [6], F g F g page 305.), so we have R (U, V)W = R (U, V)W, where R and R are the g F curvature tensors of F and g, respectively. Let R := R = R be the curvature tensor of F (or g). Also for the flag curvature we have [18]: gY(R(U, Y)Y, U) K(P, Y) = , (6) 2 gY(Y, Y).gY(U, U) − g Y(Y, U) 2 ∂ 2 where gY(U, V) = 1/2 (F (Y + sU + tV))|s=t=0. ∂s∂t By a direct computation for F we get g(X, Y).g(Y, V).g(Y, U) gY(U, V) = g(U, V) + g(X, U).g(X, V) − × 3/2 g(Y, Y) 1 { ×√ g(X, U).g(Y, V) + g(X, Y).g(U, V) + g(Y, Y) } + g(X, V).g(Y, U) . (7) Since {Y, U} is an orthonormal basis of P with respect to < ., . >, by using the formula (7) we have: 2 gY(Y, Y).gY(U, U) − gY(Y, U) = (1+ < X, Y >) (1− < X, Y >). (8) Also we have: gY(R(U, Y)Y, U) = < R(U, Y)Y, U > + < X, R(U, Y)Y > . < X, U > + + < X, Y > . < R(U, Y)Y, U > + + < X, U > . < Y, R(U, Y)Y >, (9) now let α=< X, R(U, Y)Y >, θ =<Y, R(U, Y)Y > and γ =< R(U, Y)Y, U >. On the flag curvature of invariant Randers metrics 5 By using Püttmann’s formula (see Lemma 1) and some computations we have: ( ) α = 1/4 < [φU, Y] + [U, φY], [Y, X] >0 + < [U, Y], [φY, X] + [Y, φX] >0 + −1 + 3/4 < [Y, U], [Y, X]m > +1/2 < [U, φX] + [X, φU], φ ([Y, φY]) >0 − −1 − 1/4 < [U, φY] + [Y, φU], φ ([Y, φX] + [X, φY]) >0, (10) θ = 0, (11) and γ = 1/2 < [φU, Y] + [U, φY], [Y, U] >0 +3/4 < [Y, U], [Y, U]m > + −1 + < [U, φU], φ ([Y, φY]) >0 − −1 −1/4 < [U, φY] + [Y, φU], φ ([Y, φU] + [U, φY]) >0 . (12) Substituting (7), (8), (9), (10), (11) and (12) in the (6) completes the proof. ⊔⊓ Remark In the previous theorem, If we let H = {e} and m = g then we can obtain a formula for the flag curvature of the left invariant Randers metrics of Berwald types arising from a left invariant Riemannian metric g and a left ˜ invariant vector field X on Lie group G. If the invariant Randers metric arises from a bi-invariant Riemannian metric on a Lie group then we can obtain a simpler formula for the flag curvature, we give this formula in the following theorem. Theorem 2 Suppose that g0 is a bi-invariant Riemannian metric on a Lie group ˜ ˜ ˜ ˜ G and X is a left invariant vector field on G such that g0(X, X) < 1 and X is parallel with respect to g0. Then we can define a left invariant Randers metric F as follows: √ ( ) ˜ F(x, Y) = g0(x)(Y, Y) + g0(x) Xx, Y . Assume that (P, Y) is a flag in TeG such that {Y, U} is an orthonormal basis of P with respect to < ., . >0. Then the flag curvature of the flag (P, Y) in TeG is given by <[Y, [U,Y]], X>0 .< X,U >0 +<[Y, [U,Y]], U >0 (1+< X, Y>0) K(P, Y) = . 2 4(1+ < X, Y >0) (1− < X, Y >0) ˜ Proof Since X is parallel with respect to g0 the curvature tensors of g0 and F coincide. On the other hand for g0 we have R(X, Y)Z = 1/4[Z, [X, Y]], therefore by substituting R in (6) and using (7) the proof is completed. ⊔⊓ 6 H.R.S. Moghaddam 3 Invariant Randers Metrics on Lie Groups In this section we study the left invariant Randers metrics on Lie groups and, in some special cases, find some results about the dimension of Lie groups which can admit invariant Randers metrics. These conclusions are obtained by using Yasuda–Shimada theorem. The Yasuda–Shimada theorem is one of important theorems which characterize the Randers spaces. In the year 2001, Shen’s examples of Randers manifolds with constant flag curvature motivated Bao and Robles to determine necessary and sufficient