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Special projective Lichnérowicz–Obata theorem for Randers spaces
C. R. Acad. Sci. Paris, Ser. I 351 (2013) 927–930
Contents lists available at ScienceDirect
C. R. Acad. Sci. Paris, Ser. I www.sciencedirect.com
Differential geometry
Special projective Lichnérowicz–Obata theorem for Randers spaces Le théorème projectif resteint de Lichnérowicz–Obata sur les espaces de Randers Mehdi Rafie-Rad a,b a b
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P.O. Box 47416-1467, Babolsar, Iran
a r t i c l e
i n f o
Article history: Received 24 July 2013 Accepted after revision 16 October 2013 Available online 13 November 2013 Presented by the Editorial Board
a b s t r a c t It is proved that either every special projective vector field V on a Randers space ( M , F = α + β) is a conformal vector field of the Riemannian metric α 2 − β 2 , or F is of isotropic S-curvature. This result is applied to establish a projective Lichnérowicz–Obata-type result on the closed manifolds with generic Randers metrics. © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
r é s u m é On prouve que, soit chaque champ projectif de vecteurs sur un espace de Randers ( M , F = α + β) est conforme à la métrique riemanienne α 2 − β 2 , soit F est à S-courbure isotrope. Ce résultat est appliqué à l’établissement d’un théorème de type de Lichnérowicz–Obata sur les variétés fermées de Randers. © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
1. Introduction The projective Lichnérowicz–Obata theorem in Riemannian geometry has been recently extended to closed Randers spaces in [4], cf. Corollary 1.5. However, this result seems to be incomplete since, unlike its Riemannian prototype, it does not imply the positivity of the flag curvature of the metric. A suggestion for arriving to the case of positive flag curvature is to consider only a sub-class of projective geometry in order to establish a reduced Lichnérowicz–Obata-type theorem. As it will be presented, the special projective geometry—which has been recently discussed in [5–7] for Randers metrics—is a good candidate for such a purpose, since this is an immediate extension of the Riemannian projective geometry. The results would imply that the special projective Randers geometry may refer to study one of the following cases: (a) conformal transformations of an appropriate Riemannian space, (b) isometries of a Randers space, or (c) Randers spaces of isotropic S-curvature. We prove the result for the pure Randers metrics: Theorem 1.1. Let us suppose that ( M , F = α + β) is a Randers space of dimension n 2. Then, at least one of the following statements holds:
E-mail address: rafi[email protected]. 1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.crma.2013.10.012
928
M. Rafie-Rad / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 927–930
(i) every special projective vector field on ( M , F ) is a conformal vector field of the Riemannian metric α 2 − β 2 , (ii) F is of isotropic S-curvature. Theorem 1.1 implies the following result: Theorem 1.2. Let us suppose that ( M , F = α + β) is a closed and connected Randers space of dimension n 2 and V is a special projective vector field of F . Then, at least one of the following statements holds: (i) V is a conformal vector field for the Riemannian metric α 2 − β 2 , (ii) there is a Randers metric Fˆ such that V is a Killing vector field for Fˆ , (iii) after an appropriate rescaling, F is of the following local form:
F (x, y ) =
| y |2 + |x|2 | y |2 − x, y 2 f k yk − x , 2 1 + |x| 1 − f 2 (x)
y ∈ TxM ∼ = Rn ,
(1)
where f is an eigenfunction of the standard Laplacian satisfying f = nf and maxx∈ M | f | < 1. In particular, F is of positive flag curvature. All manifolds are assumed to be smooth and connected, the natural coordinates on the tangent manifold T M are denoted by (xi , y i ) and the derivations with respect to xk and yk are denoted by the subscripts xk and yk , respectively. Moreover, we deal with pure and positive definite Randers metrics. 2. Special projective Finsler geometry Two Finsler metrics F and F˜ on M are said to be projectively equivalent if they have the same forward geodesics. A Finsler metric F is said to be locally projectively flat if, at any point x ∈ M, there is a neighborhood U such that F and the Euclidean metric are projectively equivalent on U . Given a Finsler space ( M , F ), a diffeomorphism φ : M −→ M is called a projective transformation if F andφ ∗ F are projectively equivalent.
