ON MATSUMOTO’S STATEMENT OF BERWALD’S THEOREM ON PROJECTIVE FLATNESS

on the occasion of his 85^thbirthday

ON MATSUMOTO’S STATEMENT OF BERWALD’S THEOREM ON PROJECTIVE FLATNESS

PETER L. ANTONELLI, SOLANGE F. RUTZ and CARLOS E. HIRAKAWA

This work presents a correction on Matsumoto’s statement concerning Berwald’s theorem involving projectively flat 2-dimensional Finsler spaces with rectilinear coordinates. A required condition for the correct statement is presented along with two different proofs, one by spray theory and other by Finsler theory, that Matsumoto’s version is faulty.

AMS 2010 Subject Classification: 53B40, 53C22.

Key words: Finsler and projective geometries, spray theory, Berwald spaces, prin- cipal scalar.

1. INTRODUCTION

In at least two published works of M. Matsumoto on 2-dimensional Berwald manifolds with squared principal scalar I² = 9/2, Berwald’s classic result is misunderstood, [1], [2]. In pursuit of models of evolution of eukary- ote cells in plants and animals, we used his versions of Berwald’s theorem.

Much to our surprise our models led us to a counterexample of Matsumoto’s statement of this theorem. We are here going to present two entirely different proofs that Matsumoto’s version is faulty. One proof is by spray theory and the other purely Finsler theoretic. We begin with some preliminaries in the first section and give the spray result in second section and the Finsler ap- proach in the third and last section. The appendix contains a short description of the biological modeling problem that gave rise to the counterexample. We end this introduction with Matsumoto’s Faulty Statement:

“A 2-dimensional positive definite non-Minkowski Finsler space is pro- jectively flat if and only if the squared principal scalar I² = 9/2 and the metric function can be transformed into the (β)²/γ, where β and γ are 1-forms.”

REV. ROUMAINE MATH. PURES APPL.,57(2012),1, 17-33

The correct version of Berwald’s theorem requires, perhaps after a coordinate transformation, that the metric to be of the form

[dx¹+Z(x¹, x²)dx²]²/dx²

and Z is smooth and satisfies the relationx¹+Zx² =M(Z), whereM is an arbitrary smooth function of Z.

2. PROJECTIVE GEOMETRY 2.1. Local sprays

Consider a smooth connected n-manifold Mⁿ and select a trivializing chart (U, h) onMⁿfor the slit tangent bundle ˜T Mⁿ(i.e., with the zero section removed). A (local) spray G in (U, h) is a system of ode’s

(1) d²xⁱ

ds² + 2Gⁱ

x,dx ds

= 0, i= 1, . . . , n,

where thenfunctionsGⁱareC^ooonU inxⁱ, . . . , xⁿand in dx¹/ds, . . . ,dxⁿ/ds (off the zero section), are otherwise continuous and are second-degree posi- tively homogeneous in the dxⁱ/ds. The path parameter s is special. For a general parameter t along solutions of Eq. (1) we have

(2) x¨ⁱ+ 2Gⁱ(x,x) =˙ s⁰⁰ s⁰x˙ⁱ,

where s⁰ := ds/dt,s⁰⁰:= (s⁰)⁰, ˙xⁱ:= dxⁱ/dtand ¨xⁱ:= d²xⁱ/dt².

Considerψ(x,x), a smooth scalar function on ˜˙ T Mⁿ, which is first-degree positively homogeneous in ˙x¹, . . . ,x˙ⁿ.

The quantities

¨

xⁱ+ 2Gⁱ

˙

xⁱ = x¨^j + 2G^j

˙

x^j , ∀i, j∈ {1, . . . , n}

remain unchanged by the transformation

(3) Gⁱ→G¯ⁱ :=Gⁱ+ψ·x˙ⁱ,

which sends the spray Gin (U, h) to spray ¯Gin (U, h). That is, there exists a diffeomorphism which smoothly maps solutions ofGinto solutions of ¯G. Such a mapping is called the projective transformation of Gonto ¯Gin (U, h).

One obtains from thespray parameter s(i.e., one which makes the RHS of Eq. (2) vanish) a new spray parameter determined by ψ. Namely,

(4) s¯=A+B·

Z

e^2/(n+1)

R

γψ(x,dx/d˜t)d˜tds,

where ˜tis any parameter along any path γ, that is a solution ofG, and A,B are constants of integration.

