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The (α,β) -metrics of scalar flag curvature
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The (α, β)-metrics of scalar flag curvature Xinyue Cheng 1 School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, PR China
a r t i c l e
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Article history: Received 10 November 2013 Received in revised form 6 February 2014 Available online xxxx Communicated by Z. Shen MSC: 53B40 53C60
a b s t r a c t One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study and characterize the (α, β)-metrics of scalar flag curvature. When the dimension of the manifold is greater than 2, we classify Randers metrics of weakly isotropic flag curvature (that is, Randers metrics of scalar flag curvature with isotropic S-curvature). Further, we characterize and classify (α, β)-metrics of scalar flag curvature with isotropic S-curvature. Finally, we conclude that the non-trivial regular (α, β)-metrics of scalar flag curvature with isotropic S-curvature on an n-dimensional manifold M (n ≥ 3) must be Randers metrics. © 2014 Elsevier B.V. All rights reserved.
Keywords: Finsler metric (α, β)-metric Randers metric Flag curvature S-curvature
1. Introduction Let F be a Finsler metric on an n-dimensional C ∞ manifold M . Put gij (x, y) :=
1 2 F (x, y) yi yj . 2
∂ For any non-zero vector y = y i ∂x i ∈ Tx M , F induces an inner product gy on Tx M as follows
gy (u, v) := gij (x, y)ui v j , ∂ i ∂ where u = ui ∂x i ∈ Tx M , v = v ∂xi ∈ Tx M . The geodesics of F are characterized locally by the following system of second order ordinary differential equations
E-mail address: [email protected]. Supported by the National Natural Science Foundation of China (11371386) and European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 317721. 1
http://dx.doi.org/10.1016/j.difgeo.2014.04.011 0926-2245/© 2014 Elsevier B.V. All rights reserved.
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dx d2 xi (t) i x(t), = 0, + 2G dt2 dt where Gi :=
1 il 2 g F xk yl y k − F 2 xl 4
(1)
and (g ij ) = (gij )−1 . Gi are called the geodesic coefficients of F . Let σ = σ(t) (a ≤ t ≤ b) be a geodesic on a Finsler manifold (M, F ). Let H(t, s) be a variation of σ such that each curve σs (t) := H(t, s) (a ≤ t ≤ b) is a geodesic. Let J(t) :=
∂H (t, 0). ∂s
Then the vector field J(t) is a Jacobi field along σ satisfying the Jacobi equation: Dσ˙ Dσ˙ J(t) + Rσ˙ J(t) = 0. Here, R denotes the Riemann curvature of F . Locally, for any x ∈ M and y ∈ Tx M \{0}, the Riemann ∂ k curvature Ry = Ri k ∂x is defined by i ⊗ dx Ri k = 2
∂Gi ∂ 2 Gi m ∂Gi ∂Gm ∂ 2 Gi − y + 2Gm m k − m . k m k ∂x ∂x ∂y ∂y ∂y ∂y ∂y k
(2)
The flag curvature of (M, F ) is the function K = K(x, y, P ) of a two-dimensional plane called “flag” P ⊂ Tx M and a “flagpole” y ∈ P \{0} defined by K(x, y, P ) :=
gy (Ry (u), u) , gy (y, y)gy (u, u) − [gy (u, y)]2
where P = span{y, u}. When F is a Riemannian metric, K(x, y, P ) = K(x, P ) is independent of the flagpole y and is just the sectional curvature. A Finsler metric F is said to be of scalar flag curvature if the flag curvature is independent of the flag P , K = K(x, y). F is said to be of weakly isotropic flag curvature if K=
3θ + σ(x), F
where σ = σ(x) is a scalar function and θ is a 1-form on M . When K = constant, F is said to be of constant flag curvature. The flag curvature governs the Jacobi equation and the second variation of length. Due to this, there exists a description of the flag curvature in terms of the dynamics of the geodesic flow. In fact, the generator of the geodesic flow of a Finsler metric is a second order differential equation. It may be observed [12] that, in the much more general context of second order differential equations, there is a natural operator which plays the role of the flag curvature. (α, β)-metrics form an important class of Finsler metrics which can be expressed in the form β , F = αφ α
where α = aij (x)y i y j is a Riemann metric and β = bi (x)y i is a 1-form on a manifold. It has been proved that F = αφ(β/α) is a regular (or almost regular) Finsler metric if and only if βα < b0 (or βα ≤ b0 ) and φ = φ(s) is a positive C ∞ function on (−b0 , b0 ) satisfying the following condition [1,11]:
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φ(s) − sφ (s) + ρ2 − s2 φ (s) > 0,
|s| ≤ ρ < bo .
3
(3)
In particular, when φ = 1 + s, the Finsler metric F = α + β is called a Randers metric. Randers metrics were first introduced by physicist G. Randers in 1941 from the standpoint of general relativity. Randers metrics form an important and ubiquitous class of Finsler metrics with a strong presence in both the theory and applications of Finsler geometry. Studying Randers metrics is an important step to understand general Finsler metrics. Randers metrics can also naturally be characterized as the solution of the Zermelo navigation problem. In fact, a Finsler metric F is a Randers metric if and only if it is the solution of Zermelo navigation problem on a Riemann space (M, h) under the influence of a force field W with |W |h < 1, where |W |h denotes the length of W with respect to Riemannian metric h. Given the
∂ Riemannian metric h = hij (x)y i y j and the vector field W := W i (x) ∂x i with |W |h < 1 on a manifold M , the solution of the following navigation problem
y h x, − W F is just a Randers metric F = α + β, where α = aij =
=1
(4)
aij (x)y i y j and β = bi (x)y i . Explicitly,
Wi W j hij + , λ λ λ
bi = −
Wi . λ
Here, Wi := hij W j and λ := 1 − |W |2h . In this case, we have
F =
λh2 + W02 W0 − , λ λ
W0 := Wi y i .
