Dedicated to Professor Radu Miron
on the occasion of his 85th birthday
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, principal scalar.
1. INTRODUCTION
In at least two published works of M. Matsumoto on 2-dimensional
Berwald manifolds with squared principal scalar I 2 = 9/2, Berwald’s classic
result is misunderstood, [1], [2]. In pursuit of models of evolution of eukaryote 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 approach 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 projectively flat if and only if the squared principal scalar I 2 = 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
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Peter L. Antonelli, Solange F. Rutz and Carlos E. Hirakawa
2
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 relation x1 + Zx2 = M (Z), where M is an
arbitrary smooth function of Z.
2. PROJECTIVE GEOMETRY
2.1. Local sprays
Consider a smooth connected n-manifold M n and select a trivializing
chart (U, h) on M n for the slit tangent bundle T̃ M n (i.e., with the zero section
removed). A (local) spray G in (U, h) is a system of ode’s
d2 xi
dx
i
(1)
+ 2G x,
= 0, i = 1, . . . , n,
ds2
ds
where the n functions Gi are C oo on U in xi , . . . , xn and in dx1 /ds, . . . , dxn /ds
(off the zero section), are otherwise continuous and are second-degree positively homogeneous in the dxi /ds. The path parameter s is special. For a
general parameter t along solutions of Eq. (1) we have
(2)
ẍi + 2Gi (x, ẋ) =
s00 i
ẋ ,
s0
where s0 := ds/dt, s00 := (s0 )0 , ẋi := dxi /dt and ẍi := d2 xi /dt2 .
Consider ψ(x, ẋ), a smooth scalar function on T̃ M n , which is first-degree
positively homogeneous in ẋ1 , . . . , ẋn .
The quantities
ẍj + 2Gj
ẍi + 2Gi
=
, ∀i, j ∈ {1, . . . , n}
ẋi
ẋj
remain unchanged by the transformation
(3)
Gi → Ḡi := Gi + ψ · ẋi ,
which sends the spray G in (U, h) to spray Ḡ in (U, h). That is, there exists a
diffeomorphism which smoothly maps solutions of G into solutions of Ḡ. Such
a mapping is called the projective transformation of G onto Ḡ in (U, h).
One obtains from the spray parameter s (i.e., one which makes the RHS
of Eq. (2) vanish) a new spray parameter determined by ψ. Namely,
Z
R
(4)
s̄ = A + B · e2/(n+1) γ ψ(x,dx/dt̃)dt̃ ds,
3
On Matsumoto’s statement of Berwald’s theorem on projective flatness
19
where t̃ is any parameter along any path γ, that is a solution of G, and A, B
are constants of integration.
We can see the effect of this projective change, or time-sequencing change,
by considering the canonical spray connection coefficients in (U, h):
(5)
Gij := ∂˙j Gi ,
Gijk := ∂˙k Gij ,
where ∂˙l indicates partial differentiation with respect to ẋl . The transformation of coordinates from (U, h) to (Ū , h̄), i.e., from x1 , . . . , xn to x̄1 , . . . , x̄n ,
has the effect [1],
(6)
∂ 2 x̄i
∂ x̄r ∂ x̄s i
∂ x̄i r
G
−
Ḡ
=
.
∂xj ∂xk rs ∂xr jk ∂xj ∂xk
Because Gi are homogeneous of the second degree in ẋl , we have the equivalent
expression for Eq. (1)
dx dxj dxk
d2 xi
i
+ Gjk x,
= 0.
(7)
ds2
ds ds ds
Upon time-sequencing change ψ of Eq. (7), we have by differentiation in (U, h)
(8)
Ḡijk = Gijk + δji ψk + δki ψj + ẋi ∂˙k ψj ,
where ψl = ∂˙l ψ. Note that the 3-index G’s are the local coefficients of the
Berwald connection (see [6]) where they are called spray connection coefficients.
Define
1
(9)
Πi := Gi −
Ga ẋi , Πij := ∂˙j Πi , Πijk := ∂˙k Πij ,
n+1 a
for a given spray G in (U, h). It is easy to see that
(10)
Πijk = Gijk −
1
δji Gaak + δki Gaaj + ẋi Daajk
n+1
and that
Πaak = 0.
Dijkl := ∂˙l Gijk , called the (non-projective) Douglas tensor, transforms as a
classical fourth-rank tensor. Its importance lies in the fact that Gijk are independent of ẋl if and only if Dijkl = 0. That is, the vanishing of tensor D is
necessary and sufficient for G to be a quadratic spray, as in classical affine geometry and its specialization to Riemannian geometry. If Gijk are constants in
(U, h), then we say (7) is a constant spray and (U, h) is an adapted coordinate
system.
20
Peter L. Antonelli, Solange F. Rutz and Carlos E. Hirakawa
4
Furthermore, Πijk remains unchanged when G is projectively mapped onto
Ḡ. Π is called the normal spray connection in (U, h) for G. Its spray curves
are solutions of
(11)
j
k
d2 xi
i dx dx
+
Π
= 0.
jk
ds̄2
ds̄ ds̄
Remark. (1) s̄ remains unchanged under coordinate transformations (U, h)
→ (Ū , h̄), whose Jacobians lie in SL(n, R), the real unimodular group on Rn ,
and only those (i.e., the structural group of T̃ M n is reduced from GL+ (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) Πijk transforms as a classical connection (i.e., like Gijk above) if and
only if transformations have constant Jacobian determinant.
(3) Πijk is a tensor if and only if the structural group of T̃ M n is reduced
to the transformation of coordinates (U, h) → (Ū , h̄) of the form