We can see the effect of this projective change, ortime-sequencing change, by considering the canonical spray connection coefficients in (U, h):

(5) Gⁱ_j := ˙∂jGⁱ, Gⁱ_jk := ˙∂_kGⁱ_j,

where ˙∂_l indicates partial differentiation with respect to ˙x^l. The transforma- tion of coordinates from (U, h) to ( ¯U ,¯h), i.e., from x¹, . . . , xⁿ to ¯x¹, . . . ,x¯ⁿ, has the effect [1],

(6) ∂x¯^r

∂x^j

∂x¯^s

∂x^k

G¯ⁱ_rs= ∂x¯ⁱ

∂x^rG^r_jk− ∂²x¯ⁱ

∂x^j∂x^k.

BecauseGⁱare homogeneous of the second degree in ˙x^l, we have the equivalent expression for Eq. (1)

(7) d²xⁱ

ds² +Gⁱ_jk

x,dx ds

dx^j ds

dx^k ds = 0.

Upon time-sequencing changeψof Eq. (7), we have by differentiation in (U, h) (8) G¯ⁱ_jk =Gⁱ_jk+δ_jⁱψ_k+δ_kⁱψj + ˙xⁱ∂˙_kψj,

where ψ_l = ˙∂_lψ. Note that the 3-index G’s are the local coefficients of the Berwald connection (see [6]) where they are called spray connection coeffi- cients.

Define

(9) Πⁱ:=Gⁱ− 1

n+ 1G^a_ax˙ⁱ, Πⁱ_j := ˙∂jΠⁱ, Πⁱ_jk := ˙∂kΠⁱ_j, for a given spray Gin (U, h). It is easy to see that

(10) Πⁱ_jk =Gⁱ_jk− 1

n+ 1 δⁱ_jG^a_ak+δ_kⁱG^a_aj+ ˙xⁱD^aajk

and that

Π^a_ak = 0.

Dⁱ_jkl := ˙∂_lGⁱ_jk, called the (non-projective) Douglas tensor, transforms as a classical fourth-rank tensor. Its importance lies in the fact that Gⁱ_jk are in- dependent of x˙^l if and only if Dⁱ_jkl = 0. That is, the vanishing of tensor D is necessary and sufficient forGto be aquadratic spray, as in classical affine geo- metry and its specialization to Riemannian geometry. If Gⁱ_jk are constants in (U, h), then we say (7) is aconstant sprayand (U, h) is an adapted coordinate system.

Furthermore, Πⁱ_jk remainsunchanged when G is projectively mapped onto G. Π is called the¯ normal spray connection in (U, h) for G. Its spray curves are solutions of

(11) d²xⁱ

d¯s² + Πⁱ_jkdx^j d¯s

dx^k d¯s = 0.

Remark.(1) ¯sremainsunchangedunder coordinate transformations (U, h)

→ ( ¯U ,h), whose Jacobians lie in¯ SL(n,R), the real unimodular group onRⁿ, and only those(i.e., the structural group of ˜T Mⁿ is reduced fromGL⁺(n,R), the nonsingular real n×n matrices with positive determinant, to SL(n,R)).

Since, D= 0 for the system (∗∗) (see Appendix), normal coordinates exist at any point we care to consider in production space.

(2) Πⁱ_jk transforms as a classical connection (i.e., likeGⁱ_jk above) if and only if transformations have constant Jacobian determinant.

(3) Πⁱ_jk is a tensor if and only if the structural group of ˜T Mⁿ is reduced to the transformation of coordinates (U, h)→( ¯U ,h) of the form¯

¯

xⁱ = aⁱ_jx^j+bⁱ c_kx^k+h and















b¹ aⁱ_j ...

bⁿ c₁. . . c_n h















(n+1)×(n+1) constant matrix with nonzero determinant. This is theclassical projective group.

Following the procedure of KCC Theory and performing path-deviation for spray Eq. (11), we obtain the analogue of the usual “geodesic” deviation equation

(12) D²uⁱ

d¯s² +W_jⁱu^j = 0, where

(13) W_jⁱ = 2∂jΠⁱ−∂rΠⁱ_jx˙^r+ 2Πⁱ_jrΠ^r−Πⁱ_rΠ^r_j.

This occurs as follows: We aregiven the local spray Π in (U, h) and let xⁱ(¯s;η) be a smooth 1-parameter family of solutions with initial conditions xⁱ(0;η), ˙xⁱ(0). Since a spray will have a solution through any pointp∈U and in any direction, these are called arbitrary smooth initial conditions.

By Taylor’s theorem,

xⁱ(¯s;η) =xⁱ(¯s) +ηuⁱ(¯s) +η²(. . .)

and substituting this into Πⁱ_jk(x,x) passage to the limit˙ η → 0, yields the variational equations

(14) d²uⁱ

d¯s² +∂_lΠⁱ_jk(x,x)u˙ ^ldx^j d¯s

dx^k

d¯s + 2Πⁱ_jk(x,x)˙ dx^k d¯s

du^j d¯s = 0.

Defining the projective covariant differential operationas, for example, (15) Aⁱ_/l :=∂_lAⁱ+ Πⁱ_jl(x,x)A˙ ^j,

and

(16) DAⁱ

d¯s :=Aⁱ_/ldx^l d¯s,

with similar formulas holding for higher order tensors.

Using this we can rewrite Eq. (14) as (17) D

d¯s duⁱ

d¯s + Πⁱ_ru^r

+Πⁱ_l du^l

d¯s + Π^l_ru^r

+

2∂rΠⁱ−dΠⁱ_r

d¯s −Πⁱ_lΠ^l_r

u^r= 0, which is precisely Eq. (12) because of Eq. (11) and the second degree homo- geneity of Πⁱ(x,x) in ˙˙ x and Eq. (9).

Now, following Berwarld’s technique, define

(18) W_jkⁱ := 1

3

∂˙_kW_jⁱ−∂˙_jW_kⁱ and (Weyl’s Projective Curvature)

(19) W_jklⁱ := ˙∂_lW_jkⁱ .

This four-index quantity actually is atensor. However, theprojective covariant derivative of a tensor is not necessarily a tensor.

We are now able to state the two main theorems of local projective dif- ferential geometry, [14].

Theorem A.There is a coordinate chart(U, h) onMⁿ,n≥3, such that Πⁱ_jk = 0, if and only if, W_jklⁱ = 0, and∂˙_lΠⁱ_jk := Πⁱ_jkl= 0. The tensor,Πⁱ_jkl, is called the (projective) Douglas tensor.

Theorem B.There is a coordinate chart(U, h)onM²such thatΠⁱ_jk = 0, if and only if, Πⁱ_jkl = 0 and ρ_jkl= 0, where ρ_jkl:= r_jk/l−r_jl/k, r_jk :=B^h_jkh, where

Bⁱ_jkl=∂lΠⁱ_jk+ Π^s_jkΠⁱ_sl+ Πⁱ_jlsΠ^s_mkx˙^m−(k/l).

The symbol, −(k/l), means to repeat all terms that come before but interchange k and l and put a minus in front of the whole expression.

Remark. The four-index tensorB is analogous to the usual curvature of a spray except that Gⁱ_j,Gⁱ_jk are replaced by Πⁱ_j, Πⁱ_jk.

The condition Πⁱ_jk = 0 for alli, j, k∈ {1, . . . , n}in some (U, h) coordinate system is the so-called condition of projective flatness. We now consider the n = 2 case of a normal spray connection curves (11) associated with a given constant spray. We know from Eq. (10) that

Π¹₁₁=−Π²₂₁ and Π²₂₂=−Π¹₁₂.

We can therefore set Π¹₁₁= ¯α₁, Π²₂₂= ¯β₁, Π¹₂₂= ¯α₂ and Π²₁₁= ¯β₂ in Eq. (11), which becomes

(20)

d²x¹ d¯s² + ¯α₁

dx¹ d¯s

2

−2 ¯β₁dx¹ d¯s

dx² d¯s + ¯α₂

dx² d¯s

2

= 0, d²x²

d¯s² + ¯β1

dx² d¯s

2

−2 ¯α1

dx¹ d¯s

dx² d¯s + ¯β2

dx¹ d¯s

2

= 0.

Now,

(21) ρ₁₂₁ = Π¹₁₂r₁₁+ Π²₁₂r₂₁−Π¹₁₁r₁₂−Π²₁₁r₂₂ from Theorem B. But,

(22) r12= Π¹₁₁Π²₂₂−Π²₁₁Π¹₂₂=r21. Also,

(23) Π²₁₂r₂₁−Π¹₁₁r₁₂=−Π¹₁₁[2Π¹₁₁Π²₂₂−Π¹₂₂Π¹₁₁−Π¹₂₂Π²₁₁].

Furthermore,

(24) r₂₂= 2[Π¹₂₂Π¹₂₂−Π²₂₂Π²₂₂], r₁₁= 2[−Π¹₁₁Π¹₁₁+ Π²₂₂Π²₁₁], so that substitution of Eqs. (22) and (24) into Eq. (21), yields

(25) ρ121= 0,

by using Eq. (23). Similarly, one can prove that

(26) ρ212= 0.

It is now clear thatρ_jkl= 0. Also, Πⁱ_jkl= 0 because in this constant connection case the normal spray is quadratic since, in general,

Πⁱ_jkl=Dⁱ_jkl−P 1

n+ 1δ_jⁱD^a_akl

− 1

n+ 1yⁱ∂˙_aD^a_jkl,

whereP means a sum of the three terms obtained by the cyclic permutation of j,k,l. Therefore, Πⁱ_jk = 0 in some coordinate chart ( ¯U ,¯h). We have therefore, proved the following theorem.

Theorem C (Part I). Every two-dimensional constant spray is projec- tively flat, [5].

Remark. It is not true that there is a projective time-sequencing change from, say,

(27)

d²x¹

ds² =−2α₂dx¹ ds

dx² ds +α1

"

dx² ds

2

− dx¹

ds 2#

, d²x²

ds² =−2α₁dx¹ ds

dx² ds +α₂

"

dx¹ ds

2

− dx²

ds 2#

to d²x¹/d¯s² = 0, d²x²/d¯s² = 0, by assuming that α₁, α₂ are not zero. The reason is that Eq. (10) implies

Π¹₂₂6= 0, Π²₁₁6= 0,

since Dⁱ_jkl = 0 holds for Eq. (25). Theorem C states only that there is some coordinate system ( ¯U ,¯h) for which ¯Πⁱ_jk in Eq. (10), vanish. This is where the tensor character of Theorems A and B play an important role.

Theorem C (Part II). In every dimension ≥ 3 there exists a constant spray which is not projectively flat.

Proof. Consider the n-dimensional conformally flat Riemannian metric (g_ij) = e^2φ(x)·(δ_ij), withφ(x) =α_ixⁱ,α_iconstants. It is a well known fact that the Riemannian scalar curvatuteRis never constant and vanishes if and only if n= 2. Yet, the (geodesic) spray of this metric has constant coefficients. But, in Riemannian geometry, projective flatness is equivalent to constant sectional curvatures. Therefore,Rmust be a constant as well, and the proof is complete (see [6]).

Remark. There exist two-dimensional projectively flat Finsler metrics which are not of constant curvature [1], [9]. Obviously, these cannot be Rie- mannian metrics.

We have given the metric (∗) and geodesics equations (∗∗) (see Appen- dix). Now, let φbe a smooth function of x¹ andx² and denote ∂_iφ asφ_i and likewise φ_ijk =∂i∂j∂_kφ, etc.

Theorem D.For the Finsler metric(∗), the geodesics are rectilinear for some coordinates x¯ⁱ if and only iff ρ121 = 0, where

ρ121 = λ(2λ+ 1)

3(λ+ 1) [φ121+λφ1φ12], and ρ₂₁₂= 0, where

ρ₂₁₂ = λ(λ−1) 3(λ+ 1)

φ₂₁₂− λ

λ+ 1φ₂φ₁₂

.

Remarks. (1)ρ121 = 0⇔λ=−1/2, sinceφ is arbitrary and also

(2)ρ212= 0⇔λ= 1, since, again, φ is an arbitrary smooth function of x¹,x².

But, from Remarks (1) and (2) above, have a contradiction if bothρ₁₂₁= 0 and ρ212 = 0. We conclude that (∗) is not projectively flat, so cannot have rectilinear geodesics.

We give now a different proof, one using Berwald’s formulaIKs=−3K_b, characterizing projectively flat 2-dimensional Finsler spaces (see [9] for defini- tions)

φ=−α₁x¹+ (λ+ 1)α2x²+ν3x¹x², F¯= e^φ·(y²)^1+1/λ

(y¹)^1/λ , 2G¹=−λφ₁(y¹)², 2G² = λ

λ+ 1φ₂(y²)², I²= (λ+ 2)²

λ+ 1 ,

√g¯= e^2φ·

√λ+ 1 λ ·

y² y¹

1+2/λ

, m¹ =− l2

√¯g, m² = l1

√g¯, li = ˙∂iF .¯

Notation:

δ_iK =∂_iK−G^r_i∂˙_rK, G¹₁=−λφ₁y¹, G²₂ = λ

λ+ 1φ₂y², G¹₂ = 0 =G²₁, lⁱ =yⁱ/F ,¯ l1 =−1

λe^φ y²

y¹

1+1/λ

, l2= λ+ 1 λ

y² y¹

1/λ

e^φ, m¹=−√

λ+ 1 y¹

y²

1+1/λ

e^−φ, m² =− 1

√λ+ 1e^−φ y¹

y² 1/λ

, K = λ²

λ+ 1·ν3· y¹

y²

1+2/λ

·e^−2φ, (δiK)mⁱ =K_b,

δ_iK =∂_iK−G^r_i∂˙_rK,

∂iK =−2φ_iK,

∂˙_iK = λ(λ+ 2)

λ+ 1 ·ν₃·e^−2φ· 1 y² ·

y¹ y²

2/λ

,

∂˙2K=−λ(λ+ 2)

λ+ 1 ·ν3·e^−2φ· y¹

y²

1+2/λ

· 1 y², δ₁K=λKφ₁,

δ2K=− λ

λ+ 1φ2K, K_b =−λKe^−φ

y¹ y²

1/λ√

λ+ 1φ1

y¹

y² − φ2

(λ+ 1)^3/2

, K_s= (δ_iK)lⁱ,

Ks=λKe^−φ y¹

y² 1/λ

φ1

y¹

y² − φ2

λ+ 1

.

Projective flatness holds iff IK_s = −3K_b. Since φ_i are functions of x¹ and x² only, this identity implies both λ = 1 and λ = −1/2, a contradiction of K 6= 0, φ₁, φ₂6= 0.This is an example of a 2-dimensional Bewarld space with I² = 9/2 and with K 6= 0 which is not projectively flat. Berwald provided many examples which are projectively flat [9].

On page 838 of theHandbook of Finsler Geometry, M. Matsumoto claims that a Berwald 2-space with I² = 9/2 andF =γ²/β, whereγ and β are inde- pendent 1-forms, must be projectively flat (with rectilinear extremals). This isnotBerwald’s theorem. It has been misunderstood by Prof. Matsumoto. In- deed, takingγ = e^φ(x)·(y²), β =y¹,withφ(x) =−α₁x¹+(λ+1)α₂x²+ν₃x¹x², λ = 1, then I² = 9/2, from above, and noting F leads to a positive definite gij on a suitable positively conical region of ˙T M²,the slit tangent bundle, we exhibit a conter-example.

Now, following Berwald,

(28) ds=F(x,dx) = (dx¹+Z(x¹, x²)dx²)² dx²

must satisfy

d dγ

Z

γ

ds= 0.

Furthermore, from [1], vol. II, page 801, we have (29) ∂_x¹∂˙_x²F−∂_x²∂˙_x¹F = 0

as a necessary and sufficient condition for projectively flat with rectilinear coordinates. Using (28) and (29), one easily arrives at

(30) Z∂_x1Z−∂_x2Z= 0.

The solution Z = constant yieldsR= 0, but there are plenty of non-constant solutions Z(x, y). By integration we arrive at

(31) x¹+x²Z=M(Z),

where M(Z) is an arbitrary smooth function ofZ. Berwald also showed that

(32) R = M⁰⁰(Z)

(M⁰(Z)−x²)³ ·

y² y¹+Zy²

3

.

Therefore, those F_s for which M⁰⁰(Z)>0,M⁰(Z)−x² >0 will be candidates for projective flat with rectilinear coordinates. The system equations are now written as

(33)









 d²x¹

dt² +dx¹ dt =

Φ(x¹, x²)dx² dt

dx¹ dt , d²x²

dt² +dx² dt =

Φ(x¹, x²)dx² dt

dx² dt ,

where we have used the parameter transformations=A−B exp(−t), >0, the Douglas tensor and

(34) φ(x¹, x²) = 1

M⁰(Z)−x².

We knowD= 0 for (33). In the next section we prove that the expression (34) cannot be linear nor quadratic in x¹ and x².

3. THEOREM E

Proposition 1. Zx = Φ(x, y) does not admit local extrema. We use notation x=x¹,y=x², ˙x¹=y¹ and ˙x² =y².

Proof. We know x+yZ =M(Z), with M⁰⁰(Z)6= 0. Clearly, (35) yZxx =M⁰⁰(Z)·(Zx)²+M⁰(Z)·Zxx,

so that

Z_xx = M⁰⁰(Z)·(Z_x)² y−M⁰(Z) . But, clearly, 1 +yZx=M⁰(Z)Zx, so

Z_x= 1 M⁰(Z)−y. Therefore,

(36) Zxx =−M⁰⁰(Z)·(Zx)³ and, since any local extrema must satisfy

(Z_x)_x = 0 = (Z_x)_y

we would have Z_x = 0 at any such point (x_o, y_o). This contradicts Z_x = 1/(M⁰(Z)−y) never zero.

Proposition 2. Zx = Φ(x, y) is not a linear nor a quadratic form in x, y.

Proof. (A) Suppose Φ = ax+by+c. Fix any x =x_o and solve to get yo = −^a_bxo − ^c_b so that Φ(xo, yo) = 0. If b = 0, take x = xo = −c/a. If a=b= 0,then Φ =c=Z_x and (35) implies

M⁰⁰(Z)·c² = 0

so c = 0 (since M⁰⁰(Z) 6= 0). Therefore, in all cases of Φ linear we see that Zx= 0 at some point, which is impossible.

(B) Suppose Φ =a_ijxⁱx^j+b_ixⁱ+c(with x¹ =x,x² =y and summation convention). Clearly, Φx= 0 = Φy has alwaysat leastone solution (unique, if det(a_ij)6= 0). Hence, Z_xx = 0 and (36) implyZ_x= 0 at this point.

Proposition 3. The Berwald I² = 9/2 geodesics are

(37)









 d²x¹

ds² =

Φ(x¹, x²)dx² ds

dx¹ ds, d²x²

ds² =

Φ(x¹, x²)dx² ds

dx² ds.

They are projectively flat (without any change of coordinates)and Φ(x¹, x²) is never linear nor quadratic.

Proof. Proposition 2. From (36) we know Z_xx =−M⁰⁰(Z)(Z_x)³ and since

K=M⁰⁰(Z)·(Z_x)³· y˙

˙ x+Zy˙

3

, we have

K =−Z_xx y˙

˙ x+Zy˙

3

. Taking ∂_x of K we get

K_x = y˙

˙ x+Zy˙

3

·

−Z_xxx+3Z_xxZ_xy˙

˙ x+Zy˙

. Therefore, K_x = 0 at (x_o, y_o) = (x, y)⇔

Z_xxx = 3Z_xxZ_x ( ˙x/y˙+Z(x, y)), or, from (36),

Z_xxx =−3M⁰⁰(Z)(Z_x)⁴

˙

x/y˙+Z(x, y).

Note

∂_x˙K= 1

˙

y∂_{x/˙ y˙}K, ∂_y˙K =− x˙

( ˙y)² ·∂_{x/˙ y˙}K⇒∂_x˙K, ∂_y˙K never zero.

The left-hand side is just∂x∂x∂xZ(x, y) and is independent of ˙x,y˙or ˙x/y,˙ since Z(x, y) is such, by definition. But, since M⁰⁰(Z) 6= 0 andZ_x 6= 0 everywhere in (x, y)-space, the derivative of the right-hand side with respect to ˙x/y˙ is

3M⁰⁰(Z)(Zx)⁴ ( ˙x/y˙+Z(x, y))²

and is never zero – contradiction. Hence, Kx is never zero in (x, y)-space and (likewise,K_y 6= 0 ∀(x, y)). We have proved

Theorem E.K(x, y,x,˙ y)˙ as a function onT M˜ ², the slit tangent bundle, cannot have local extremals on M².

The only 2-dimensional Riemannian space with rectilinear extremals and K >0 is the sphere, all points of which are local extremals sinceK= constant.

4. APPENDIX

According to the Ancestral Commune Theory of Carl Woese, life started as a loose conglomerate of many types of proto-cells, some 4000 MYBP. A billion years later trading loose bits of RNA and DNA between these pre- cursors gave way to Darwinian natural selection. An anaerobic thermophile bacterium was present in the primordial soup and possessed a nuclear genome.

This evolved Prokaryote was subsequently parasitized by a bacterium that had a flagellum for swimming and could process oxygen for its energy. It is called a mitochondrian, while the symbiocosm is called a Eukaryote. It is the common ancestor of all creatures accept the Prokaryotes themselves. This theory has been fully validated via molecular genetics and is credited to Lynn Margulis.