α = ai j y i y j is a Riemannian metric and β = bi (x) y i is a 1-form defined on M such that βx := sup y ∈ T x M \{0} β( y )/α ( y ) < 1. Then the function F = α + β is a Finsler metric on M, which is called a Randers metric. The i and G i , and the Levi-Civita connection of α is geodesic spray coefficients of α and F are denoted respectively by the G α Suppose that
denoted by ∇ . The covariant derivation of β is given by (∇ j b i ) dx j := db i − b j θi j , where θi j := Γ˜ik dxk denote the associj
ated connection forms. Let us stipulate the following conventions: r i j :=
s j := b i si j and e i j := r i j + b i s j + b j si , e 00 := e i j y i y j , s0 := si y i and si 0 := are of the following form:
i Gi = Gα +
e 00 2F
− s0 y i + α si 0 .
1 (∇ j bi + ∇i b j ), si j := 12 (∇ j bi − ∇i b j ), si j := aih shj , 2 i s j y j . Then the geodesic spray coefficients G i of F
(2)
It is well known that a Randers metric F = α + β on M is locally projectively flat if and only if α is of constant sectional curvature and if β is closed. The locally projectively flat Randers metrics with isotropic S-curvature has been characterized by Chen, Mo and Shen in [2], cf. Theorem 1.3 and Theorem 1.4. A projective transformation φ : M −→ M is said to be special if it preserves the E-curvature; in this case, φ changes the geodesic spray coefficients as G˜ i (x, y ) = G i (x, y ) + P (x, y ) y i , where P = P i (x) y i . The complete lift of any vector field V on
i M is given by Vˆ = V i ∂ i + yk ∂ Vk ∂ i . The Lie derivative operator with respect to the vector field V is denoted by L Vˆ . It is
∂x
∂x ∂ y
well known that, L Vˆ y i = 0, L Vˆ dxi = 0 and the differential operators L Vˆ , ∂ i , the exterior differential operator d and ∂ i ∂x ∂y commute within any natural coordinates system on tangent manifold. The vector field V is called a projective vector field, if there is a function P on TM0 , called the projective factor, such that L Vˆ G i = P y i , see [1]. In this case, given any appropriate t, the local flow {φt } associated with V is a projective transformation. A projective vector field V is said to be special if the projective factor P (x, y ) is lift of a 1-form on M, i.e. P (x, y ) = P i (x) y i . Notice that, on the Riemannian spaces, given any projective vector field V , the projective factor P (x, y ) is linear with respect to y, while this property is a non-Riemannian feature in a Finslerian background. The projective vector fields have several characterizations in the contexts, see Ref. [1] for some such results. The following characterization is useful in the sequel: Theorem 2.1. (See [5–7].) A vector field V is projective on a Randers space ( M , F = α + β) if and only if V is projective on ( M , α ) and L Vˆ (α si 0 ) = 0. Given any vector field V , let us stipulate the notation t 00 = L Vˆ α 2 . Now, we prove the following characterization of special projective vector fields on Randers spaces:
M. Rafie-Rad / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 927–930
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Lemma 2.2. A vector field V on a Randers space ( M , F = α + β) is special projective if and only if there is a 1-form P = P i (x) y i such that the following equations hold: i (1) 8α 2 β L Vˆ G α + (2α 2 L Vˆ e 00 − e 00 t 00 − 8α 2 β(L Vˆ s0 − P )) y i = 0,
i + (2β L Vˆ e 00 − 2e 00 L Vˆ β − 4(α 2 + β 2 )(L Vˆ s0 − P )) y i = 0. (2) 4(α 2 + β 2 )L Vˆ G α
Proof. A vector field V on ( M , F ) is special projective if and only if there is a 1-form P = P i (x) y i on M such that L Vˆ G i = P y i . By (2) and Theorem 2.1, this is equivalent to:
e 00 L Vˆ G αi + − s0 y i = P y i . 2F
(3)
After expanding the terms, Eq. (3) is equivalent to the identities below:
i
0 = L Vˆ G α +
= L Vˆ G αi + = L Vˆ G αi +
e 00 2F
L Vˆ e 00
− s0 y i − P y i = L Vˆ G αi + L Vˆ yi −