conditions for a Randers manifold to have constant flag curvature. Shen’s examples showed that the original version of Yasuda–Shimada theorem (1977) is wrong. Then Bao and Robles corrected the Yasuda–Shimada theorem (1977) and gave the correct version of this theorem, Yasuda–Shimada theorem (2001) (see [5]; for a comprehensive history of Yasuda–Shimada theorem see [4]). Suppose that M is an n-dimensional manifold endowed with a Riemannian metric g = (gij(x)) and a nowhere zero 1-form b = (bi(x)) such that ‖b‖ = ij b i(x)b j(x)g (x) < 1. We can define a Randers metric on M as follows √ i j i F(x, Y) = gij(x)Y Y + bi(x)Y . (13) i i Next, we consider the 1-form β = b (b j|i − bi| j)dx , where the covariant deriv- ative is taken with respect to Levi–Civita connection to M. Now we give the Yasuda–Shimada theorem from [4]. Theorem 3 (Yasuda–Shimada; see [4]) Let F be a strongly convex non- Riemannian Randers metric on a smooth manifold M of dimension n ⩾ 2. Let gij be the underlying Riemannian metric and bi the drift 1-form. Then: (+) F satisfies β = 0 and has constant positive flag curvature K if and only if: – b is a non-parallel Killing field of g with constant length; – the Riemann curvature tensor of g is given by ( ) ( ) 2 Rhijk = K 1 − ‖b‖ ghkgij − ghjgik + ( ) + K gijb hb k − gikb hb j − ( ) − K ghjb ib k − ghkb ib j − − bi| jb h|k + bi|kb h| j + 2b h|ib j|k (0) F satisfies β = 0 and has zero flag curvature ⇔ it is locally Minkowskian. (–) F satisfies β = 0 and has constant negative flag curvature if and only if: – b is a closed 1-form; 2 – b i|k = 1/2σ(gik − bib k), with σ = −16K; – g has constant negative sectional curvature 4K, that is, Rhijk = 4K(gijghk − gikghj). On the flag curvature of invariant Randers metrics 7 Since any Randers manifold of dimension n = 1 is a Riemannian manifold from now on we consider n > 1. An immediate conclusion of Yasuda–Shimada theorem is the following corollary. Corollary 1 There is no non-Riemannian Randers metric of Berwald type with β = 0 and constant positive flag curvature. Now by using the results of [8] we obtain the following conclusions. n Theorem 4 Let F = (M, F, gij, b i) be an n-dimensional parallelizable Randers manifold of constant positive flag curvature with β = 0 on M and complete Riemannian metric g = (gij). Then the dimension of M must be 3 or 7. Proof By using Theorem 2.2 of [8] M is diffeomorphic with a sphere of m dimension n = 2k + 1. But a sphere S is parallelizable if and only if m = 1, 3 or 7 (see [1]). Therefore n = 3 or 7. ⊔⊓ A family of Randers metrics of constant positive flag curvature on Lie 3 group S was studied by D. Bao and Z. Shen (see [7]). They produced, for each K > 1, an explicit example of a compact boundaryless (non-Riemannian) Randers spaces that has constant positive flag curvature K, and which is not 3 projectively flat, on Lie group S . In the following we give some results about the dimension of Lie groups which can admit Randers metrics of constant positive flag curvature. These results show that the dimension 3 is important. Corollary 2 There is no Randers Lie group of constant positive flag curvature with β = 0, complete Riemannian metric g = (gij) and n ̸= 3. Proof Any Lie group is parallelizable, so by attention to Theorem 4 and the 7 7 condition n ̸= 3, n must be 7. Since G is diffeomorphic to S