(5)
We call the couple (h, W ) the navigation data of Randers metric F = α + β. In Finsler geometry, there are some important quantities which all vanish for Riemannian metrics. Hence they are said to be non-Riemannian. Let F be a Finsler metric on an n-dimensional manifold M . Let {bi } be a basis for Tx M and {ω i } be the basis for Tx∗ M dual to {bi }. Define the Busemann–Hausdorff volume form by dVBH := σBH (x)ω 1 ∧ · · · ∧ ω n , where σBH (x) :=
Vol(Bn (1)) . Vol{(y i ) ∈ Rn |F (x, y) < 1}
Here Vol{·} denotes the Euclidean volume function on subsets in Rn .
If F = gij (x)y i y j is a Riemannian metric, then σBH (x) =
det gij (x) .
However, in general, for a Finsler metric F , σBH (x) =
det(gij (x, y)). Define
det(gij (x, y)) . τ (x, y) := ln σBH (x) τ = τ (x, y) is well-defined, which is called the distortion of F . The distortion τ characterizes the geometry of tangent space (Tx M, Fx ). It is well-known that a Finsler metric F is Riemannian if and only if τ (x, y) = 0.
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It is natural to study the rate of change of the distortion along geodesics. For a vector y ∈ Tx M \ {0}, let σ = σ(x) be the geodesic with σ(0) = x and σ˙ = y. Put S(x, y) :=
d τ σ(t), σ(t) ˙ |t=0 . dt
Equivalently, S(x, y) := τ;m (x, y)y m , where “;” denotes the horizontal covariant derivative with respect to F . S = S(x, y) is called the S-curvature of Finsler metric F . The S-curvature S(x, y) measures the rate of change of (Tx M, Fx ) in the direction y ∈ Tx M . It is easy to see that, for any Berwald metric, S = 0. In particular, S = 0 for Riemannian metrics [11]. Hence, S-curvature is a non-Riemannian quantity. We say that F is of isotropic S-curvature if there exists a scalar function c(x) on M such that S(x, y) = (n + 1)c(x)F (x, y), equivalently, τ;m (x, y)y m = (n + 1)c(x). F (x, y)
(6)
Eq. (6) means that the rate of change of the tangent space (Tx M, Fx ) along the direction y ∈ Tx M at each x ∈ M is independent of the direction y but just dependent of the point x. If c(x) =constant, we say that F has constant S-curvature. 2. S-curvature of (α, β)-metrics The S-curvature S = S(x, y) is one of the most important non-Riemannian quantities in Riemann–Finsler geometry. Recent study shows that the S-curvature plays a very important role in Riemann–Finsler geometry (see [7,16]). The following theorem is very important for our classification theorems. Theorem 2.1. (See [5].) Let (M, F ) be an n-dimensional Finsler manifold of scalar flag curvature. Suppose that the S-curvature is isotropic, S = (n + 1)c(x)F , where c(x) is a scalar function on M . Then there is a scalar function σ(x) on M such that K=
3cxm y m + σ(x). F (x, y)
In this case, F is of weakly isotropic flag curvature. The converse of Theorem 2.1 does not hold in general. However, for a Randers metric F = α + β of scalar flag curvature, F is of isotropic S-curvature, S = (n + 1)c(x)F , if and only if F is of weakly isotropic flag curvature, K=
3cxm y m + σ, F
where σ = σ(x) is a scalar function on M (see [9]). For an (α, β)-metric F = αφ(β/α) on an n-dimensional manifold M , let Φ := − Q − sQ {nΔ + 1 + sQ} − b2 − s2 (1 + sQ)Q , where b := βx α and Δ := 1 + sQ + (b2 − s2 )Q and Q := φ /(φ − sφ ). We have the following
(7)
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Lemma 2.2. (See [10].) A regular (α, β)-metric F = αφ(β/α) is a Riemann metric if and only if Φ = 0. Let rij := rj := bi rij ,
1 (bi|j + bj|i ), 2 sj := bi sij ,
sij :=
1 (bi|j − bj|i ), 2
eij := rij + bi sj + bj si ,
where bi|j denote the covariant derivatives of β with respect to α. According to [13], the S-curvature of Randers metric F = α + β is given by e00 − (s0 + ρ0 ) , S = (n + 1) 2F
where ρ := ln 1 − βx 2α and e00 := eij y i y j , s0 := si y i and ρ0 := ρxm y m . In [6], the author and Z. Shen have proved that a Randers metric F = α + β is
of isotropic S-curvature, S = (n + 1)c(x)F , if and only if β satisfies eij = 2c(aij −bi bj ). Further, let F = k1 α2 + k2 β 2 +k3 β be an (α, β)-metric of Randers type, where k1 > 0, k2 and k3 = 0 are constants. We have proved that F is of isotropic S-curvature, S = (n + 1)c(x)F , if and only if β satisfies rij + τ (bi sj + bj si ) = where τ := is given by
k32 k12
(8)
− k2 [7]. More general, we have obtained the formula of S-curvature for (α, β)-metrics which S=
where Ψ :=
2c(x)(1 + k2 b2 )k12 (aij − bi bj ), k3
Q 2Δ
2Ψ −
f (b) Φ (r0 + s0 ) − α−1 (r00 − 2αQs0 ), bf (b) 2Δ2
and π f (b) := π0
sinn−2 t dt
sinn−2 t 0 φ(b cos t)n
dt
.
See [7]. We have characterized the (α, β)-metrics of non-Randers type with isotropic S-curvature in [7]. In particular, for regular (α, β)-metrics, we have the following theorem. Theorem 2.3. (See [7].)
Let F = αφ(β/α) be a regular (α, β)-metric on an n-dimensional manifold M . Suppose that F = k1 α2 + k2 β 2 + k3 β for any constants k1 > 0, k2 and k3 . Then F is of isotropic S-curvature, S = (n + 1)c(x)F , if and only if one of the following holds (a) β satisfies rij = ε b2 aij − bi bj ,
sj = 0,
(9)
where ε = ε(x) is a scalar function, and φ = φ(s) satisfies Φ = −2(n + 1)k
φΔ2 , b2 − s2
where k is a constant. In the case, S = (n + 1)cF with c = kε.