2F
e 00 L Vˆ F 2F 2
e 00 2F
y i − L Vˆ s0 y i − P y i
y i − L Vˆ s0 y i − P y i
t
00 + L Vˆ β i 1 i y − L Vˆ s0 y i − P y i = Rat + α Irrat i , y i − e 00 2α 2 2 2F 4α F
L Vˆ e 00 2F
where,
i Rati = 8α 2 β L Vˆ G α + 2α 2 L Vˆ e 00 − e 00 t 00 − 8α 2 β(L Vˆ s0 − P ) y i ,
2
2
i
Irrat = 4 α + β L Vˆ G α + 2β L Vˆ e 00 − 2e 00 L Vˆ β − 4 α + β (L Vˆ s0 − P ) y i . i
2
2
(4) (5)
Hence, V is a special projective vector field if and only if Rat i = 0 and Irrati = 0, for every i = 1, . . . , n. This completes the proof. 2 3. Proof of main theorems Proof of Theorem 1.1. Let us suppose that V is an arbitrary special projective vector field on ( M , F = α + β). From Lemma 2.2, there is a 1-form P = P i (x) y i on M such that Rat i = 0 and Irrati = 0, for any index i; Notice that, Rat i and Irrat i are given in (4) and (5). Now, it follows that:
0 = Rati − β Irrat i
= 4 α 2 − β 2 β L Vˆ G αi + 2 α 2 − β 2 L Vˆ e 00 y i − e 00 L Vˆ α 2 − β 2 y i − 4β α 2 − β 2 (L Vˆ s0 − P ) y i = α 2 − β 2 Q i − e 00 L Vˆ α 2 − β 2 y i (i = 1, . . . , n),
i where, Q i = {4β L Vˆ G α + 2L Vˆ e 00 y i − 4β(L Vˆ s0 − P ) y i }. Given any point x ∈ M, the irreducible polynomial (α 2 − β 2 ) ∈
R[ y 1 , . . . , yn ] divides the polynomials e 00 L Vˆ (α 2 − β 2 ) y i (i = 1, . . . , n). Notice that (α 2 − β 2 ) cannot divide y i for any index i. Given any special projective vector field V , if (α 2 − β 2 ) divides L Vˆ (α 2 − β 2 ), then it follows that V is a conformal vector field of the Riemannian metric (α 2 − β 2 ) and this proves (i) in Theorem 1.1. Otherwise, (α 2 − β 2 ) divides e 00 ; in this case, F is of isotropic S-curvature, cf. [3], and this proves (ii). 2 Proof of Theorem 1.2. Suppose that V is a special projective vector field on ( M , F = α + β) which is not a conformal vector field of the Riemannian metric α 2 − β 2 . By Theorem 1.1, F is of isotropic S-curvature. Moreover, by a result in [4], cf. Corollary 1.5, there is a Randers metric Fˆ such that V is either a Killing vector field of Fˆ or F is locally projectively flat Theorem 1.4, after an and α has positive constant sectional curvature. In the latter case, by a result in [2], cf. the case (c) in appropriate rescaling, F is locally isometric to the Randers metric given by F (x, y ) = α (x, y ) − f xk yk / 1 − f (x)2 , where f is an eigenfunction of the standard Laplacian corresponding to the eigenvalue λ = n with maxx∈ M | f (x)| < 1. Moreover, the flag curvature and the S-curvature of F are of the following forms:
K(x, y ) =
1 4
+
3F (x, − y ) 4(1 − f (x)2 ) F (x, y )
f (x) S(x, y ) = (n + 1) F (x, y ). 2 1 − f (x)2
,
It can be checked now that we have K > 0.
2
930
M. Rafie-Rad / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 927–930
Acknowledgement I would like to express my sincere to the Institute for Research in Fundamental Sciences (IPM) where this work was supported in part under grant No. 91530039. References [1] [2] [3] [4] [5] [6] [7]
H. Akbar-Zadeh, Champs de vecteurs projectifs sur le fibré unitaire, J. Math. Pures Appl. 65 (1986) 47–79. X. Chen, X. Mo, Z. Shen, On the flag curvature of Finsler metrics of scalar curvature, J. Lond. Math. Soc. 68 (2003) 762–780. X. Chen, Z. Shen, Randers metrics with special curvature properties, Osaka J. Math. 40 (2003) 87–101. V.S. Matveev, On projective equivalence and pointwise projective relation of Randers metrics, Int. J. Math. 23 (9) (2012) 1250093 (14 pages). M. Rafie-Rad, Some new characterizations of projective Randers metrics with constant S-curvature, J. Geom. Phys. 9 (4) (2012) 272–278. M. Rafie-Rad, Special projective algebra of Randers metrics of constant S-curvature, Int. J. Geom. Methods Mod. Phys. 6 (2) (2012). M. Rafie-Rad, B. Rezaei, On the projective algebra of Randers metrics of constant flag curvature, SIGMA 7 (2011) 085 (12 pages).