It is therefore obvious that the most ancient symbiocosm is the Eukaryote cell. For plant species, one has the chloroplasts as well as mitochondia, both of which occur in variable small numbers across the plant and animal species spectrum. Using the Volterra-Hamilton method we have compared the evolu- tionary theories of Woese and Margulis and because we treated both within a single logical framework, in spite of the fact that the communes of proto- cells in Woese’s theory predate the evolved Eukaryote symbiocosm, sensible comparison was achieved. We are now going to recount the detailed mathe- matical material.

Using the Volterra-Hamilton systems as a logical method we have com- pared (see [12] for recent advances in endosymbiosis) the evolutionary theories

of Carl Woese and Lynn Margulis and found the former suffers from robust in- stability while the later is robustly stable in its production processes [13], [18].

Nⁱ = density of i-th bacterial population, assumed to satisfy classical logistic dynamics

dNⁱ

dt =λNⁱ(1−α_(i)Nⁱ),

with pre-symbiont condition (all lambdas equal), i= 1,2, . . . , n and repeated indices are summed except if there is a parenthesis. Since our model will describe ecology and chemical production (in the form of modular bits of RNA), we introduce the Volterra production equation

dxⁱ

dt =k_(i)Nⁱ.

We require that, for either model, our evolved system has the form d²xⁱ

ds² +Gⁱ_jkdx^j ds

dx^k ds = 0,

where the n³ coefficients are constants or involve xⁱ. This class of dynamical systems will encompass both the primeval and the evolved systems, and serve for modeling either Margulis’ or Woese’ theories.

In order to accommodate our ergonomics, i.e., division of labour, we require the production parameter to be given by the cost of production func- tional, ds = F(x,dx) > 0. Moreover, F is to be positively homogeneous of degree 1 in dx= (dx¹, . . . ,dxⁿ),that is, for any positive constantc,

F

x, c dx dt

=c F

x,dx dt

,

so that ds/dt, the rate of production in the symbiocosm, depends on indi- vidual bacterial rates dxⁱ/dt through the cost F(x,dx/dt). The arc-length s = Rt

t0F(x,dx/dt)dt represents the total production of the symbiocosm in the interval (t2−t1) along a given curvex(t) = (x¹(t), . . . , xⁿ(t)).The homo- geneity of F means that, if all the individual rates dxⁱ/dtare magnified by a factor of c,then ds/dtis so magnified. This forces s to be independent of the time measure.

We have to introduce the expression H_s = (1/2)F²(x,dx/dt), which yields Euler-Lagrange equations. Note that H_s is therefore positively homo- geneous of degree 2 in dx/dt, thus, multiplying each dxⁱ/dt by a positive constant c implies that H_s is multiplied by c². Furthermore, H_s defines two classes of systems, namely, the Riemannian class, where Hs is quadratic in dx/dt, and the Finsler (non-Riemannian) class, where H_s is not quadratic, but is homogeneous of degree 2 in dx/dt.Here are two examples, one for each

class. For the former, quadratic case, we may have Hs= 1

2 e^2αixi

"

dx¹ dt

2

+· · ·+ dxⁿ

dt 2#

, while for the latter, non-quadratic but homogeneous case, we have

(∗) H_s= 1

2 e^2φ(x)(dx²/dt)^2+2/λ (dx¹/dt)^2/λ ,

where we have taken n= 2, λ is a positive constant and φ(x) is an arbitrary polynomial on x¹ andx².We will see in the following sections that the former applies to Woese’s theory, while the latter applies to Margulis’ theory.

To solve the problem we need to use the techniques of Finsler geometry.

Our main result is that, for Hs given as above, with φ(x) = −α₁x¹ + (l+ 1)α₂x² +ν₃x¹x², and λ > 0, α_i > 0 and ν₃ non-zero, the Euler-Lagrange equations are

(∗∗)









 dy¹

ds +λ(α₁−ν₃x²)·(y¹)² = 0, dy²

ds +λ

α2+ ν3

λ+ 1x¹

·(y²)²= 0.

Note that, if ν₃ = 0, then the original double logistic system are obtained.

(Where ds= e^λtdtmust be employed to transform to realtimetfrom parame- ter s.) Moreover, Liapunov stability of this system is completely determined by the sign of the curvature

K= λ² λ+ 1ν3

y¹ y²

1+2/λ

·exp(−2[−α₁x¹+ (λ+ 1)α2x²+ν3x¹x²]).

If ν₃>0,then stability results, while the reverse is true forν₃<0.Geodesics of 2-dimension positively curved spaces, such as a sphere, will remain close generally in x-space, the system being, therefore, stable, while for those with a negative or zero curvature, as a trumpet surface or a plane, respectively, will not, yielding unstable systems. The parameter ν3 is called the exchange parameter. Thus, system (∗∗) has stable production. Index # 1 indicates the parasite and # 2, the host.