and S can not admit any Lie group structure, hence the proof is completed. ⊔⊓ Similar to the [15] for the sectional curvature of the left invariant Rie- mannian metrics on Lie groups, we compute the flag curvature of the left invariant Randers metrics on Lie groups in the following theorem. Theorem 5 Let G be a compact Lie group with Lie algebra g, g0 a bi-invariant Riemannian metric on G, and g any left invariant Riemannian metric on G such that < X, Y >=< φX, Y >0 for a positive definite endomorphism φ: g −→ g. Assume that X ∈ g is a vector such that < X, X >< 1 and F is the Randers ˜ metric arising from X and g as follows: √ ( ) ˜ F(x, Y) = g(x)(Y, Y) + g(x) Xx, Y , ˜ where X is the left invariant vector field corresponding to X, and we have ˜ assumed X is parallel with respect to g. Let {e1, · · · , en} ⊂ g be a g-orthonormal 8 H.R.S. Moghaddam basis for g. Then the flag curvature of F for the flag P = span{ei, e j}(i ̸= j) at the point (e, ei), where e is the unit element of G, is given by the following formula: X j. < R(e j, ei)ei, X > +(1 + Xi). < R(e j, ei)ei, e j > K(P = span{ei, e j}, ei) = , 2 (1 + Xi) (1 − Xi) k where X = X ek, ( < R(e j, ei)ei, X > = − 1/4 < [φe j, ei], [ei, X] >0 + < [e j, φei], [ei, X] >0 ) + < [e j, ei], [φei, X] >0 + < [e j, ei], [ei, φX] >0 + + 3/4 < [e j, ei], [ei, X] > − ( ) −1 − 1/2 < [e j, φX] + [X, φe j], φ [ei, φei] >0 + ( ) −1 + 1/4 < [e j, φei] + [ei, φe j], φ [ei, φX] + [X, φei] >0 and ( ) < R(e j, ei)ei, e j > = − 1/2 < [φe j, ei], [ei, e j] >0 + < [e j, φei], [ei, e j] >0 + ( ) −1 + 3/4 < [e j, ei], [ei, e j] >−< [e j, φe j], φ [ei, φei] >0 + ( ) −1 + 1/4 < [e j, φei] + [ei, φe j], φ [ei, φe j] + [e j, φei] >0 . Proof By using Theorem 1, the proof is clear. ⊔⊓ Now we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field. Theorem 6 There is no left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field on connected Lie groups with a perfect Lie algebra, that is, a Lie algebra g for which the equation [g, g] = g holds. Proof If a left invariant vector field X is parallel with respect to a left invariant Riemannian metric g then, by using Lemma 4.3 of [9], g(X, [g, g]) = 0. Since g is perfect therefore X must be zero. ⊔⊓ Corollary 3 There is not any left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field on semisimple connected Lie groups. Corollary 4 If a Lie group G admits a left invariant non-Riemannian Randers metric of Berwald type F arising from a left invariant Riemannian metric g and a left invariant vector field X then for sectional curvature of the Riemannian metric g we have K(X, u) ⩾ 0 for all u, where equality holds if and only if u is orthogonal to the image [X, g]. On the flag curvature of invariant Randers metrics 9 Proof Since F is of Berwald type, X is parallel with respect to g. By using Lemma 4.3 of [9], ad(X) is skew-adjoint, therefore by Lemma 1.2 of [15] we have K(X, u) ⩾ 0. ⊔⊓ References 1. Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72(2), 20–104 (1960) 2. Antonelli, P.L., Ingarden, R.S., Matsumoto, M.: The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology. Kluwer, Dordrecht (1993) 3. Asanov, B.S.: Finsler Geometry, Relativity and Gauge Theories. Kluwer, Dordrecht (1985) 4. Bao, D.: Randers space forms. Period. Math. Hungar. 48(1), 3–15 (2004) 5. Bao, D., Robles, C.: On randers spaces of constant flag curvature. Rep. Math. Phys. 51(1), 9–42 (2003) 6. Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann–Finsler Geometry. Springer- Verlag, Berlin (2000) 3 7. Bao, D., Shen, Z.: Finsler metrics of constant positive curvature on lie group S . J. Lond. Math. Soc. 66(2), 453–467 (2002) 8. Bejancu, A., Farran H.R.: Randers manifolds of positive constant curvature. Internat. J. Math. Math. Sci. 2003(18), 1155–1165 (2003) 9. Brown, N., Finck, R., Spencer, M., Tapp K., Wu, Z.; Invariant metrics with nonnegative curvature on compact lie groups. Canad. Math. Bull. 50(1), 24–34 (2007) 10. Esrafilian, E., Salimi Moghaddam, H.R.: Flag curvature of invariant Randers metrics on homogeneous manifolds. J. Phys. A: Math. Gen. 39, 3319–3324 (2006) 11. Esrafilian, E., Salimi Moghaddam, H.R.: Induced invariant Finsler metrics on quotient groups. Balkan J. Geom. Appl. 11(1), 73–79 (2006) 12. Deng, S., Hou, Z.: Invariant Finsler metrics on homogeneous manifolds. J. Phys. A: Math. Gen. 37, 8245–8253 (2004) 13. Deng, S., Hou, Z.: Invariant Randers metrics on homogeneous Riemannian manifolds. J. Phys. A: Math. Gen. 37, 4353–4360 (2004) 14. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry Vol. 2. Interscience Publishers, John Wiley & Sons (1969) 15. Milnor, J.: Curvatures of left invariant metrics on lie groups. Adv. Math. 21, 293–329 (1976) 16. Nomizu, K.: Invariant affine connections on homogeneous spaces. Amer. J. Math. 76, 33–65 (1954) 17. Püttmann, T.: Optimal pinching constants of odd dimensional homogeneous spaces. Invent. Math. 138, 631–684 (1999) 18. Shen, Z.: Lectures on Finsler Geometry. World Scientific (2001) Math Phys Anal Geom (2008) 11:11–51 DOI 10.1007/s11040-008-9038-7 Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies M. Cafasso Received: 26 November 2007 / Accepted: 19 February 2008 / Published online: 1 April 2008 © Springer Science + Business Media B.V. 2008 Abstract We propose a method for computing any Gelfand-Dickey τ function defined on the Segal-Wilson Grassmannian manifold as the limit of block Toeplitz determinants associated to a certain class of symbols W(t; z). Also truncated block Toeplitz determinants associated to the same symbols are shown to be τ functions for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from the point of view of inte- grable systems and block Toeplitz operator theory. Examples of applications to algebro-geometric solutions are given. Keywords Block Toeplitz determinants · Integrable hierarchies · Grassmannians · KP · Riemann-Hilbert problems Mathematics Subject Classifications (2000) 37K10 · 47B35 1 Introduction This paper deals with the applications of block Toeplitz determinants and their asymptotics to the study of integrable hierarchies. Asymptotics of block Toeplitz determinants and their applications to physics is a developing field of research; in recent years it has been shown how to compute some physically relevant quantities (e.g. correlation functions) studying asymptotics of some block Toeplitz determinants (see [27–29]). In particular in [27] and [28] the authors, for the first time, showed effective computations for the case of block Toeplitz determinants with symbols that do not have half truncated Fourier M. Cafasso (B) SISSA-International School for Advanced Studies, Via Beirut 2/4, 34014 Grignano, Italy e-mail: