On characterizations of Randers norms in a Minkowski space

On characterizations of Randers norms in a Minkowski space XIAOHUAN MO AND LIBING HUANG Key Laboratory of Pure and Applied Mathematics School of Mathematical Sciences Peking University, Beijing 100871, China E-mail: moxh@pku.edu.cn January 7, 009 Abstract Let V be a vector space with Minkowski norm. We find a geometric quantity on V, ) which characterizes the Randers norm. We also give a different proof of Matsumoto-Hōjō theorem. Key words and phrases: Randers norm, Minkowski space, Matsumoto torsion, main scalar, affine normal field 1991 Mathematics Subject Classification: 58E0. 1 Introduction Let V be a vector space. A Minkowski norm on V in the form := + β is called the Randers norm where is an Euclidean norm and β is a linear functional on V [11]. Randers norms form an important and rich class of Minkowski norms. They were first studied by physicist, G.Randers, in 1941 [11] from the standard point of general relativity. Since then, many geometers have made efforts in investigation on the geometric properties of Randers norms. See [, 6, 7, 8] for some recent developments. This work is supported by the National Natural Science oundation of China 10771004 1

An important approach in discussing Randers metric is navigation representation []. It tells us that a geometric characterization for Randers norm is its indicatrix is an ellipsoid whose center is not necessarily at the origin). To characterize Randers norms among Minkowski norms, M. Matsumoto introduces the following quantity. or y V \{0}, u, v, w V, define M y u, v, w) := A y u, v, w) 1 n + 1 {I yu)h y v, w) + I y v)h y w, u) + I y w)h y u, v)} where h y, A y and I y are the angular metric, the Cartan torsion and the mean Cartan torsion respectively see Section ). Clearly, M = 0 for all two-dimensional Minkowski norms. Matsumoto and Hōjō proved the following [7, 8, 14]: Theorem 1[7, 8]) Let be a Minkowski norm on a vector space V of dimension n 3. The Matsumoto torsion M = 0 if and only if is a Randers norm. Theorem 1 tells us that the Matsumoto torsion gives a measure of the failure of an n-dimensional Minkowski norm to be a Randers norm if n 3. A natural task for us is to give a geometric quantity on a Minkowski plane which characterizes the Randers plane. In this paper we find the desired quantity. Now let us describe our construction. Let χ := e τ 3 1 + 3 I 9 I) where τ, I, I are the distortion, the main scalar and the derivative of I along ) e := detg ij ) y y 1 y 1 y respectively see Section ). Then we have the following Theorem Let is a Minkowski norm on a plane V. Then is a Randers norm if and only if χ is constant along the indicatrix. Then we generalize χ to higher dimensional Minkowski norm see Theorem 3.). Hence we give a new geometric quantity characterizing Randers norm which is available to two dimensional Minkowski norm.

Our approach of proving Theorem and Theorem 3. is partially the affine geometry. We also give an alternative proof of Matsumoto-Hōjō theorem in the spirit of affine geometry. By using this fundamental result, Mo-Shen established a global rigidity theorem for negatively curved insler metrics on a compact manifold of dimension n 3 [9]. Preliminaries Let be a Minkowski norm on a vector space V of dimension n. Set V 0 = V \{0}. or a vector y V 0, let where u, v, w V. A y u, v, w) := y) 4 3 r s t [ y + ru + sv + tw)] r=s=t=0 We call the family A := {A y y V 0 } the Cartan torsion. The Cartan torsion gives a measure of the failure of a Minkowski norm to be an Euclidean norm. Let {b i } be a basis for V. The Minkowski norm = y i b i ) is a function of y i ). Let g ij y) := 1 [ ] y i y jy), gij y)) = g ij y)) 1. Define the mean value of the Cartan torsion by I y u) := n g ij y)a y u, b i, b j ). i=1 We call the family I := {I y y V 0 } the mean Cartan torsion. Euclidean norms can be characterized by the mean Cartan torsion [4]. Let h ij y) := y) y i y jy). or y = y i b i V 0, the angular metric is a bilinear symmetric form defined by Let h y b i, b j ) := h ij y). M y u, v, w) := A y u, v, w) 1 n + 1 {I yu)h y v, w) + I y v)h y w, u) + I y w)h y u, v)} 3

where h y, A y and I y are the angular metric, the Cartan torsion and the mean Cartan torsion respectively. Each M y is a symmetric trilinear form on V. This quantity is introduced by M. Matsumoto [7]. We call the family M := {M y y V 0 } the Matsumoto torsion. It follows from the homogeneity of that M y y, v, w) = M y u, y, w) = M y u, v, y) = 0. 1) Clearly, M = 0 for all two-dimensional Minkowski norms. Matsumoto proves that every Randers norm satisfies that M y = 0. Later on, Matsumoto and Hōjō proved [8] that the converse is true too. That is, the Matsumoto torsion gives a measure of the failure of a Minkowski norm to be a Randers norm if n 3 see Theorem 1). Define or a basis {b i } for V and let Both σ σ := VolB n ) Vol{y i ) R n y i b i ) < 1}. τ = ln detgij y)) σ. and detg ij y)) are transformed in the way as the basis {b i } changes. Thus τ = τy) is well-defined, which is called the distortion of [1, 13]. Let be a Minkowski norm on a vector space V of dimension n. Set [y] + := {λy λ > 0}. Then induces an immersion S : S V, defined by S [y] + ) := y y) S S S) often refereed to as the indicatrix) where S := {[y] + y V 0 }. The indicatrix for the Minkowski norm is a smooth, strictly convex hypersurface enclosing the origin [5]. We consider a section L ΓT S ) defined by Ly) := y. Then L is transversal to S S). Take the transversal field ι = L. Then ι is an equiaffine unit normal field of S i.e. it has vanishing transversal connection form) and S is a centro-affine immersion [3, 10]. Denote the affine fundamental form and the induce affine connection of ι by h and respectively. Then h is the cubic form induced by the transversal field ι, equivalently, the transversal connection form of ι vanishes. In particular, it is fully symmetric. urthermore, h resp. 1 h) is just the same as the 4

angular metric resp. the Cartan torsion) induced by [3]. We define φ : S R + by φ = exp τ n + 1 where τ is the distortion of. Denote the Riemannian connection of h by ˆ. Then we have where A : ΓT S) ΓT S) ΓT S) is defined by ) u v = ˆ u v Au, v) 3) h Au, v), w) = Au, v, w), u, v, w ΓT S). 4) Define a vector field Z on S by Z := dφ) where : T S T S is the standard musical ) isomorphism obtained from the Riemannian metric h. Set ι := φ ι + S Z. Then we have the following: Lemma.1 ι is an affine normal field, i.e. i) ι is equiaffine; ii) θ coincides with the volume element of h where θ and h are the volume element and the affine fundamental form of ι respectively. Proof. It is easy to see D u ι = S u) for u ΓT S), i.e. the transversal connection form of ι is zero where D is the covariant differentiation in V [10]. It follows that D u ι = D u φ ι + S Z) = uφ) ι + φ D u ι + D u S Z) = uφ) ι φ S u) + D u S Z). 5) On the other hand, by using the formula of Gauss and the definition of Z, we have D u S Z) = S u Z) + hu, Z)ι = S u Z) uφ) ι. 6) Plugging 6) into 5) yields D u ι = φ S u) + S u Z). 5

It follows that ι is equiaffine. Let {e } be an orthonormal frame of h for S and {ω } be the coframe dual to {e }. Then the volume element of h is det h β ) ω 1 ω n 1 where h β = he, e β ) and det h β ) = detφ 1 h β ) = φ n 1) detδ β ) = φ n 1). 7) The volume element of ι is defined by θe 1,, e n 1 ) := dµs e 1,, S e n 1, ι) 8) where dµ := σ θ 1 θ n and {θ i } is the dual basis of {b i }. Let d µ be the volume form with respect to g = g ij y)θ i θ j. It is easy to see and τ = log d µ dµ 9) d µs e 1,, S e n 1, ι) = 1. 10) By using 7)-10) and ), we obtain θe 1,, e n 1 ) = e τ d µs e 1,, S e n 1, ι) = e τ d µs e 1,, S e n 1, φ ι + S Z) = e τ φd µs e 1,, S e n 1, ι) = φ n+1 φ = φ n 1 = det h β ). It follows that θ = det h β ) ω 1 ω n 1. 6

Lemma. Let s be the affine shape operator determined by the formulas of Gauss and Weingraten. Then we have the following s = φid Z. 11) Proof. Lemma.1 tells us that ι is an affine normal field. Hence ι induces its Blaschke structure, h, s). rom Proposition.5 in [10] we have s = φs Z + τ )Z 1) where τ is the transversal connection form of ι. Note that ι is equiaffine, ones obtain that τ = 0. 13) or centro-affine hypersurface S [cf. [10, page 37, Example.] and [3, page 8]), we have s = Id. 14) Plugging 13) and 14) into 1) yields 11). Let {e } be an orthonormal frame of h for S. Then the mean Cartan torsion is given by I y u) = A y u, e, e ). We denote its component by I := Ie ). In particular, for Minkowski plane, we have e := e 1 = detg ij ) and I := I 1 is called the main scalar. y ) y 1 y 1 y 7

3 Two dimensional Randers norm Now we consider a Minkowski plane V, ). Let χ := e τ 3 1 + 3 I 9 I) where τ, I, I are the distortion, the main scalar and the derivative of I along e := detg ij ) respectively. Then we have the following y ) y 1 y 1 y Theorem 3.1 Let is a Minkowski norm on a plane V. Then is a Randers norm if and only if χ is constant along the indicatrix. Proof Assume that = + β is a Randers norm where = a ij y i y j and β = b i y i. It is easy to show [1, page 89] where g ij := 1 detg ij ) = ) y i y j. It follows that ) 3 deta ij) 15) ln detg ij ) = 3 ln + 1 ln deta ij). Hence the distortion of is given by detgij ) τ = ln = 3 ln + ln detaij ) σ σ = ln [ ) 3 [detaij )] 1 σ 1 ]. 16) Let f : V \{0} R be a function. Then ef) is a function on V \{0}. We denote simply it by f. Then the main scalar I is given by I = τ = = 3 3 ln + ln detaij ) 17) ) ln ) = 3 ) σ from 16). On the other hand, a direct calculation yields [1, page 9 d)]) I = 9 ) β β 4 18) 8

where β = b i b j a ij, a ij ) = a ij ) 1. We have the following: β ) = 0, 19) ) β ) ) = 1 = = 3 I by 17)), 0) 1 + β β =. 1) By using 18), 19) and 0), we have [ II = 9 ) ) ) ] 4 β β + β β [ = 9 4 1 ) ) ) ] /) β β β β [ = 9 4 ) ) β β ) β ) ] [ = 9 ) ) ] 4 β β + β ) [ = 9 ) ) ] 4 β β + β 3 I [ ) ] = 3 I β β + β. Without loss of generality, we can assume that is non-euclidean. Thus I 0. It follows that I = 3 4 [ β β Together with 18) and ) we have [ ) ] 1 + 3 I 9 I = 1 1 β β + β = 1 β + β β = β = 1 β ). On the other hand, from 16) ones obtain e τ = Taking this together with 3) we obtain ) + β ]. ) 1 ) β β 3) ) 3 [detaij )] 1 σ 1. 4) χ : = e τ 3 1 + 3 I 9 I) = [deta ij)] 1 3 σ 3 1 β ) = [deta ij )] 1 3 σ 3 1 β ) = constant. 9

Conversely, we assume χ = constant. Denote the affine shape operator of ι := φ ι + S Z by s see Lemma.). Then s = φid Z. 5) Put Z = λe. 6) rom 5) and 6) we have λ = φ = 3 φτ = φi. 7) 3 It follows that λ = 3 φ I + φi ) = 3 φ 3 I + I ). 8) By using 6) and 7) one obtains IZ) = I λe) = λi = 3 φi. 9) where I is the mean Cartan torsion. Together with 8) we have χ = φ + λ + IZ). 30) On the other hand, the mean) affine curvature of s is given by trace s) = h φid Z)e), e) = he, e)φ h e Z, e)) ) = φ h ˆ e Z Ae, Z), e = φ I) + II) 31) where and ) I) = h ˆ e Z, e ) = h ˆ e λe), e) by 4)) ) = h eλ)e, e) h ˆ e e, e = eλ)he, e) λ ˆ e = λ 3) II) = h Ae, Z), e) = Ae, Z, e) = IZ). 33) 10

Plugging 3) and 33) into 31) yields trace s) = φ + λ + IZ) = χ = constant. By Theorem 9.8 in [10], the indicatrix S of is an ellipse whose center is not necessary at the origin). Hence = + β ia a Randers norm. The theory of plane curves can by viewed as a special case of hypersurface theory. In this case the affine shape operator s is simply κid where κ is the affine curvature. In the proof of Theorem 3.1, we deal with plane curve by using hypersurface theory. Now we are going to generalize this result to higher dimensional case. Theorem 3. Let is a Minkowski norm on an n-dimensional vector space V. Then is a Randers norm if and only if χ is constant where [ χ := e τ n+1 n 1 + n 1) ] I, I n + 1 n + 1) where I, is the convariant derivative of I with respect to h. 34) Proof Let {e } be an orthonormal frame of h for S. We put φ = e φ). Then Z = Σφ e. 35) By using 3) and 11), we have trace s) = Σ h se ), e ) = Σ h φid Z)e ), e ) = Σ h φ e, e ) Σ h e Z, e ) ) = Σ h e, e ) φ Σ h ˆ e Z Ae, Z), e ) = n 1)φ Σ h ˆ e Z, e + Σ h Ae, Z), e ). 36) rom 4), we have Σ h Ae, Z), e ) = Ae, Z, e ) = IZ) 37) 11

and ) Σ h ˆ e Z, e = divz = div ˆ φ = φ = Σφ,. 38) By using ) we have and By 35) and 39) we get φ = τ n + 1 φ = I n + 1 φ 39) φ, = ) n + 1 φ I + φi, ) = φ I + I, φ. 40) n + 1 n + 1 IZ) = I Σφ e ) = Σφ I = n + 1 φ Σ I. 41) Plugging 37) and 38) into 36), and using 40), 41) and 34) yields trace s) = n 1)φ + Σ φ, + IZ) = n 1)φ + 4φ Σ n+1) I + n+1 φ Σ I, [ n+1 φ Σ I = φ n 1 + n+1 I, n 1) ] n+1) I = χ. It follows that χ is constant if and only if the indicatrix S of is an ellipsoid, equivalently, is a Randers norm eg. [10, Theorem 9.7 or Theorem 9.8]). 4 An alternative proof of Matsumoto-Hōjō theorem In this section we are going to give an alternative proof of Theorem 1 see Section 1). irst of all we give the following Proposition 4.1 Let V, ) be an n-dimensional Minkowski space and M its Matsumoto torsion. Then for any y V 0 the cubic form at y induced by the affine normal field ι := φ ι + ι Z is just the same as φ 1 M y. Proof. We denote the corresponding objects with respect to the affine normal field ι by adding a tilde. Then the affine fundamental form and the induce affine connection change as follows [10] hu, v) = 1 hu, v), 4) φ 1

and u v = u v 1 hu, v)z 43) φ where u, v T y S and y V 0. Plugging 43) into 4) yields u v = u v hu, v)z. 44) Denoted the cubic form induced by the affine normal field ι by C. Note that ι is equiaffine. Then Cu, v, w) = u h)v, w) = u hv, w)) h u v, w) hv, u w) = u φ 1 hv, w) ) h u v hu, ) v)z, w h v, u w hu, ) w)z = uφ 1 )hv, w) + φ 1 [uhv, w)] φ 1 h u v, w) hu, v) hz, w) φ 1 hv, u w) hu, w) hv, Z) = uφ 1 )hv, w) + φ 1 A y u, v, w) φ [hu, v)hz, w) + hu, w)hv, Z)] 45) from 4) and 44) where u, v, w T y S. Note that uφ 1 ) = φ uφ) = φ 1 d log φ)u). 46) Recall that the vector field Z is defined by Z := dφ) where : T S T S is the musical isomorphism with respect to the affine fundamental form h. It follows that φ 1 hz, w) = φ 1 dφw) = d log φ)w). 47) Substituting 46) and 47) into 45) yields Cu, v, w) = φ 1 [A y u, v, w) d log φu)h y v, w) d log φv)h y w, u) d log φw)h y u, v)]. 48) It is easy to see that the mean Cartan torsion I y and the distortion τ is related by I y = dτ. rom which together with ) we obtain d log φ = n + 1 dτ = 13 I y n + 1. 49)

Recall that h is just the same as the angular metric induced by the Minkowski norm Ref. [3]). Plugging it into 48) yields Cu, v, w) = φ 1 {A y u, v, w) 1 n+1 [I yu)h y v, w) +I y v)h y w, u) + I y w)h y u, v)]} = φ 1 M y u, v, w). 50) It follows that C = φ 1 M y. Proof of Theorem 1. The Minkowski norm induces its indicatrix S : S V, defined by S [y] + ) := y y). Because ι is an affine normal field, it induce a Blaschke structure on the hypersurface S S). By the classical Maschke-Pick-Berwald theorem, the indicatrix S of is an ellipsoid, equivalently, is a Randers norm if the cubic form C induced by the affine normal field ι vanishes identically when n 3. Now our conclusion is an immediate consequence of 1) and Proposition 4.1. 14

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