Contents lists available at ScienceDirect
C. R. Acad. Sci. Paris, Ser. I www.sciencedirect.com
Differential geometry
Special projective Lichnérowicz–Obata theorem for Randers spaces Le théorème projectif resteint de Lichnérowicz–Obata sur les espaces de Randers Mehdi Rafie-Rad a,b a b
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P.O. Box 47416-1467, Babolsar, Iran
a r t i c l e
i n f o
Article history: Received 24 July 2013 Accepted after revision 16 October 2013 Available online 13 November 2013 Presented by the Editorial Board
a b s t r a c t It is proved that either every special projective vector field V on a Randers space ( M , F = α + β) is a conformal vector field of the Riemannian metric α 2 − β 2 , or F is of isotropic S-curvature. This result is applied to establish a projective Lichnérowicz–Obata-type result on the closed manifolds with generic Randers metrics. © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
r é s u m é On prouve que, soit chaque champ projectif de vecteurs sur un espace de Randers ( M , F = α + β) est conforme à la métrique riemanienne α 2 − β 2 , soit F est à S-courbure isotrope. Ce résultat est appliqué à l’établissement d’un théorème de type de Lichnérowicz–Obata sur les variétés fermées de Randers. © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
1. Introduction The projective Lichnérowicz–Obata theorem in Riemannian geometry has been recently extended to closed Randers spaces in [4], cf. Corollary 1.5. However, this result seems to be incomplete since, unlike its Riemannian prototype, it does not imply the positivity of the flag curvature of the metric. A suggestion for arriving to the case of positive flag curvature is to consider only a sub-class of projective geometry in order to establish a reduced Lichnérowicz–Obata-type theorem. As it will be presented, the special projective geometry—which has been recently discussed in [5–7] for Randers metrics—is a good candidate for such a purpose, since this is an immediate extension of the Riemannian projective geometry. The results would imply that the special projective Randers geometry may refer to study one of the following cases: (a) conformal transformations of an appropriate Riemannian space, (b) isometries of a Randers space, or (c) Randers spaces of isotropic S-curvature. We prove the result for the pure Randers metrics: Theorem 1.1. Let us suppose that ( M , F = α + β) is a Randers space of dimension n 2. Then, at least one of the following statements holds:
E-mail address: rafi[email protected]. 1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.crma.2013.10.012
928
M. Rafie-Rad / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 927–930
(i) every special projective vector field on ( M , F ) is a conformal vector field of the Riemannian metric α 2 − β 2 , (ii) F is of isotropic S-curvature. Theorem 1.1 implies the following result: Theorem 1.2. Let us suppose that ( M , F = α + β) is a closed and connected Randers space of dimension n 2 and V is a special projective vector field of F . Then, at least one of the following statements holds: (i) V is a conformal vector field for the Riemannian metric α 2 − β 2 , (ii) there is a Randers metric Fˆ such that V is a Killing vector field for Fˆ , (iii) after an appropriate rescaling, F is of the following local form:
F (x, y ) =
| y |2 + |x|2 | y |2 − x, y 2 f k yk − x , 2 1 + |x| 1 − f 2 (x)
y ∈ TxM ∼ = Rn ,
(1)
where f is an eigenfunction of the standard Laplacian satisfying f = nf and maxx∈ M | f | < 1. In particular, F is of positive flag curvature. All manifolds are assumed to be smooth and connected, the natural coordinates on the tangent manifold T M are denoted by (xi , y i ) and the derivations with respect to xk and yk are denoted by the subscripts xk and yk , respectively. Moreover, we deal with pure and positive definite Randers metrics. 2. Special projective Finsler geometry Two Finsler metrics F and F˜ on M are said to be projectively equivalent if they have the same forward geodesics. A Finsler metric F is said to be locally projectively flat if, at any point x ∈ M, there is a neighborhood U such that F and the Euclidean metric are projectively equivalent on U . Given a Finsler space ( M , F ), a diffeomorphism φ : M −→ M is called a projective transformation if F andφ ∗ F are projectively equivalent.
α = ai j y i y j is a Riemannian metric and β = bi (x) y i is a 1-form defined on M such that βx := sup y ∈ T x M \{0} β( y )/α ( y ) < 1. Then the function F = α + β is a Finsler metric on M, which is called a Randers metric. The i and G i , and the Levi-Civita connection of α is geodesic spray coefficients of α and F are denoted respectively by the G α Suppose that
denoted by ∇ . The covariant derivation of β is given by (∇ j b i ) dx j := db i − b j θi j , where θi j := Γ˜ik dxk denote the associj
ated connection forms. Let us stipulate the following conventions: r i j :=
s j := b i si j and e i j := r i j + b i s j + b j si , e 00 := e i j y i y j , s0 := si y i and si 0 := are of the following form:
i Gi = Gα +
e 00 2F
− s0 y i + α si 0 .
1 (∇ j bi + ∇i b j ), si j := 12 (∇ j bi − ∇i b j ), si j := aih shj , 2 i s j y j . Then the geodesic spray coefficients G i of F
(2)
It is well known that a Randers metric F = α + β on M is locally projectively flat if and only if α is of constant sectional curvature and if β is closed. The locally projectively flat Randers metrics with isotropic S-curvature has been characterized by Chen, Mo and Shen in [2], cf. Theorem 1.3 and Theorem 1.4. A projective transformation φ : M −→ M is said to be special if it preserves the E-curvature; in this case, φ changes the geodesic spray coefficients as G˜ i (x, y ) = G i (x, y ) + P (x, y ) y i , where P = P i (x) y i . The complete lift of any vector field V on
i M is given by Vˆ = V i ∂ i + yk ∂ Vk ∂ i . The Lie derivative operator with respect to the vector field V is denoted by L Vˆ . It is
∂x
∂x ∂ y
well known that, L Vˆ y i = 0, L Vˆ dxi = 0 and the differential operators L Vˆ , ∂ i , the exterior differential operator d and ∂ i ∂x ∂y commute within any natural coordinates system on tangent manifold. The vector field V is called a projective vector field, if there is a function P on TM0 , called the projective factor, such that L Vˆ G i = P y i , see [1]. In this case, given any appropriate t, the local flow {φt } associated with V is a projective transformation. A projective vector field V is said to be special if the projective factor P (x, y ) is lift of a 1-form on M, i.e. P (x, y ) = P i (x) y i . Notice that, on the Riemannian spaces, given any projective vector field V , the projective factor P (x, y ) is linear with respect to y, while this property is a non-Riemannian feature in a Finslerian background. The projective vector fields have several characterizations in the contexts, see Ref. [1] for some such results. The following characterization is useful in the sequel: Theorem 2.1. (See [5–7].) A vector field V is projective on a Randers space ( M , F = α + β) if and only if V is projective on ( M , α ) and L Vˆ (α si 0 ) = 0. Given any vector field V , let us stipulate the notation t 00 = L Vˆ α 2 . Now, we prove the following characterization of special projective vector fields on Randers spaces:
M. Rafie-Rad / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 927–930
929
Lemma 2.2. A vector field V on a Randers space ( M , F = α + β) is special projective if and only if there is a 1-form P = P i (x) y i such that the following equations hold: i (1) 8α 2 β L Vˆ G α + (2α 2 L Vˆ e 00 − e 00 t 00 − 8α 2 β(L Vˆ s0 − P )) y i = 0,
i + (2β L Vˆ e 00 − 2e 00 L Vˆ β − 4(α 2 + β 2 )(L Vˆ s0 − P )) y i = 0. (2) 4(α 2 + β 2 )L Vˆ G α
Proof. A vector field V on ( M , F ) is special projective if and only if there is a 1-form P = P i (x) y i on M such that L Vˆ G i = P y i . By (2) and Theorem 2.1, this is equivalent to:
e 00 L Vˆ G αi + − s0 y i = P y i . 2F
(3)
After expanding the terms, Eq. (3) is equivalent to the identities below:
i
0 = L Vˆ G α +
= L Vˆ G αi + = L Vˆ G αi +
e 00 2F
L Vˆ e 00
− s0 y i − P y i = L Vˆ G αi + L Vˆ yi −
2F
e 00 L Vˆ F 2F 2
e 00 2F
y i − L Vˆ s0 y i − P y i
y i − L Vˆ s0 y i − P y i
t
00 + L Vˆ β i 1 i y − L Vˆ s0 y i − P y i = Rat + α Irrat i , y i − e 00 2α 2 2 2F 4α F
L Vˆ e 00 2F
where,
i Rati = 8α 2 β L Vˆ G α + 2α 2 L Vˆ e 00 − e 00 t 00 − 8α 2 β(L Vˆ s0 − P ) y i ,
2
2
i
Irrat = 4 α + β L Vˆ G α + 2β L Vˆ e 00 − 2e 00 L Vˆ β − 4 α + β (L Vˆ s0 − P ) y i . i
2
2
(4) (5)
Hence, V is a special projective vector field if and only if Rat i = 0 and Irrati = 0, for every i = 1, . . . , n. This completes the proof. 2 3. Proof of main theorems Proof of Theorem 1.1. Let us suppose that V is an arbitrary special projective vector field on ( M , F = α + β). From Lemma 2.2, there is a 1-form P = P i (x) y i on M such that Rat i = 0 and Irrati = 0, for any index i; Notice that, Rat i and Irrat i are given in (4) and (5). Now, it follows that:
0 = Rati − β Irrat i
= 4 α 2 − β 2 β L Vˆ G αi + 2 α 2 − β 2 L Vˆ e 00 y i − e 00 L Vˆ α 2 − β 2 y i − 4β α 2 − β 2 (L Vˆ s0 − P ) y i = α 2 − β 2 Q i − e 00 L Vˆ α 2 − β 2 y i (i = 1, . . . , n),
i where, Q i = {4β L Vˆ G α + 2L Vˆ e 00 y i − 4β(L Vˆ s0 − P ) y i }. Given any point x ∈ M, the irreducible polynomial (α 2 − β 2 ) ∈
R[ y 1 , . . . , yn ] divides the polynomials e 00 L Vˆ (α 2 − β 2 ) y i (i = 1, . . . , n). Notice that (α 2 − β 2 ) cannot divide y i for any index i. Given any special projective vector field V , if (α 2 − β 2 ) divides L Vˆ (α 2 − β 2 ), then it follows that V is a conformal vector field of the Riemannian metric (α 2 − β 2 ) and this proves (i) in Theorem 1.1. Otherwise, (α 2 − β 2 ) divides e 00 ; in this case, F is of isotropic S-curvature, cf. [3], and this proves (ii). 2 Proof of Theorem 1.2. Suppose that V is a special projective vector field on ( M , F = α + β) which is not a conformal vector field of the Riemannian metric α 2 − β 2 . By Theorem 1.1, F is of isotropic S-curvature. Moreover, by a result in [4], cf. Corollary 1.5, there is a Randers metric Fˆ such that V is either a Killing vector field of Fˆ or F is locally projectively flat Theorem 1.4, after an and α has positive constant sectional curvature. In the latter case, by a result in [2], cf. the case (c) in appropriate rescaling, F is locally isometric to the Randers metric given by F (x, y ) = α (x, y ) − f xk yk / 1 − f (x)2 , where f is an eigenfunction of the standard Laplacian corresponding to the eigenvalue λ = n with maxx∈ M | f (x)| < 1. Moreover, the flag curvature and the S-curvature of F are of the following forms:
K(x, y ) =
1 4
+
3F (x, − y ) 4(1 − f (x)2 ) F (x, y )
f (x) S(x, y ) = (n + 1) F (x, y ). 2 1 − f (x)2
,
It can be checked now that we have K > 0.
2
930
M. Rafie-Rad / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 927–930
Acknowledgement I would like to express my sincere to the Institute for Research in Fundamental Sciences (IPM) where this work was supported in part under grant No. 91530039. References [1] [2] [3] [4] [5] [6] [7]
H. Akbar-Zadeh, Champs de vecteurs projectifs sur le fibré unitaire, J. Math. Pures Appl. 65 (1986) 47–79. X. Chen, X. Mo, Z. Shen, On the flag curvature of Finsler metrics of scalar curvature, J. Lond. Math. Soc. 68 (2003) 762–780. X. Chen, Z. Shen, Randers metrics with special curvature properties, Osaka J. Math. 40 (2003) 87–101. V.S. Matveev, On projective equivalence and pointwise projective relation of Randers metrics, Int. J. Math. 23 (9) (2012) 1250093 (14 pages). M. Rafie-Rad, Some new characterizations of projective Randers metrics with constant S-curvature, J. Geom. Phys. 9 (4) (2012) 272–278. M. Rafie-Rad, Special projective algebra of Randers metrics of constant S-curvature, Int. J. Geom. Methods Mod. Phys. 6 (2) (2012). M. Rafie-Rad, B. Rezaei, On the projective algebra of Randers metrics of constant flag curvature, SIGMA 7 (2011) 085 (12 pages).