In the Ancestral Commune model we disallow explicit xⁱ in the coeffi- cients, but allow the number of species to be large. This is our model of a

“loose conglomerate of diverse bacterial species”. Neither do we allow the coefficients to depend on the populations’ sizes, as this would be inclusion of social interactions. As before, we will try for a quadratic cost functional as the simplest possible. Using the fundamental theorem of Volterra-Hamilton systems to arrive at the equation above as then-dimensional functional, using

the additional assumption that our community issimple. This means precisely that each bacterial species in the conglomerate exchanges chemical informa- tion with at least one other. If this were not assumed, then the fundamental theorem ensures that the conglomerate splits into simple sub-communities, each of the above type, i.e., having the previously stated, quadratic H_s, as cost functional, with n varying from one community to another, totaling the original number of species. The dynamic equations which are generated by the cost functional through the calculus of variations, are actually Euler-Lagrange equations, with coefficients Gⁱ_jk as

(∗ ∗ ∗)

Gⁱ_ii=α_i, Gⁱ_jk = 0, i6=j6=k, Gⁱ_ij =Gⁱ_ji =αj, i6=j, Gⁱ_jj =−α_i, i6=j.

These coefficients of interaction completely characterize our model of Ancestral Commune. It is especially significant that the Gⁱ_jk,with all indexes different, vanishes. For example, for a 3 species conglomerate, G¹₂₃, G²₁₃, G³₁₂all vanish, which means there are no higher-order interactions, so that species 2 and 3 do not have an interaction which influences species 1,etc. This is a consequence of our model. It is consistent with the “loose conglomerate” of Woese.

Another consequence is that the scalar curvature R is given by R=−(n−1)(n−2)exp(−2α_ixⁱ)[(α1)²+· · ·+ (αn)²].

As one can notice from the above equation, R increases quadratically with n (all other quantities being constant), becoming ever more negative, and so unstable, the larger the commune. The Riemann scalar curvature R and Berwald’s Gaussian curvatureKplay an important role in stochastic problems in curved spaces.

The stochastic version of the above theory comparison have been pub- lished [7]. The results are essentially the same: Woese’s theory suffers from instability in the chemical exchanges while Margulis does not. We had to employ the theory of noise in Finsler spaces due Antonelli and Zastawniak [8].

Representing the Serial Endosymbiosis Theory by a succession of stages, let us consider the initial stage, the time when a bacterium parasitizes a Prokaryote. A simple model, based on the Volterra-Hamilton system for this stage, can be given considering

Gⁱ_ii=α_i, Gⁱ_jj = 0, i6=j, α_i >0, Gⁱ_ij =Gⁱ_ji =−β_i, G^j_ij =G^j_ji =βj, i6=j, βi, βj >0,

with i, j = 1,2. We will search for a model for the change of state between parasitism and endosymbiosis. To do so, we will use the biological concept of heterochrony and mathematical concepts from projective geometry.

Heterochrony, considered an important evolutionary process, is defined as a time-sequencing change in the ontogenetic process, which describes the origin and development of the individual from the fertilized egg to its adult form. In other words, heterochrony can be understood as evolutionary changes in an individual caused by changes in the pace of development. In our context, we will use this concept as an analogy for the evolutionary process of the Margulis Theory in which the evolutionary transition between stages occurs haphazardly. For the modeling of heterochrony we will use the projective geometry, implying a time-sequencing change in the parametertbetween these stages.

The above parasitic system is projectively flat by Theorem C, so there is a time-sequencing change from it to the geodesic system of Theorem E. The system (∗∗) above has no time-sequencing change, coming from or going to, the parasitic one, as it is not projectively flat. Our model of the Endosymbiosis is therefore any one allowed by Theorem E.

Acknowledgements. Dr. Antonelli was partially supported by research grant from the state of Pernambuco: BFP-0043-1.01/11.

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Received 2 February 2012 University of Alberta

Department of Math. Sci.

Edmonton, Canada and

Visiting Prof. UFPE, Mat. Dept.

Recife, PE, Brazil peter.antonelli@gmail.com UFPE, Mat. Dept., Recife, PE, Brazil

solange.rutz@gmail.com UERJ, IME, Rio de Janeiro, RJ, Brazil

carlos-mat@oi.com.br