(10)
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(b) β satisfies rij = 0,
sj = 0.
(11)
In the case, S = 0, regardless of the choice of a particular φ. It is easy to see that (11) implies (9), while (9) implies that b := βx α is a constant. In fact, from Propositions 5.3 and 6.3 in [7], we can further obtain the following revised version of Theorem 2.3. Theorem 2.4. A regular (α, β)-metric F = αφ(β/α) of non-Randers type on an n-dimensional manifold M is of isotropic S-curvature, S = (n + 1)cF , if and only if β satisfies rij = 0,
sj = 0.
(12)
In this case, S = 0, regardless of the choice of a particular φ = φ(s). Proof. We just need to prove that case (a) and case (b) in Theorem 2.3 are equivalent under the regularity condition. According to the Propositions 5.3 and 6.3 in [7], if an (α, β)-metric F = αφ(β/α) of non-Randers type is of isotropic S-curvature, S = (n + 1)cF , then β satisfies rij = b2 aij − bi bj ,
sj = 0,
(13)
where = (x) is a scalar function, and φ = φ(s) satisfies b2 − s2 Φ = −2(n + 1)cφΔ2 .
(14)
By the regularity of F = αφ(β/α), b := βα < b0 and |s| ≤ b. By (3), we have φ(s) > 0,
φ(s) − sφ (s) > 0,
∀|s| < b0 .
Hence, letting s approximate b in (14) yields c = 0 because we have Δ(b) = 1 + bQ(b) =
φ(b) >0 φ(b) − bφ (b)
and the regularity of φ on interval (−b0 , b0 ) guarantees that Φ does not approximate the infinity when s approximates b. Thus we conclude that = 0 by (14) again. Then we get (12) from (13). 2 3. Randers metrics of scalar flag curvature It is well-known that a Riemannian metric α of constant sectional curvature μ is locally isometric to the following standard metric αμ on the unit ball B n ⊂ Rn or the whole Rn for μ = −1, 0, +1:
|y|2 − (|x|2 |y|2 − x, y 2 ) , 1 − |x|2 α0 (x, y) = |y|, y ∈ Tx Rn ∼ = Rn ,
|y|2 + (|x|2 |y|2 − x, y 2 ) α+1 (x, y) = , 1 + |x|2
α−1 (x, y) =
y ∈ Tx Bn ∼ = Rn ,
(15) (16)
y ∈ Tx R n ∼ = Rn .
(17)
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Unfortunately, we cannot obtain such concise and nice classification theorem for Finsler metrics of constant/scalar flag curvature in Finsler geometry. Hence, it is one of the most important problems in Finsler geometry to classify Finsler metrics of scalar flag curvature. However, this problem still remains open even in the constant flag curvature case. Up to now, the representative works for this problem focus mainly on Randers metrics or (α, β)-metrics. In 2003, Z. Shen classified locally projectively flat Randers metrics with constant flag curvature and obtain a new family of Randers metrics of negative constant flag curvature [15]. Further, X. Mo, Z. Shen and the author classified locally projectively flat Randers metrics with isotropic S-curvature [5]. On the other hand, motivated by Zermelo’s navigation idea, D. Bao, C. Robles and Z. Shen classified Randers metrics of constant flag curvature by using techniques of algebra, topology and PDE [3]. Note that locally projectively flat Finsler metrics are always of scalar flag curvature and that any Randers metric of constant flag curvature must be of constant S-curvature [2]. Hence we are in the position to give the following classification theorem for Randers metrics of scalar flag curvature whose S-curvatures are isotropic (the class of such kind of Randers metrics contains all projectively flat Randers metrics with isotropic S-curvature and Randers metrics of constant flag curvature). Theorem 3.1. (See [8].) Let F = α+β be a Randers metric on a manifold M of dimension n with navigation data (h, W ). Assume that n ≥ 3. Then F is of scalar flag curvature K = K(x, y) and of isotropic S-curvature, S = (n + 1)cF , if and only if at any point, there is a local coordinate system in which h, c and W are given by
|y|2 + μ(|x|2 |y|2 − x, y 2 ) h= , 1 + μ|x|2
(18)
δ + a, x c=
, 1 + μ|x|2
|x|2 a 2 + xQ + b + μ b, x x, W = −2 δ 1 + μ|x| + a, x x −
1 + μ|x|2 + 1
(19) (20)
where δ, μ are constants, Q = (qj i ) is an anti-symmetric matrix and a, b are constant vectors in Rn . In this case, the flag curvature is given by K=
3cm y m + σ, F
that is, F is of weakly isotropic flag curvature. Here σ = μ − c2 − 2cm W m . This classification theorem extends all classification results on Randers metrics before. Example 3.1. In (18)–(20), let μ = 0, δ = 0, Q = 0 and b = 0. We get h = |y|,
c = a, x ,
W = −2 a, x x + |x|2 a.
With the above navigation data (h, W ), the Randers metric in (5) is given by
(1 − |a|2 |x|4 )|y|2 + (|x|2 a, y − 2 a, x x, y )2 F = 1 − |a|2 |x|4 −
|x|2 a, y − 2 a, x x, y . 1 − |a|2 |x|4
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By a direct computation, one can easily verify that F is of isotropic S-curvature and scalar flag curvature, namely, S = (n + 1)cF,
K=
3cm y m + σ, F
where c = a, x and σ = 3 a, x 2 − 2|a|2 |x|2 . 4. (α, β)-metrics of scalar flag curvature Now we consider (α, β)-metrics of non-Randers type of scalar flag curvature whose S-curvatures are isotropic. By Theorem 2.1, this class of (α, β)-metrics must be of weakly isotropic flag curvature. Assume that a Finsler metric F is of scalar flag curvature K = K(x, y). We have the following Bianchi identity [11] Ji;m y m + KF 2 Ii = −
n+1 2 F K·i , 3
(21)
where I := Ii dxi and J := Ji dxi denote the mean Cartan torsion and the mean Landsberg curvature of F respectively and Ji := Ii;m y m . Here, K·i := Kyi . Based on (21), we can first get the following lemma. Lemma 4.1. (See [4].) Let F = αφ(β/α) be an (α, β)-metric on an n-dimensional manifold M (n ≥ 3).
Suppose that F is not a Finsler metric of Randers type, that is F = k1 α2 + k2 β 2 + k3 β, where k1 > 0, k2 and k3 are scalar functions on M . If F is of scalar flag curvature and of constant S-curvature, then β must be closed. Further, we can prove the following theorem. Theorem 4.2. (See [4].) Let F = αφ(β/α) be a regular (α, β)-metric on an n-dimensional manifold M (n ≥ 3). Suppose that F is not a Finsler metric of Randers type. Then F is a Finsler metric of scalar flag curvature with vanishing S-curvature if and only if the flag curvature K = 0 and F is a Berwald metric. In this case, F is a locally Minkowski metric. We must mention that Theorem 4.2 does not hold any longer for Finsler metrics of Randers type. We can find Randers metrics with K = 0 and S = 0 which are not Berwaldian. Example 4.1. (See [14].) Let n ≥ 2 and Ω := x = (x, y, x ¯) ∈ R2 × Rn−2 | x2 + y 2 < 1 , where x ¯ ∈ Rn−2 . For any y = (u, v, y¯) ∈ Tx Ω = Rn and x = (x, y, x ¯) ∈ Ω, define
F (x, y) :=
(−yu + xv)2 + |y|2 (1 − x2 − y 2 ) − (−yu + xv) , 1 − x2 − y 2
(22)
where | · | denotes the standard Euclidean norm. The Finsler metric F is a Randers metric on Ω. It is easy to show that K = 0 and S = 0 for F . However, F is not a locally Minkowski metric, and then, F is not a Berwald metric. Now, from Theorem 2.4 and Theorem 4.2, we are in the position to obtain the following classification theorem for regular (α, β)-metrics of non-Randers type of scalar flag curvature whose S-curvatures are isotropic.
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Theorem 4.3. Let F = αφ(β/α) be a regular (α, β)-metric of non-Randers type on an n-dimensional manifold M (n ≥ 3). Then F is of scalar flag curvature and of isotropic S-curvature if and only if the flag curvature K = 0 and F is a Berwald metric. In this case, F is a locally Minkowski metric. In sum, by Theorem 3.1 and Theorem 4.3, we have classified the regular (α, β)-metrics of scalar flag curvature on an n-dimensional manifold M (n ≥ 3) under the extra condition that S-curvatures are isotropic. If we regard locally Minkowski metrics as so-called trivial (α, β)-metrics, then, by Theorem 3.1 and Theorem 4.3, we have the following theorem. Theorem 4.4. The non-trivial regular (α, β)-metrics of scalar flag curvature with isotropic S-curvature on an n-dimensional manifold M (n ≥ 3) must be Randers metrics. In this case, the metrics are of weakly isotropic flag curvature and are determined by navigation data (h, W ) and (18), (19) and (20). For our final aim, we have the following open problems: (1) (S.S. Chern, 1995) Develop Minkowski geometry, or more generally, Finsler geometry of constant flag curvature. Develop theory of discontinuous groups of isometries. (2) Classify Randers metrics of scalar flag curvature. (3) Classify (α, β)-metrics of scalar flag curvature. Finally, we must say that the present knowledge of existence and classification of Finsler metrics of scalar/constant flag curvature is still very preliminary. References [1] S. Bácsó, X. Cheng, Z. Shen, Curvature properties of (α, β)-metrics, in: Advanced Studies in Pure Mathematics, vol. 48, Math. Soc. of Japan, 2007, pp. 73–110. [2] D. Bao, C. Robles, Ricci and flag curvatures in Finsler geometry, in: A Sampler of Riemann–Finsler Geometry, in: Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, 2004, pp. 197–259. [3] D. Bao, C. Robles, Z. Shen, Zermelo navigation on Riemannian manifolds, J. Differ. Geom. 66 (2004) 377–435. [4] X. Cheng, On (α, β)-metrics of scalar flag curvature with constant S-curvature, Acta Math. Sin. Engl. Ser. 26 (9) (2010) 1701–1708. [5] X. Cheng, X. Mo, Z. Shen, On the flag curvature of Finsler metrics of scalar curvature, J. Lond. Math. Soc. 68 (2) (2003) 762–780. [6] X. Chen(g), Z. Shen, Randers metrics with special curvature properties, Osaka J. Math. 40 (2003) 87–101. [7] X. Cheng, Z. Shen, A class of Finsler metrics with isotropic S-curvature, Isr. J. Math. 169 (1) (2009) 317–340. [8] X. Cheng, Z. Shen, Randers metrics of scalar flag curvature, J. Aust. Math. Soc. 87 (2009) 359–370. [9] X. Cheng, Z. Shen, Finsler Geometry – an approach via Randers spaces, Springer, Heidelberg and Science Press, Beijing, 2012. [10] X. Cheng, H. Wang, M. Wang, (α, β)-metrics with relatively isotropic mean Landsberg curvature, Publ. Math. (Debr.) 72 (3–4) (2008) 475–485. [11] S.S. Chern, Z. Shen, Rieman–Finsler Geometry, Nankai Tracts in Mathematics, vol. 6, World Scientific, 2005. [12] P. Foulon, Géométrie des équations différentielles du second ordre, Ann. Inst. Henri Poincaré, a Phys. Théor. 45 (1986) 1–28. [13] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic, Dordrecht, 2001. [14] Z. Shen, Finsler metrics with K = 0 and S = 0, Can. J. Math. 55 (1) (2003) 112–132. [15] Z. Shen, Projectively flat Randers metrics with constant flag curvature, Math. Ann. 325 (2003) 19–30. [16] Z. Shen, Finsler manifolds with nonpositive flag curvature and constant S-curvature, Math. Z. 249 (3) (2005) 625–639.
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Differential Geometry and its Applications www.elsevier.com/locate/difgeo
The (α, β)-metrics of scalar flag curvature Xinyue Cheng 1 School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, PR China
a r t i c l e
i n f o
Article history: Received 10 November 2013 Received in revised form 6 February 2014 Available online xxxx Communicated by Z. Shen MSC: 53B40 53C60
a b s t r a c t One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study and characterize the (α, β)-metrics of scalar flag curvature. When the dimension of the manifold is greater than 2, we classify Randers metrics of weakly isotropic flag curvature (that is, Randers metrics of scalar flag curvature with isotropic S-curvature). Further, we characterize and classify (α, β)-metrics of scalar flag curvature with isotropic S-curvature. Finally, we conclude that the non-trivial regular (α, β)-metrics of scalar flag curvature with isotropic S-curvature on an n-dimensional manifold M (n ≥ 3) must be Randers metrics. © 2014 Elsevier B.V. All rights reserved.
Keywords: Finsler metric (α, β)-metric Randers metric Flag curvature S-curvature
1. Introduction Let F be a Finsler metric on an n-dimensional C ∞ manifold M . Put gij (x, y) :=
1 2 F (x, y) yi yj . 2
∂ For any non-zero vector y = y i ∂x i ∈ Tx M , F induces an inner product gy on Tx M as follows
gy (u, v) := gij (x, y)ui v j , ∂ i ∂ where u = ui ∂x i ∈ Tx M , v = v ∂xi ∈ Tx M . The geodesics of F are characterized locally by the following system of second order ordinary differential equations
E-mail address: [email protected]. Supported by the National Natural Science Foundation of China (11371386) and European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 317721. 1
http://dx.doi.org/10.1016/j.difgeo.2014.04.011 0926-2245/© 2014 Elsevier B.V. All rights reserved.
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dx d2 xi (t) i x(t), = 0, + 2G dt2 dt where Gi :=
1 il 2 g F xk yl y k − F 2 xl 4
(1)
and (g ij ) = (gij )−1 . Gi are called the geodesic coefficients of F . Let σ = σ(t) (a ≤ t ≤ b) be a geodesic on a Finsler manifold (M, F ). Let H(t, s) be a variation of σ such that each curve σs (t) := H(t, s) (a ≤ t ≤ b) is a geodesic. Let J(t) :=
∂H (t, 0). ∂s
Then the vector field J(t) is a Jacobi field along σ satisfying the Jacobi equation: Dσ˙ Dσ˙ J(t) + Rσ˙ J(t) = 0. Here, R denotes the Riemann curvature of F . Locally, for any x ∈ M and y ∈ Tx M \{0}, the Riemann ∂ k curvature Ry = Ri k ∂x is defined by i ⊗ dx Ri k = 2
∂Gi ∂ 2 Gi m ∂Gi ∂Gm ∂ 2 Gi − y + 2Gm m k − m . k m k ∂x ∂x ∂y ∂y ∂y ∂y ∂y k
(2)
The flag curvature of (M, F ) is the function K = K(x, y, P ) of a two-dimensional plane called “flag” P ⊂ Tx M and a “flagpole” y ∈ P \{0} defined by K(x, y, P ) :=
gy (Ry (u), u) , gy (y, y)gy (u, u) − [gy (u, y)]2
where P = span{y, u}. When F is a Riemannian metric, K(x, y, P ) = K(x, P ) is independent of the flagpole y and is just the sectional curvature. A Finsler metric F is said to be of scalar flag curvature if the flag curvature is independent of the flag P , K = K(x, y). F is said to be of weakly isotropic flag curvature if K=
3θ + σ(x), F
where σ = σ(x) is a scalar function and θ is a 1-form on M . When K = constant, F is said to be of constant flag curvature. The flag curvature governs the Jacobi equation and the second variation of length. Due to this, there exists a description of the flag curvature in terms of the dynamics of the geodesic flow. In fact, the generator of the geodesic flow of a Finsler metric is a second order differential equation. It may be observed [12] that, in the much more general context of second order differential equations, there is a natural operator which plays the role of the flag curvature. (α, β)-metrics form an important class of Finsler metrics which can be expressed in the form β , F = αφ α
where α = aij (x)y i y j is a Riemann metric and β = bi (x)y i is a 1-form on a manifold. It has been proved that F = αφ(β/α) is a regular (or almost regular) Finsler metric if and only if βα < b0 (or βα ≤ b0 ) and φ = φ(s) is a positive C ∞ function on (−b0 , b0 ) satisfying the following condition [1,11]:
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φ(s) − sφ (s) + ρ2 − s2 φ (s) > 0,
|s| ≤ ρ < bo .
3
(3)
In particular, when φ = 1 + s, the Finsler metric F = α + β is called a Randers metric. Randers metrics were first introduced by physicist G. Randers in 1941 from the standpoint of general relativity. Randers metrics form an important and ubiquitous class of Finsler metrics with a strong presence in both the theory and applications of Finsler geometry. Studying Randers metrics is an important step to understand general Finsler metrics. Randers metrics can also naturally be characterized as the solution of the Zermelo navigation problem. In fact, a Finsler metric F is a Randers metric if and only if it is the solution of Zermelo navigation problem on a Riemann space (M, h) under the influence of a force field W with |W |h < 1, where |W |h denotes the length of W with respect to Riemannian metric h. Given the
∂ Riemannian metric h = hij (x)y i y j and the vector field W := W i (x) ∂x i with |W |h < 1 on a manifold M , the solution of the following navigation problem
y h x, − W F is just a Randers metric F = α + β, where α = aij =
=1
(4)
aij (x)y i y j and β = bi (x)y i . Explicitly,
Wi W j hij + , λ λ λ
bi = −
Wi . λ
Here, Wi := hij W j and λ := 1 − |W |2h . In this case, we have
F =
λh2 + W02 W0 − , λ λ
W0 := Wi y i .
(5)
We call the couple (h, W ) the navigation data of Randers metric F = α + β. In Finsler geometry, there are some important quantities which all vanish for Riemannian metrics. Hence they are said to be non-Riemannian. Let F be a Finsler metric on an n-dimensional manifold M . Let {bi } be a basis for Tx M and {ω i } be the basis for Tx∗ M dual to {bi }. Define the Busemann–Hausdorff volume form by dVBH := σBH (x)ω 1 ∧ · · · ∧ ω n , where σBH (x) :=
Vol(Bn (1)) . Vol{(y i ) ∈ Rn |F (x, y) < 1}
Here Vol{·} denotes the Euclidean volume function on subsets in Rn .
If F = gij (x)y i y j is a Riemannian metric, then σBH (x) =
det gij (x) .
However, in general, for a Finsler metric F , σBH (x) =
det(gij (x, y)). Define
det(gij (x, y)) . τ (x, y) := ln σBH (x) τ = τ (x, y) is well-defined, which is called the distortion of F . The distortion τ characterizes the geometry of tangent space (Tx M, Fx ). It is well-known that a Finsler metric F is Riemannian if and only if τ (x, y) = 0.
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It is natural to study the rate of change of the distortion along geodesics. For a vector y ∈ Tx M \ {0}, let σ = σ(x) be the geodesic with σ(0) = x and σ˙ = y. Put S(x, y) :=
d τ σ(t), σ(t) ˙ |t=0 . dt
Equivalently, S(x, y) := τ;m (x, y)y m , where “;” denotes the horizontal covariant derivative with respect to F . S = S(x, y) is called the S-curvature of Finsler metric F . The S-curvature S(x, y) measures the rate of change of (Tx M, Fx ) in the direction y ∈ Tx M . It is easy to see that, for any Berwald metric, S = 0. In particular, S = 0 for Riemannian metrics [11]. Hence, S-curvature is a non-Riemannian quantity. We say that F is of isotropic S-curvature if there exists a scalar function c(x) on M such that S(x, y) = (n + 1)c(x)F (x, y), equivalently, τ;m (x, y)y m = (n + 1)c(x). F (x, y)
(6)
Eq. (6) means that the rate of change of the tangent space (Tx M, Fx ) along the direction y ∈ Tx M at each x ∈ M is independent of the direction y but just dependent of the point x. If c(x) =constant, we say that F has constant S-curvature. 2. S-curvature of (α, β)-metrics The S-curvature S = S(x, y) is one of the most important non-Riemannian quantities in Riemann–Finsler geometry. Recent study shows that the S-curvature plays a very important role in Riemann–Finsler geometry (see [7,16]). The following theorem is very important for our classification theorems. Theorem 2.1. (See [5].) Let (M, F ) be an n-dimensional Finsler manifold of scalar flag curvature. Suppose that the S-curvature is isotropic, S = (n + 1)c(x)F , where c(x) is a scalar function on M . Then there is a scalar function σ(x) on M such that K=
3cxm y m + σ(x). F (x, y)
In this case, F is of weakly isotropic flag curvature. The converse of Theorem 2.1 does not hold in general. However, for a Randers metric F = α + β of scalar flag curvature, F is of isotropic S-curvature, S = (n + 1)c(x)F , if and only if F is of weakly isotropic flag curvature, K=
3cxm y m + σ, F
where σ = σ(x) is a scalar function on M (see [9]). For an (α, β)-metric F = αφ(β/α) on an n-dimensional manifold M , let Φ := − Q − sQ {nΔ + 1 + sQ} − b2 − s2 (1 + sQ)Q , where b := βx α and Δ := 1 + sQ + (b2 − s2 )Q and Q := φ /(φ − sφ ). We have the following
(7)
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Lemma 2.2. (See [10].) A regular (α, β)-metric F = αφ(β/α) is a Riemann metric if and only if Φ = 0. Let rij := rj := bi rij ,
1 (bi|j + bj|i ), 2 sj := bi sij ,
sij :=
1 (bi|j − bj|i ), 2
eij := rij + bi sj + bj si ,
where bi|j denote the covariant derivatives of β with respect to α. According to [13], the S-curvature of Randers metric F = α + β is given by e00 − (s0 + ρ0 ) , S = (n + 1) 2F
where ρ := ln 1 − βx 2α and e00 := eij y i y j , s0 := si y i and ρ0 := ρxm y m . In [6], the author and Z. Shen have proved that a Randers metric F = α + β is
of isotropic S-curvature, S = (n + 1)c(x)F , if and only if β satisfies eij = 2c(aij −bi bj ). Further, let F = k1 α2 + k2 β 2 +k3 β be an (α, β)-metric of Randers type, where k1 > 0, k2 and k3 = 0 are constants. We have proved that F is of isotropic S-curvature, S = (n + 1)c(x)F , if and only if β satisfies rij + τ (bi sj + bj si ) = where τ := is given by
k32 k12
(8)
− k2 [7]. More general, we have obtained the formula of S-curvature for (α, β)-metrics which S=
where Ψ :=
2c(x)(1 + k2 b2 )k12 (aij − bi bj ), k3
Q 2Δ
2Ψ −
f (b) Φ (r0 + s0 ) − α−1 (r00 − 2αQs0 ), bf (b) 2Δ2
and π f (b) := π0
sinn−2 t dt
sinn−2 t 0 φ(b cos t)n
dt
.
See [7]. We have characterized the (α, β)-metrics of non-Randers type with isotropic S-curvature in [7]. In particular, for regular (α, β)-metrics, we have the following theorem. Theorem 2.3. (See [7].)
Let F = αφ(β/α) be a regular (α, β)-metric on an n-dimensional manifold M . Suppose that F = k1 α2 + k2 β 2 + k3 β for any constants k1 > 0, k2 and k3 . Then F is of isotropic S-curvature, S = (n + 1)c(x)F , if and only if one of the following holds (a) β satisfies rij = ε b2 aij − bi bj ,
sj = 0,
(9)
where ε = ε(x) is a scalar function, and φ = φ(s) satisfies Φ = −2(n + 1)k
φΔ2 , b2 − s2
where k is a constant. In the case, S = (n + 1)cF with c = kε.
