Average Methods and Their Applications in Differential Geometry I

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− ∂h = δi hm + myihm 1 . j ∂yj Therefore ∞ m ∞ JvG1 J G ∂h exp J G := = v m = eh(v) δi + vi = J T v 1 m! m! j ∂yj v mX=0 mX=0 v and thus det exp JvG1 = det JvT . Since

traceJvG1 e = detexp JvG1, we have by the zero homogenity of the function h that n ∂h trace J G = nh(v)+ vk = nh(v) v 1 ∂yk Xk=1 v n and, consequently, det JvT = ϕ (v) as was to be proved.

2. Minkowski functionals and associated objects I. Let K Rn be a convex body containing the origin in its interior and consider a⊂ non-zero element v Rn. The value L(v) of the Minkowski functional induced by K is deﬁned∈ as the only positive real number such that v/L(v) ∂K, where ∂K is the boundary of K; for the general theory of convex sets∈ and Minkowski spaces see [9] and [15]. AVERAGE METHODS AND THEIR APPLICATIONS ... 3

Definition 1. The function L is called a Finsler-Minkowski functional if each non-zero element v Rn has an open neighbourhood such that the restricted function is of class∈ at least 4 and the Hessian matrix C ∂2E g = ij ∂yj∂yi of the energy function E := (1/2)L2 is positive definite. II. [1, 10] Since the differentiation decreases the degree of homogenity in a systematical way we have that gij’s are positively homogeneous of degree zero. To express the infinitesimal measure of changing the inner product it is usual to introduce the (lowered) Cartan tensor by the components 3 1 ∂gij 1 ∂ E (1) ijk = = . C 2 ∂yk 2 ∂yk∂yj∂yi k It is totally symmetric and y ijk = 0 because of the zero homogenity of gij. ij C k kl Using its inverse g we can introduce the quantities ij = g ijl to express the covariant derivative C C k i ∂Y i j k ∂ X Y = X + X Y ∇ ∂yi Cij ∂yk with respect to the Riemannian metric g. The formula is based on the standard Lévi-Civita process and repeated pairs of indices are automatically summed as usual. The curvature tensor Q can be written as Ql = l s l s . ijk CskCij −CsjCik Since ∂E ∂L yig = = L , i.e. g(C,Y )= Y E = L(YL), ij ∂yj ∂yj we have that the Liouville vector field is an outward-pointing unit normal −1 to the indicatrix hypersurface ∂K := L (1). On the other hand X C = X which means that the indicatrix is a totally umbilical hypersurface∇ and div C = n. III. Consider the volume form 1 n dµ = det gij dy . . . dy . ∧ ∧ Let f be a homogeneous functionp of degree zero. We have by Euler’s theorem that div (fC)= fdiv C + Cf = nf. Using the divergence theorem (with the Liouville vector field as an outward- pointing unit normal to the indicatrix hypersurface) for the vector field X = fC it follows that 1 (2) f dµ = f µ, ZK n Z∂K where n i−1 i 1 i−1 i+1 n µ = ιC dµ = det gij ( 1) y dy . . . dy dy . . . dy − ∧ ∧ ∧ ∧ p Xi=1 AVERAGE METHODS AND THEIR APPLICATIONS ... 4 denotes the induced volume form on the indicatrix hypersurface. Using integration we can introduce the following average inner products

(3) γ1(v, w) := g(v, w) µ and γ2(v, w)= m(v, w) µp Z∂K Z∂K on the vector space, where m(v, w)= g(v, w) (VL)(WL) − is the angular metric tensor and ∂ ∂ ∂ ∂ V = v1 + . . . + vn and W = w1 + . . . + wn . ∂y1 ∂yn ∂y1 ∂yn Furthermore

γ3(v, w)= γ1(v, w) γ2(v, w)= (VL)(WL) µp. − Z∂K

The basic version γ1 of the average inner products was introduced in [16]. For further processes to construct inner products by averaging we can refer to [3]. In what follows we take one more step forward by introducing associated Randers-Minkowski functionals. The linear form of the perturbation will play a crucial role in the applications (see Funk metrics and Brickell’s theorem). Proposition 1. The linear isometry group of the Minkowski vector space is a subgroup of the orthogonal group with respect to each of the average inner products γ1, γ2 and γ3. Proof. Suppose that the linear transformation Φ: Rn Rn preserves the Minkowskian lenght of the elements in the space. First→ of all compute the Euclidean volume of K: −1 −1 volE(K)= 1= detΦ = detΦ volE(K) ZK ZΦ(K) | | | | because K is invariant under Φ. Therefore detΦ = 1. ± Since the energy is also invariant under Φ we have, by diﬀerentiating the relationships L(Φ) = L and E(Φ) = E, respectively, that k ∂L ∂L − ∂E ∂E − ∂Φ M k = Φ 1 and M k = Φ 1, where M k = i ∂yk ∂yi ◦ i ∂yk ∂yi ◦ i ∂yi denotes the matrix of the linear transformation. The second order partial derivatives give that Φ is an isometry with respect to both the Riemann- Finsler metric and the angular metric tensor: k l −1 k l −1 M M gkl = gij Φ and M M mkl = mij Φ . i j ◦ i j ◦ −1 Since the absolute value of det Φ is 1 we have that det gij = det gij Φ . Finally ◦

γ1(v, w) := g(v, w) µ = n g(v, w) dµ = n g(v, w) det gij = Z Z Z ∂K K K p AVERAGE METHODS AND THEIR APPLICATIONS ... 5

−1 −1 −1 = n g(v, w) Φ det gij Φ detΦ = Z ◦ ◦ | | Φ(K) p

= n g(Φ(v), Φ(w)) det gij = n g(Φ(v), Φ(w)) dµ = Z Z K p K = g(Φ(v), Φ(w)) µ = γ1(Φ(v), Φ(w)). Z∂K The computation is similar in cases of both the angular metric tensor and the diﬀerence metric γ = γ γ . 3 1 − 2 Corollary 1. If the linear isometry group of the Minkowski vector space is irreducible then there exists a number 0 <λ< 1 such that γ2 = λγ1 and, consequently, λ γ = (1 λ)γ and γ = γ . 3 − 1 2 1 λ 3 − IV. Using Lemma 1 we have the following practical method to calculate integrals of the form ∂K fµ, where the integrand f is a zero homogeneous function. Consider theR mapping v v T (v) := ϕ(v)v, where ϕ(v)= | | , 7→ L(v) as a diﬀeomorphism (away from the origin); v denotes (for example) the usual Euclidean norm of vectors in Rn. Then | | n n f dµ = ϕ f ϕ det gij ϕ = ϕ f dµ Z Z −1 ◦ ◦ Z K T (K) p B because of the zero homogenity of the integrand. Therefore

(4) ϕnf dµ = f dµ ZB ZK and, by equation (2),

(5) ϕnfµ = f µ. Z∂B Z∂K V. As one of further associated objects consider the linear functional

1 ∂ n ∂ β(v) := VLµ, where V = v 1 + . . . + v n . Z∂K ∂y ∂y Remark 1. Using the divergence theorem we have that

div LV (VL)C dµ = 0 Z − K because the vector ﬁeld LV (VL)C is tangential to the indicatrix hypersurface: − LV (VL)C L = L(VL) (VL)L = 0. − − Since C(VL) = 0 (2) div (LV ) dµ = div ((VL)C) dµ = n VLdµ = VLµ. ZK ZK ZK Z∂K On the other hand i div (LV )= VL + Lv Ci, AVERAGE METHODS AND THEIR APPLICATIONS ... 6

jk where Ci = g Cijk. Since div (LV ) is a zero homogeneous function

i n div (LV ) dµ = div (LV ) µ = VL + Lv Ci µ ZK Z∂K Z∂K and thus i v LCi µ = (n 1) VLµ = (n 1)β(v). Z∂K − Z∂K − Deﬁnition 2. The indicatrix body K is called balanced if the assotiated linear functional β is identically zero. Proposition 2. If the linear isometry group of the Minkowski vector space is irreducible then the indicatrix body is balanced. Proof. Using the previous notations

β(v)= VLµ = n VLdµ = n VL det gij = Z Z Z ∂K K K p

i ∂L −1 −1 −1 k i ∂L = n v Φ det gij Φ detΦ = n v M det gij = Z ∂yi ◦ ◦ | | Z k ∂yi Φ(K) p K p k i ∂L k i ∂L = n v Mk i dµ = v Mk i µ = β(Φ(v)). ZK ∂y Z∂K ∂y Therefore the null-space of the linear functional β is an invariant subspace under the linear isometry group of the Minkowski vector space. Since the group is irreducible we have that β = 0 as was to be stated. VI. Since 2 β2(v)= VLµ 1 µ (VL)2 µ = Area (∂K) (VL)2 µ Z∂K ≤ Z∂K Z∂K Z∂K and 1 1 (VL)2 = g2(C,V ) g(C,C) g(V,V )= g(V,V ) L2 ≤ L2 we have that β2(v) Area (∂K) γ (v, v) Area (∂K) γ (v, v). ≤ 3 ≤ 1 The inequalities are strict because C and V are linearly independent away from the points v and v. Introducing the weighted inner products − 1 Γ (v, w)= γ (v, w) i Area (∂K) i (i = 1, 2, 3) we can write that β 2 (v) < Γ (v, v) < Γ (v, v). Area (∂K) 3 1 Therefore β the supremum norm of the weighted linear functional < 1 Area (∂K) with respect to each of the unit balls of Γ1 and Γ3. AVERAGE METHODS AND THEIR APPLICATIONS ... 7

Corollary 2. The functionals β(v) β(v) L (v) := Γ (v, v)+ and L (v) := Γ (v, v)+ 1 1 Area (∂K) 3 3 Area (∂K) p p are Finsler-Minkowski functionals which are called associated Randers-Min- kowski functionals to the Minkowski vector space. Theorem 1. (The main theorem). If L is a Finsler-Minkowski functional with a balanced indicatrix body of dimension at least three and the L´evi-Civita connection has zero curvature then L is a norm coming from an inner product, i.e.∇ the Minkowski vector space reduces to a Euclidean one. The original version was proved by F. Brickell [2] using the stronger condition of absolute homogenity (the symmetry of K with respect to the origin) instead of the balanced indicatrix body. The preparations of the proof in- volve the theory of the Funk spaces and the adaptation of the setting of average objects to Finsler spaces in general.

3. Finsler spaces I. Let M be a differentiable manifold with local coordinates u1, ..., un on U M. The induced coordinate system on the tangent manifold consists of the⊂ functions x1 := u1 π,...,xn = un π and y1 := du1,...,yn = dun, ◦ ◦ where π : T M M is the canonical projection. A Finsler structure on a differentiable→ manifold M is a smoothly varying family F : T M R of Finsler-Minkowski functionals in the tangent spaces satisfying the following→ conditions: each non-zero element v T M has an open neighbourhood such that • the restricted function is∈ of class at least 4 in all of its variables x1, ..., xn and y1, ..., yn, C the Hessian matrix of the energy function E := (1/2)F 2 with respect • to the variables y1,...,yn is positive definite. II. Let f : T M R be a zero homogeneous function and let us define the average-valued function→

Af (p) := fµp, Z∂Kp where ∂Kp is the indicatrix hypersurface belonging to the Finsler-Minkow- ski functional of the tangent space. Integration is taken with respect to the orientation induced by the coordinate vector ﬁelds ∂/∂y1,...,∂/∂yn. It can be easily seen that the integral

−1 −1 1 n f dµp = f y det gij y dy ...dy Z Z ◦ ◦ K y(K) p is independent of the choice of the coordinate system (orientation). Actually, the orientation is convenient but not necessary to make integrals of functions sense [18]. AVERAGE METHODS AND THEIR APPLICATIONS ... 8

In what follows we compute the partial derivatives of the function Af . Fol- lowing IV in section 2 we choose an auxiliary Euclidean structure at each point in the coordinate neighbourhood U induced by the coordinate vector ﬁelds as an orthonormal system of tangent vectors. Since it is an Euclidean structure there is no need to distinguish the diﬀerent base points. Using equations (2), (4) and (5)

1 2 n 2 n (y ) + . . . + (y ) Af = n ϕ f det gij dy, where ϕ = . Z p F B p Therefore ∂Af ∂ n = n ϕ f det gmn dy = ∂ui p ∂xi ZB p n ∂ ln F ∂f ∂ ln √det gmn = n ϕ nf + + f det gmn dy ∂xi ∂xi ∂xi ZB − p ∂ ln F ∂f ∂ ln √det gmn = nf i + i + f i µp Z∂Kp − ∂x ∂x ∂x using equations (2), (4) and (5) again. Diﬀerentiating det gmn as the com- position of the function D := det and the usual coordinates of the matrix M := gij we have that

∂ det gmn ∂D ∂gmn i = (M) i = ∂x ∂gmn ∂x

m+n th th ∂gmn = ( 1) det M without its m row and n column i = − ∂x ∂g = (det g )gmn mn ij ∂xi and, consequently, ∂ ln √det g 1 ∂g (6) mn = gmn mn ∂xi 2 ∂xi The last step is to divide the partial derivatives of the integrand into vertical and horizontal parts [1], see also [5] and [6]. k III. Using compatible collections Gi of functions on local neighbourhoods of the tangent manifold let us define the (horizontal) vector fields ∂ ∂ (7) Xh = Gk . i ∂xi − i ∂yk Definition 3. The horizontal distribution h is a collection of subspaces h spanned by the vectors Xi as the base point runs through the non-zero el- k ements of the tangent manifold. If the functions Gi are positively homogeneous of degree 1 then the distribution is called homogeneous. In case of ∂Gk ∂Gk (8) i = j ∂yj ∂yi we say that h is torsion-free. The horizontal distribution is conservative if the derivatives of F vanishes into the horizontal directions AVERAGE METHODS AND THEIR APPLICATIONS ... 9

It is well-known from the calculus that (8) is a necessary and suﬃcient condition for the existence of functions Gk such that ∂Gk = Gk. ∂yi i Using (7) the integrand can be written into the form ∂ ln F ∂f ∂ ln √det g nf + + f mn = nfXhlnF + Xhf+ − ∂xi ∂xi ∂xi − i i

1 mn h k ∂ ln F k ∂f k 1 mn ∂gmn +f g X gmn nfG + G + fG g . 2 i − i ∂yk i ∂yk i z2 }|∂yk{ On the other hand ∂ ∂ ln F ∂ ∂ ln F ∂f div f Gk Gk C = fdiv Gk Gk C +Gk , i ∂yk − i ∂yk i ∂yk − i ∂yk i ∂yk where ∂ ∂ ln F ∂ ∂Gk ∂ ln F div Gk Gk C = Gkdiv + i Gk div C = i ∂yk − i ∂yk i ∂yk ∂yk − i ∂yk k k ∂Gi k ∂ ln F = G k + nG . i C ∂yk − i ∂yk Using the divergence theorem

k ∂ k ∂ ln F div f Gi k Gi k C µp = 0 Z∂Kp ∂y − ∂y and thus k ∂Af h h 1 mn h ∂Gi i = nfXi lnF + Xi f + f g Xi gmn + f k µp ∂u Z∂Kp − 2 ∂y In terms of index-free expressions

h h ′ c (9) XAf = nfX lnF + X f + f ˜ (X ) µp, Z∂Kp − C ′ where ˜ is the semibasic trace of the second Cartan tensor C ′ g( (Xc, Xc), Xv)= C i j k 1 h v v h v v v h v = Xi g(Xj , Xk ) g([Xi , Xj ], Xk ) g(Xj , [Xi , Xk ]) 2 − − associated to h, Xv, Xc and Xh are the vertical, complete and horizontal lifts of the vector ﬁeld X on the base manifold. Especially ∂ ∂ ∂ ∂ Xv := , Xc := and Xh = Gk . i ∂yi i ∂xi i ∂xi − i ∂yk Corollary 3. If the horizontal distribution is conservative then we have the reduced formula

h ˜′ c (10) XAf = X f + f (X ) µp. Z∂Kp C AVERAGE METHODS AND THEIR APPLICATIONS ... 10

IV. Using the reduced formula (10) we have that

X(Y Af )=

h h ˜′ c h h ˜′ c ˜′ c h ˜′ c ˜′ c = X (Y f)+ (Y )X f+fX (Y )+ (X )Y f+f (X ) (Y ) µp. Z∂Kp C C C C C Changing the role of the vector ﬁelds X and Y

h h h ˜′ c h ˜′ c [X,Y ]Af = [X ,Y ]f + f X (Y ) Y (X ) µp. Z∂Kp C − C On the other hand (10) h ˜′ c [X,Y ]Af = [X,Y ] f + f ([X,Y ] ) µp Z∂Kp C and, consequently, we have that

c c h ˜′ c h ˜′ c ˜′ c R(X ,Y )fµp = f X (Y ) Y (X ) ([X,Y ] ) µp, Z∂Kp Z∂Kp C − C − C where R(Xc,Y c) := v[Xh,Y h] = [X,Y ]h [Xh,Y h] − − is the curvature of the horizontal distribution, see [5] and [6]. V. In case of a conservative and torsion-free horizontal distribution the result can be written into the form c c ˜′ c ˜′ c (11) R(X ,Y )fµp = f (DXh )(Y ) (DY h )(X ) µp, Z∂Kp Z∂Kp C − C where D is one of the Finsler connections of type Berwald, Cartan or Chern- Rund associated to h; [13] see also [5] and [6]. Taking f = 1 we also have that ˜′ c ˜′ c (12) (DXh )(Y ) (DY h )(X ) µp = 0 Z∂Kp C − C provided that h is conservative and torsion-free. VI. Consider the associated Riemannian metric tensors v v v v γ1(Xp,Yp)= g(X ,Y ) µp, γ2(Xp,Yp)= m(X ,Y ) µp, Z∂Kp Z∂Kp where m(Xv,Y v)= g(Xv,Y v) (XvL)(Y vL) − is the angular metric tensor and

v v γ3(Xp,Yp)= γ1(Xp,Yp) γ2(Xp,Yp)= (X F )(Y F ) µp − Z∂Kp together with the weighted versions 1 Γi(Xp,Yp)= γi(Xp,Yp), Area (∂Kp) where i = 1, 2, 3 and

Area (∂Kp)= 1 µp. Z∂Kp AVERAGE METHODS AND THEIR APPLICATIONS ... 11

Remark 2. After introducing the 1-form

v β(Xp)= X F µp Z∂Kp we have, by Corollary 2, that β(v) β(v) F (v) := Γ (v, v)+ and F (v) := Γ (v, v)+ 1 1 Area (∂K) 3 3 Area (∂K) p p are smoothly varying family of Randers-Minkowski functionals which are called the associated Randers functionals to the Finsler space.

VII. In what follows we shall use the canonical horizontal distribution of the Finsler manifold which is uniquely determined by the following conditions: it is conservative, torsion-free and homogeneous. Recall the rules of covariant derivative with respect to the Berwald, Cartan and Chern-Rund connections associated to the canonical horizontal distribution:

Berwald Cartan Chern-Rund

v c c DXv Y 0 (X , Y ) 0 C

v h v h v ′ c c h v ′ c c DXh Y [X , Y ] [X , Y ]+ ˜ (X , Y ) [X , Y ]+ ˜ (X , Y ) C C

h c c DXv Y 0 ˜(X , Y ) 0 FC

v h v h v ′ c c h v ′ c c DXh Y [X , Y ] [X , Y ]+ ˜ (X , Y ) [X , Y ]+ ˜ (X , Y ) F F FC F FC where the almost complex structure associated to h is deﬁned by the following formulas F (Xh)= Xv and (Xv)= Xh. F − F To determine the L´evi-Civita connections associated to the average Rie- mannian metrics γ1, γ2 and γ3, respectively, we shall use the Cartan connection because it is metrical, i.e. v v v v v v v X g(Y ,Z )= g(DXv Y ,Z )+ g(Y , DXv Z ),

h v v v v v v X g(Y ,Z )= g(DXh Y ,Z )+ g(Y , DXh Z ) and both the (v)v- and the (h)h-torsion vanish. The theory of the classical Finsler connections also says that the deﬂection tensor is identically zero:

DXh C = 0. For the sake of simplicity consider vector ﬁelds X, Y and Z on the base manifold such that each pair of the vector ﬁelds has a vanishing Lie bracket. AVERAGE METHODS AND THEIR APPLICATIONS ... 12

Using the standard L´evi-Civita proccess it follows by (10) that

h v v ′ c v v Xγ1(Y,Z)= X g(Y ,Z )+ ˜ (X )g(Y ,Z ) µ, Z∂K C

h v v ′ c v v Y γ1(X,Z)= Y g(X ,Z )+ ˜ (Y )g(X ,Z ) µ, Z∂K C h v v ′ c v v Zγ1(X,Y )= Z g(X ,Y )+ ˜ (Z )g(X ,Y ) µ. Z∂K C Therefore v v 1 c c c (13) γ1 (( 1)X Y,Z)= g (DXh Y ,Z )+ ρ(X ,Y ,Z ) µ, ∇ Z∂K 2 where ′ ′ ′ ρ(Xc,Y c,Zc)= ˜ (Xc)g(Y v,Zv)+ ˜ (Y c)g(Xv ,Zv) ˜ (Zc)g(Xv ,Y v). C C − C To compute the corresponding formula for γ3 we use the identity 1 Y vF = g(Y v,C). F Therefore

h v 1 v v 1 v X (Y F )= g(D h Y ,C)+ g(Y , D h C) = g(D h Y ,C) F X X F X because of the vanishing of the deﬂection. We have

v v v Z F Y F v ′ c v v Xγ3(Y,Z)= g(DXh Y , C) + g(DXh Z , C)+ ˜ (X )(Y F )(Z F ) µ, Z∂K F F C v v v Z F X F v ′ c v v Yγ3(X,Z)= g(DY h X , C) + g(DY h Z , C)+ ˜ (Y )(X F )(Z F ) µ, Z∂K F F C v v v Y F X F v ′ c v v Zγ3(X, Y )= g(DZh X , C) + g(DZh Y , C)+ ˜ (Z )(X F )(Y F ) µ, Z∂K F F C and the L´evi-Civita process results in the formula v v Z F 1 c c c (14) γ3 (( 3)X Y,Z)= g (DXh Y ,C) + δ(X ,Y ,Z ) µp, ∇ Z∂K F 2 where ′ ′ δ(Xc,Y c,Zc)= ˜ (Xc)(Y vF )(ZvF )+ ˜ (Y c)(XvF )(ZvF ) C C − ′ ˜ (Zc)(XvF )(Y vF ). C Finally γ = γ γ shows that 2 1 − 3 γ (( )X Y,Z)= γ (( )X Y,Z) γ (( )X Y,Z) . 2 ∇2 1 ∇1 − 3 ∇3 Therefore v v 1 c c c (15) γ2 (( 2)X Y,Z)= m (DXh Y ,Z )+ η(X ,Y ,Z ) µp, ∇ Z∂K 2 where ′ ′ ′ η(Xc,Y c,Zc)= ˜ (Xc)m(Y v,Zv)+ ˜ (Y c)m(Xv,Zv) ˜ (Zc)m(Xv,Y v). C C − C AVERAGE METHODS AND THEIR APPLICATIONS ... 13

4. Funk metrics Let K Rn be a convex body containing the origin in its interior and suppose that⊂ the induced function L is a Finsler-Minkowski functional on the vector space Rn. Changing the origin in the interior of K we have a smoothly varying family of Finsler-Minkowski functionals parameterized by the interior points of K: v (16) L p + = 1, Lp(v) where the script refers to the base point of the tangent vector v. I. Conformal geometry. Let U be the interior of the indicatrix body. To- gether with the Finslerian fundamental function n F : TU = U R R, vp F (vp) := Lp(v) × → 7→ the manifold U is called a Funk space (or Funk manifold). It is a special Finsler manifold. Another notations and terminology: Kp denotes the unit ball with respect to the functional Lp and ∂Kp is its boundary. Especially K0 = K and ∂K0 = ∂K. Using the Riemann-Finsler metric 1 ∂2F 2 g = , ij 2 ∂yi∂yj where the partial derivatives are taken with respect to the variable v, any tangent space (away from the origin) can be interpreted as a Riemannian manifold. So is ∂Kp equipped with the usual induced Riemannian structure. It is a totally umbilical Riemannian submanifold of the corresponding tangent space (see section 2/II).

Theorem 2. For any p U the indicatrix hypersurfaces ∂Kp and ∂K are conformal to each other as∈ Riemannian submanifolds in the corresponding tangent spaces. The conform mapping between these structures comes from the projection v ρ(vp) := p + ∂K Lp(v) ∈ of the tangent space TU. Especially

k ∂L gρ(vp)(w, z)= 1 p k gvp (w, z), − ∂u ρ(vp) where w and z are tangential to the indicatrix hypersurface ∂K at ρ(vp), i.e they are tangential to ∂Kp at vp too. Proof. Equation (16) says that L ρ = 1. In terms of the usual local coordinates of the tangent space TU◦we can write that ρj = xj + yj/F , where ρj = uj ρ is the corresponding coordinate function with respect to the standard coordinate◦ system of U. Diﬀerentiating with respect to yi we have that j j j ∂F j ∂L ∂ρ ∂L δi F y i 1 ∂L y ∂L ∂F 0= ρ = ρ − ∂y = ρ ρ . ∂uj ◦ ∂yi ∂uj ◦ F 2 F ∂ui ◦ − F 2 ∂uj ◦ ∂yi Uisng the homogenity property yj ∂L xj + ρ = L ρ = 1 F ∂uj ◦ ◦ AVERAGE METHODS AND THEIR APPLICATIONS ... 14 we have that 1 ∂L ∂L ∂F (17) 0 = ρ + xj ρ 1 . F ∂ui ◦ ∂uj ◦ − ∂yi Therefore ∂L ∂L ∂F (18) ρ = 1 xk ρ . ∂ui ◦ − ∂uk ◦ ∂yi Diﬀerentiating equation (18) with respect to the variable yj ∂2L ∂ρl ∂2L ∂ρl ∂F ∂L ∂2F ρ = xk ρ + 1 xk ρ , ∂ul∂ui ◦ ∂yj − ∂ul∂uk ◦ ∂yj ∂yi − ∂uk ◦ ∂yj∂yi where l l l ∂F ∂ρ δjF y j = − ∂y ∂yj F 2 and, by the homogenity property, yl ∂2L xl + ρ = 0. F ∂ul∂ui ◦ Therefore 1 ∂2L ∂L ∂2F ρ = 1 xk ρ F ∂uj∂ui ◦ − ∂uk ◦ ∂yj∂yi − (19) xl ∂2L ∂F xk ∂2L ∂F xkxl ∂2L ∂F ∂F ρ ρ ρ . F ∂ul∂ui ◦ ∂yj − F ∂uk∂uj ◦ ∂yi − F 2 ∂uk∂ul ◦ ∂yi ∂yj

Let w and z be tangential to the indicatrix hypersurface ∂Kp at vp, i.e. F (vp)=1 and

Wvp F = Zvp F = 0, where ∂ ∂ ∂ ∂ W = w1 + . . . + wn and Z = z1 + . . . + zn . ∂y1 ∂yn ∂y1 ∂yn Equation (18) says that w and z are tangential to the indicatrix hypersurface ∂K at ρ(vp) and, consequently, 2 i j ∂ L gρ(vp)(w, z)= w z j i . ∂u ∂u ρ(vp) which implies, by equation (19), that 2 k ∂L i j ∂ F gρ(vp)(w, z)= 1 p k w z j i = − ∂u ρ(vp) ∂y ∂y vp

k ∂L 1 p k gvp (w, z) − ∂u ρ(vp) as was to be stated. Remark 3. Note that the metric g associated to L is just the Riemann- Finsler metric of the Finsler manifold (U, F ) at the origin. Corollary 4. ∂L 1 xk ρ> 0. − ∂uk ◦ AVERAGE METHODS AND THEIR APPLICATIONS ... 15

The geometric meaning of the inequality can be seen from the homogenity property yj ∂L xj + ρ = L ρ = 1. F ∂uj ◦ ◦

Especially v is never tangential to the indicatrix hypersurface ∂K at ρ(vp). Inequality ∂L yj ∂L 0 < 1 xk ρ = ρ − ∂uk ◦ F ∂uj ◦ says that v represents an outward pointing direction relative to the indicatrix hyoersurface at ρ(vp). II. Projective geometry (Okada’s theorem) [11, 12]. After diﬀerentiating equation (16) with respect to the variable xi: ∂L ∂ρj ∂L yj ∂F ∂L yj ∂L ∂F 0= ρ = ρ δj = ρ ρ ∂uj ◦ ∂xi ∂uj ◦ i − F 2 ∂xi ∂ui ◦ − F 2 ∂uj ◦ ∂xi Using the homogenity property yj ∂L xj + ρ = L ρ = 1 F ∂uj ◦ ◦ again we have that ∂L 1 ∂L ∂F 0= ρ + xj ρ 1 . ∂ui ◦ F ∂uj ◦ − ∂xi Together with equation (17) and Corollary 4 ∂F 1 ∂F ∂F ∂F (20) 0 = F = ∂yi − F ∂xi ⇒ ∂yi ∂xi which is just Okada’s theorem for Funk spaces. In terms of diﬀerential geometric structures we can write equation (20) into the form

dhF = F dJ F, where h is the horizontal distribution detemined by the coordinate vector fields ∂/∂x1, ..., ∂/∂xn and J is the canonical almost tangent structure on the tangent space TU. Then 2 dJ dhF = dJ F dJ F + F d F = 0 ∧ J which is just the coordinate-free expression for the classical Rapcsák’s equation of projective equivalence [13]. Therefore the canonical spray ξ of the Funk space can be expressed into the following special form ∂ ∂ ξ = yi Fyi . ∂xi − ∂yi It says that the straight lines in U can be reparameterized to the geodesics of the Funk manifold [8]. If c(t) = p + tv then we need the solution of the differential equation1 ′′ ′ (21) θ = θ F (vp) −

1Equation (21) can be considered as a correction of equation (21) in [17]. AVERAGE METHODS AND THEIR APPLICATIONS ... 16

Under the initial conditions θ(0) = 0 and θ′(0) = 1. We have that

1 exp ( tF (vp)) (22) θ(t)= − − , F (vp) i.e. 1 exp ( tF (vp)) c˜(t)= p + − − v F (vp) is a geodesic of the Funk manifold. Okada’s theorem is a rule how to change derivatives with respect to xi and yi. This results in relatively simple formulas for the canonical objects of the Funk manifold. In what follows we are going to summarize some of them (proofs are straightforward calculations, [1]): 1 ∂Gk F 1 ∂F (23) Gk = ykF, Gk = = δk + yk , 2 i ∂yk 2 i 2 ∂yi

∂ ∂ ∂ F 1 ∂F ∂ (24) Xh = Gk = δk + yk i ∂xi − i ∂yk ∂xi − 2 i 2 ∂yi ∂yk for the canonical horizontal distribution. The ﬁrst and the second Cartan tensors are related as ′ 1 (25) = F C 2 C and the curvature of the canonical horizontal distribution can be expressed in the following form ∂ ∂ F ∂F ∂ ∂F ∂ (26) R , = . ∂xi ∂xi 4 ∂yi ∂yj − ∂yj ∂yi In terms of lifted vector ﬁelds 1 (27) R(Xc,Y c)= g(Xv,C)Y v g(Y v,C)Xv . 4 − III. Applications. Using relation (25) we can specialize the basic formula (10) for derivatives of integral functions. We apply this technic only in the special case of the area-function

(28) r : U R, p r(p) := A1(p)= 1 µp → 7→ Z∂Kp Theorem 3. The area function is convex. Proof. Remark 1 shows that ∂r 1 ˜ ∂ n 1 ∂F (29) i = F i µp = − i µp. ∂u p 2 Z∂Kp C ∂x 2 Z∂Kp ∂y Differentiating again 2 h ∂ r n 1 ∂ ∂F 1 ∂F ˜ ∂ (30) j i = − j i + i F j µp. ∂u ∂u p 2 Z∂Kp ∂u ∂y 2 ∂y C ∂x Here, by (24), ∂ h ∂F ∂2F 1 ∂2F = F ∂uj ∂yi ∂xj∂yi − 2 ∂yj∂yi AVERAGE METHODS AND THEIR APPLICATIONS ... 17 because of the zero homogenity of the vertical partial derivatives of the Finslerian fundamental function. Differentiating Okada’s relation (20) it follows that ∂2F ∂F ∂F ∂2F = + F ∂xj∂yi ∂yj ∂yi ∂yj∂yi and thus ∂ h ∂F ∂F ∂F 1 ∂2F = + F . ∂uj ∂yi ∂yj ∂yi 2 ∂yj∂yi The only question is how to substitute the last term in the integrand of (30) to give a positive definite expression. First of all note that the vector field ∂F ∂ ∂F ∂F T := F C ∂yi ∂yj − ∂yi ∂yj is tangential to the indicatrix hypersurface and, by the divergence theorem,

div T dµp = 0. ZKp In a more detailed form (using the zero homogenity of the vertical derivatives of the Finslerian fundamental function) ∂F ∂ ∂F ∂F ∂2F ∂F ∂F div T = F div + + F div C = ∂yi ∂yj ∂yi ∂yj ∂yj∂yi − ∂yi ∂yj ∂F ∂ ∂2F ∂F ∂F = F ˜ + F (n 1) . ∂yi C ∂xj ∂yj∂yi − − ∂yi ∂yj We have 2 ∂F ˜ ∂ ∂F ∂F ∂ F F i j dµp = (n 1) i j F j i dµp ZKp ∂y C ∂x ZKp − ∂y ∂y − ∂y ∂y and thus ∂2r n2 1 ∂F ∂F (31) j i = − i j µp. ∂u ∂u p 4 Z∂Kp ∂y ∂y The convexity follows from the positive deﬁniteness of the Hessian matrix of the area function. Remark 4. Equation (29 ) implies that n 1 dr = − β 2 and, consequently, the associated Randers functionals can be written into the following forms 2 2 F (v) := Γ (v, v)+ d log r and F (v) := Γ (v, v)+ d log r. 1 1 n 1 3 3 n 1 p − p − Thereore they are projectively equivalent to the Riemannian manifold U equipped with the weighted Riemannian metric tensors Γ1 and Γ3, respectively; see [7]. On the other hand equation (31 ) shows that the second order partial derivatives of the area function coincide the associated Riemannian metric γ3 up to a constant proportional term. Corollary 5. The Minkowskian indicatrix is ballanced if and only if the area function of the associated Funk space has a global minimizer at the origin. AVERAGE METHODS AND THEIR APPLICATIONS ... 18

To ﬁnish this section recall that the projection ρ provides a nice connection between the indicatrix hypersurfaces ∂Kp and ∂K. Using theorem 2 we have

n−1 − 2 k ∂L r(p)= µp = 1 p k µ, Z∂Kp Z∂K − ∂u where µ = µ0 is the canonical volume form associated to K. From the general theory of Minkowski functionals [1] ∂L wk L(w) ∂uk v ≤ for any nonzero element v. Substituting p U as w we have that ∈ ∂L 1 pk 1 L(p) > 0 − ∂uk ≥ − In case of w = p − ∂L 1 pk 1+ L( p) − ∂uk ≤ − and we have the following estimations:

− n−1 − n−1 2 r(p) 2 (32) 1+ L( p) 1 L(p) , − ≤ r(0) ≤ − where r(0)= 1 µ. Z∂K

5. The proof of the main theorem Theorem 4. (The main theorem). If L is a Finsler-Minkowski functional with a balanced indicatrix body of dimension at least three and the L´evi-Civita connection has zero curvature then L is a norm coming from an inner product, i.e.∇ the Minkowski vector space reduces to a Euclidean one.

Proof. Since the indicatrix body is ballanced it follows from equation (29) that the area function has zero partial derivatives at the origin. The convexity implies that it has a global minimum at 0 which is just the area of the Euclidean sphere of dimension n 1 because of the condition of the vanishing of the curvature (recall that the− dimension is at least three which means that we have a simply connected Riemannian manifold with zero curvature - it is isometric to the standard Euclidean space). If p denotes the centroid of K then the area of ∂Kp is at least r(0) and, by the generalized Santalo’s inequality [4] (for bodies with center at the origin) Kp is a Eu- clidean ball. Since K is a translate of Kp the functional L associated to K is a Randers-Minkowski functional with vanishing curvature tensor Q. This is impossible unless the translation is trivial, i.e. p is just the origin and K is an Euclidean ball as was to be stated. AVERAGE METHODS AND THEIR APPLICATIONS ... 19

Acknowledgements This work was partially supported by the European Union and the Euro- pean Social Fund through the project Supercomputer, the national virtual lab (grant no.:TAMOP-4.2.2.C-11/1/KONV-2012-0010)´ This work is supported by the University of Debrecen’s internal research project. References [1] D. Bao, S. S. Chern, Z. Shen, An introduction to Riemann-Finsler Geometry, Springer-Verlag, Berlin, 2000. [2] F. Brickell, A Theorem on Homogeneous Functions, Journal London Math. Soc., 42 (1967), 325-329. [3] M. Crampin, On the construction of Riemannian metrics for Berwald spaces by averaging, accepted for publication in Houston J. Math. [4] C. E. Duran, A volume comparasion theorem for Finsler manifolds, Proc. Amer. Math. Soc., Vol. 126 No. 10 (1998), 3079-3082. [5] J. Grifone, Structure presque-tangente et connxions I, Ann. Inst. Fourier, Grenoble 22 (1) (1972), 287-333. [6] J. Grifone, Structure presque-tangente et connxions II, Ann. Inst. Fourier, Grenoble 22 (3) (1972), 291-338. [7] M. Hashiguchi, Y. Ichijyo, Randers spaces with rectilinear geodesics, Rep. Fac. Sci. Kagoshima Univ., 13 (1980), 33-40. [8] J. Klein, A. Voutier, Formes exterieures generatrices de sprays, Ann. Inst. Fourier, Grenoble 18 (1) (1968), 241-260. [9] S. R. Lay, Convex Sets and Their Applications, John Wiley & Sons, Inc., 1982. [10] M. Matsumoto, Foundation of Finsler Geometry and Special Finsler Spaces, Kaisheisa Press, Japan, 1986. [11] T. Okada, On models of projectively flat Finsler spaces with constant negative curvature, Tensor NS 40 (1983), 117-123. [12] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic, Dor- drecht, 2001. [13] J. Szilasi, Sz. Vattamány, On the Finsler metrizabilities of spray manioflds, Period. Math. Hungar., 44 (1) (2002), 81-100. [14] J. Szilasi, Cs. Vincze, A new look at Finsler connections and Finsler manifolds, AMAPN 16 (2000), 33-63. [15] A. C. Thompson, Minkowski Geometry, Cambridge University Press, 1996. [16] Cs. Vincze, A new proof of Szabó’s theorem on the Riemann metrizability of Berwald manifolds, AMAPN, Vol. 21 No. 2 (2005), 199-204. [17] Cs. Vincze, On geometric vector fields of Minkowski spaces and their applications, J. of Diff. Geom. and its Appl., 24 (2006), 1-20. [18] F. W. Warner, Foundations of Differenetial Manifolds and Lie Groups, Graduate Texts in Mathematics, 1983.

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