b1

.. 
i xj + bi


i
a
. 
 aj
j
and
x̄i =



ck xk + h
n 
b


c1 . . . cn h
(n+1)×(n+1) constant matrix with nonzero determinant. This is the classical
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)
D2 ui
+ Wji uj = 0,
ds̄2
where
(13)
Wji = 2∂j Πi − ∂r Πij ẋr + 2Πijr Πr − Πir Πrj .
This occurs as follows: We are given the local spray Π in (U, h) and let
xi (s̄; η) be a smooth 1-parameter family of solutions with initial conditions
xi (0; η), ẋi (0). Since a spray will have a solution through any point p ∈ U and
in any direction, these are called arbitrary smooth initial conditions.
By Taylor’s theorem,
xi (s̄; η) = xi (s̄) + ηui (s̄) + η 2 (. . .)
5
On Matsumoto’s statement of Berwald’s theorem on projective flatness
21
and substituting this into Πijk (x, ẋ) passage to the limit η → 0, yields the
variational equations
j
k
d2 ui
dxk duj
i
l dx dx
i
+
∂
Π
(x,
ẋ)u
+
2Π
(x,
ẋ)
= 0.
l
jk
jk
ds̄2
ds̄ ds̄
ds̄ ds̄
Defining the projective covariant differential operation as, for example,
(14)
(15)
Ai/l := ∂l Ai + Πijl (x, ẋ)Aj ,
and
DAi
dxl
:= Ai/l
,
ds̄
ds̄
with similar formulas holding for higher order tensors.
Using this we can rewrite Eq. (14) as
l
D dui
dΠir
i du
i
i r
l r
i l
(17)
+ Πr u +Πl
+ Πr u + 2∂r Π −
− Πl Πr ur = 0,
ds̄ ds̄
ds̄
ds̄
(16)
which is precisely Eq. (12) because of Eq. (11) and the second degree homogeneity of Πi (x, ẋ) in ẋ and Eq. (9).
Now, following Berwarld’s technique, define
1
i
:= ∂˙k Wji − ∂˙j Wki
(18)
Wjk
3
and (Weyl’s Projective Curvature)
(19)
i
i
Wjkl
:= ∂˙l Wjk
.
This four-index quantity actually is a tensor. However, the projective covariant
derivative of a tensor is not necessarily a tensor.
We are now able to state the two main theorems of local projective differential geometry, [14].
Theorem A. There is a coordinate chart (U, h) on M n , n ≥ 3, such that
i = 0, and ∂˙ Πi := Πi = 0. The tensor, Πi , is
Πijk = 0, if and only if, Wjkl
l jk
jkl
jkl
called the (projective) Douglas tensor.
Theorem B. There is a coordinate chart (U, h) on M 2 such that Πijk = 0,
if and only if, Πijkl = 0 and ρjkl = 0, where ρjkl := rjk/l − rjl/k , rjk := Bhjkh ,
where
Bijkl = ∂l Πijk + Πsjk Πisl + Πijls Πsmk ẋm − (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 tensor B is analogous to the usual curvature of
a spray except that Gij , Gijk are replaced by Πij , Πijk .
22
Peter L. Antonelli, Solange F. Rutz and Carlos E. Hirakawa
6
The condition Πijk = 0 for all i, 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 = ᾱ1 , Π222 = β̄1 , Π122 = ᾱ2 and Π211 = β̄2 in Eq. (11),
which becomes
1 2
2 2
dx
dx
d2 x1
dx1 dx2
+ ᾱ1
− 2β̄1
+ ᾱ2
= 0,
2
ds̄
ds̄
ds̄ ds̄
ds̄
(20)
2 2
1 2
dx
d2 x2
dx
dx1 dx2
+ β̄1
− 2ᾱ1
+ β̄2
= 0.
2
ds̄
ds̄
ds̄ ds̄
ds̄
Now,
ρ121 = Π112 r11 + Π212 r21 − Π111 r12 − Π211 r22