(10)
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(b) β satisfies rij = 0,
sj = 0.
(11)
In the case, S = 0, regardless of the choice of a particular φ. It is easy to see that (11) implies (9), while (9) implies that b := βx α is a constant. In fact, from Propositions 5.3 and 6.3 in [7], we can further obtain the following revised version of Theorem 2.3. Theorem 2.4. A regular (α, β)-metric F = αφ(β/α) of non-Randers type on an n-dimensional manifold M is of isotropic S-curvature, S = (n + 1)cF , if and only if β satisfies rij = 0,
sj = 0.
(12)
In this case, S = 0, regardless of the choice of a particular φ = φ(s). Proof. We just need to prove that case (a) and case (b) in Theorem 2.3 are equivalent under the regularity condition. According to the Propositions 5.3 and 6.3 in [7], if an (α, β)-metric F = αφ(β/α) of non-Randers type is of isotropic S-curvature, S = (n + 1)cF , then β satisfies rij = b2 aij − bi bj ,
sj = 0,
(13)
where = (x) is a scalar function, and φ = φ(s) satisfies b2 − s2 Φ = −2(n + 1)cφΔ2 .
(14)
By the regularity of F = αφ(β/α), b := βα < b0 and |s| ≤ b. By (3), we have φ(s) > 0,
φ(s) − sφ (s) > 0,
∀|s| < b0 .
Hence, letting s approximate b in (14) yields c = 0 because we have Δ(b) = 1 + bQ(b) =
φ(b) >0 φ(b) − bφ (b)
and the regularity of φ on interval (−b0 , b0 ) guarantees that Φ does not approximate the infinity when s approximates b. Thus we conclude that = 0 by (14) again. Then we get (12) from (13). 2 3. Randers metrics of scalar flag curvature It is well-known that a Riemannian metric α of constant sectional curvature μ is locally isometric to the following standard metric αμ on the unit ball B n ⊂ Rn or the whole Rn for μ = −1, 0, +1:
|y|2 − (|x|2 |y|2 − x, y 2 ) , 1 − |x|2 α0 (x, y) = |y|, y ∈ Tx Rn ∼ = Rn ,
|y|2 + (|x|2 |y|2 − x, y 2 ) α+1 (x, y) = , 1 + |x|2
α−1 (x, y) =
y ∈ Tx Bn ∼ = Rn ,
(15) (16)
y ∈ Tx R n ∼ = Rn .
(17)
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Unfortunately, we cannot obtain such concise and nice classification theorem for Finsler metrics of constant/scalar flag curvature in Finsler geometry. Hence, it is one of the most important problems in Finsler geometry to classify Finsler metrics of scalar flag curvature. However, this problem still remains open even in the constant flag curvature case. Up to now, the representative works for this problem focus mainly on Randers metrics or (α, β)-metrics. In 2003, Z. Shen classified locally projectively flat Randers metrics with constant flag curvature and obtain a new family of Randers metrics of negative constant flag curvature [15]. Further, X. Mo, Z. Shen and the author classified locally projectively flat Randers metrics with isotropic S-curvature [5]. On the other hand, motivated by Zermelo’s navigation idea, D. Bao, C. Robles and Z. Shen classified Randers metrics of constant flag curvature by using techniques of algebra, topology and PDE [3]. Note that locally projectively flat Finsler metrics are always of scalar flag curvature and that any Randers metric of constant flag curvature must be of constant S-curvature [2]. Hence we are in the position to give the following classification theorem for Randers metrics of scalar flag curvature whose S-curvatures are isotropic (the class of such kind of Randers metrics contains all projectively flat Randers metrics with isotropic S-curvature and Randers metrics of constant flag curvature). Theorem 3.1. (See [8].) Let F = α+β be a Randers metric on a manifold M of dimension n with navigation data (h, W ). Assume that n ≥ 3. Then F is of scalar flag curvature K = K(x, y) and of isotropic S-curvature, S = (n + 1)cF , if and only if at any point, there is a local coordinate system in which h, c and W are given by
|y|2 + μ(|x|2 |y|2 − x, y 2 ) h= , 1 + μ|x|2
(18)
δ + a, x c=
, 1 + μ|x|2
|x|2 a 2 + xQ + b + μ b, x x, W = −2 δ 1 + μ|x| + a, x x −
1 + μ|x|2 + 1
(19) (20)
where δ, μ are constants, Q = (qj i ) is an anti-symmetric matrix and a, b are constant vectors in Rn . In this case, the flag curvature is given by K=
3cm y m + σ, F
that is, F is of weakly isotropic flag curvature. Here σ = μ − c2 − 2cm W m . This classification theorem extends all classification results on Randers metrics before. Example 3.1. In (18)–(20), let μ = 0, δ = 0, Q = 0 and b = 0. We get h = |y|,
c = a, x ,
W = −2 a, x x + |x|2 a.
With the above navigation data (h, W ), the Randers metric in (5) is given by
(1 − |a|2 |x|4 )|y|2 + (|x|2 a, y − 2 a, x x, y )2 F = 1 − |a|2 |x|4 −
|x|2 a, y − 2 a, x x, y . 1 − |a|2 |x|4
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By a direct computation, one can easily verify that F is of isotropic S-curvature and scalar flag curvature, namely, S = (n + 1)cF,
K=
3cm y m + σ, F
where c = a, x and σ = 3 a, x 2 − 2|a|2 |x|2 . 4. (α, β)-metrics of scalar flag curvature Now we consider (α, β)-metrics of non-Randers type of scalar flag curvature whose S-curvatures are isotropic. By Theorem 2.1, this class of (α, β)-metrics must be of weakly isotropic flag curvature. Assume that a Finsler metric F is of scalar flag curvature K = K(x, y). We have the following Bianchi identity [11] Ji;m y m + KF 2 Ii = −
n+1 2 F K·i , 3
(21)
where I := Ii dxi and J := Ji dxi denote the mean Cartan torsion and the mean Landsberg curvature of F respectively and Ji := Ii;m y m . Here, K·i := Kyi . Based on (21), we can first get the following lemma. Lemma 4.1. (See [4].) Let F = αφ(β/α) be an (α, β)-metric on an n-dimensional manifold M (n ≥ 3).
Suppose that F is not a Finsler metric of Randers type, that is F = k1 α2 + k2 β 2 + k3 β, where k1 > 0, k2 and k3 are scalar functions on M . If F is of scalar flag curvature and of constant S-curvature, then β must be closed. Further, we can prove the following theorem. Theorem 4.2. (See [4].) Let F = αφ(β/α) be a regular (α, β)-metric on an n-dimensional manifold M (n ≥ 3). Suppose that F is not a Finsler metric of Randers type. Then F is a Finsler metric of scalar flag curvature with vanishing S-curvature if and only if the flag curvature K = 0 and F is a Berwald metric. In this case, F is a locally Minkowski metric. We must mention that Theorem 4.2 does not hold any longer for Finsler metrics of Randers type. We can find Randers metrics with K = 0 and S = 0 which are not Berwaldian. Example 4.1. (See [14].) Let n ≥ 2 and Ω := x = (x, y, x ¯) ∈ R2 × Rn−2 | x2 + y 2 < 1 , where x ¯ ∈ Rn−2 . For any y = (u, v, y¯) ∈ Tx Ω = Rn and x = (x, y, x ¯) ∈ Ω, define
F (x, y) :=
(−yu + xv)2 + |y|2 (1 − x2 − y 2 ) − (−yu + xv) , 1 − x2 − y 2
(22)
where | · | denotes the standard Euclidean norm. The Finsler metric F is a Randers metric on Ω. It is easy to show that K = 0 and S = 0 for F . However, F is not a locally Minkowski metric, and then, F is not a Berwald metric. Now, from Theorem 2.4 and Theorem 4.2, we are in the position to obtain the following classification theorem for regular (α, β)-metrics of non-Randers type of scalar flag curvature whose S-curvatures are isotropic.
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Theorem 4.3. Let F = αφ(β/α) be a regular (α, β)-metric of non-Randers type on an n-dimensional manifold M (n ≥ 3). Then F is of scalar flag curvature and of isotropic S-curvature if and only if the flag curvature K = 0 and F is a Berwald metric. In this case, F is a locally Minkowski metric. In sum, by Theorem 3.1 and Theorem 4.3, we have classified the regular (α, β)-metrics of scalar flag curvature on an n-dimensional manifold M (n ≥ 3) under the extra condition that S-curvatures are isotropic. If we regard locally Minkowski metrics as so-called trivial (α, β)-metrics, then, by Theorem 3.1 and Theorem 4.3, we have the following theorem. Theorem 4.4. The non-trivial regular (α, β)-metrics of scalar flag curvature with isotropic S-curvature on an n-dimensional manifold M (n ≥ 3) must be Randers metrics. In this case, the metrics are of weakly isotropic flag curvature and are determined by navigation data (h, W ) and (18), (19) and (20). For our final aim, we have the following open problems: (1) (S.S. Chern, 1995) Develop Minkowski geometry, or more generally, Finsler geometry of constant flag curvature. Develop theory of discontinuous groups of isometries. (2) Classify Randers metrics of scalar flag curvature. (3) Classify (α, β)-metrics of scalar flag curvature. Finally, we must say that the present knowledge of existence and classification of Finsler metrics of scalar/constant flag curvature is still very preliminary. References [1] S. Bácsó, X. Cheng, Z. Shen, Curvature properties of (α, β)-metrics, in: Advanced Studies in Pure Mathematics, vol. 48, Math. Soc. of Japan, 2007, pp. 73–110. [2] D. Bao, C. Robles, Ricci and flag curvatures in Finsler geometry, in: A Sampler of Riemann–Finsler Geometry, in: Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, 2004, pp. 197–259. [3] D. Bao, C. Robles, Z. Shen, Zermelo navigation on Riemannian manifolds, J. Differ. Geom. 66 (2004) 377–435. [4] X. Cheng, On (α, β)-metrics of scalar flag curvature with constant S-curvature, Acta Math. Sin. Engl. Ser. 26 (9) (2010) 1701–1708. [5] X. Cheng, X. Mo, Z. Shen, On the flag curvature of Finsler metrics of scalar curvature, J. Lond. Math. Soc. 68 (2) (2003) 762–780. [6] X. Chen(g), Z. Shen, Randers metrics with special curvature properties, Osaka J. Math. 40 (2003) 87–101. [7] X. Cheng, Z. Shen, A class of Finsler metrics with isotropic S-curvature, Isr. J. Math. 169 (1) (2009) 317–340. [8] X. Cheng, Z. Shen, Randers metrics of scalar flag curvature, J. Aust. Math. Soc. 87 (2009) 359–370. [9] X. Cheng, Z. Shen, Finsler Geometry – an approach via Randers spaces, Springer, Heidelberg and Science Press, Beijing, 2012. [10] X. Cheng, H. Wang, M. Wang, (α, β)-metrics with relatively isotropic mean Landsberg curvature, Publ. Math. (Debr.) 72 (3–4) (2008) 475–485. [11] S.S. Chern, Z. Shen, Rieman–Finsler Geometry, Nankai Tracts in Mathematics, vol. 6, World Scientific, 2005. [12] P. Foulon, Géométrie des équations différentielles du second ordre, Ann. Inst. Henri Poincaré, a Phys. Théor. 45 (1986) 1–28. [13] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic, Dordrecht, 2001. [14] Z. Shen, Finsler metrics with K = 0 and S = 0, Can. J. Math. 55 (1) (2003) 112–132. [15] Z. Shen, Projectively flat Randers metrics with constant flag curvature, Math. Ann. 325 (2003) 19–30. [16] Z. Shen, Finsler manifolds with nonpositive flag curvature and constant S-curvature, Math. Z. 249 (3) (2005) 625–639.