(21)
from Theorem B. But,
r12 = Π111 Π222 − Π211 Π122 = r21 .
(22)
Also,
(23)
Π212 r21 − Π111 r12 = −Π111 [2Π111 Π222 − Π122 Π111 − Π122 Π211 ].
Furthermore,
(24)
r22 = 2[Π122 Π122 − Π222 Π222 ],
r11 = 2[−Π111 Π111 + Π222 Π211 ],
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 that ρjkl = 0. Also, Πijkl = 0 because in this constant connection
case the normal spray is quadratic since, in general,
1
1
i
i
i a
Πjkl = Djkl − P
δj Dakl −
y i ∂˙a Dajkl ,
n+1
n+1
where P means a sum of the three terms obtained by the cyclic permutation of
j, k, l. Therefore, Πijk = 0 in some coordinate chart (Ū , h̄). We have therefore,
proved the following theorem.
Theorem C (Part I). Every two-dimensional constant spray is projectively flat, [5].
7
On Matsumoto’s statement of Berwald’s theorem on projective flatness
23
Remark. It is not true that there is a projective time-sequencing change
from, say,
"
2 1 2 #
d2 x1
dx1 dx2
dx2
dx
= −2α2
+ α1
−
,
2
ds
ds ds
ds
ds
"
(27)
2 2 2 #
dx1
d2 x2
dx1 dx2
dx
= −2α1
+ α2
−
2
ds
ds ds
ds
ds
to d2 x1 /ds̄2 = 0, d2 x2 /ds̄2 = 0, by assuming that α1 , α2 are not zero. The
reason is that Eq. (10) implies
Π122 6= 0,
Π211 6= 0,
since Dijkl = 0 holds for Eq. (25). Theorem C states only that there is some
coordinate system (Ū , h̄) for which Π̄ijk 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
(gij ) = e2φ(x) ·(δij ), with φ(x) = αi xi , αi constants. It is a well known fact that
the Riemannian scalar curvatute R is 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, R must 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 Riemannian metrics.
We have given the metric (∗) and geodesics equations (∗∗) (see Appendix). Now, let φ be a smooth function of x1 and x2 and denote ∂i φ as φi and
likewise φijk = ∂i ∂j ∂k φ, etc.
Theorem D. For the Finsler metric (∗), the geodesics are rectilinear for
some coordinates x̄i if and only iff ρ121 = 0, where
λ(2λ + 1)
ρ121 =
[φ121 + λφ1 φ12 ] ,
3(λ + 1)
and ρ212 = 0, where
λ(λ − 1)
λ
ρ212 =
φ212 −
φ2 φ12 .
3(λ + 1)
λ+1
Remarks. (1) ρ121 = 0 ⇔ λ = −1/2, since φ is arbitrary and also
24
Peter L. Antonelli, Solange F. Rutz and Carlos E. Hirakawa
8
(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 both ρ121 =
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 formula IKs = −3Kb ,
characterizing projectively flat 2-dimensional Finsler spaces (see [9] for definitions)
φ = −α1 x1 + (λ + 1)α2 x2 + ν3 x1 x2 ,
(y 2 )1+1/λ
,
(y 1 )1/λ
λ
2G1 = −λφ1 (y 1 )2 , 2G2 =
φ2 (y 2 )2 ,
λ+1
(λ + 2)2
,
I2 =
λ+1
√
2 1+2/λ
√
λ+1
y
2φ
ḡ = e ·
·
,
λ
y1
l2
l1
m1 = − √ , m 2 = √ ,
ḡ
ḡ
li = ∂˙i F̄ .
F̄ = eφ ·
Notation:
δi K = ∂i K − Gri ∂˙r K,
λ
φ2 y 2 , G12 = 0 = G21 ,
G11 = −λφ1 y 1 , G22 =
λ+1
1+1/λ
1/λ
λ + 1 y2
1 φ y2
i
i
, l2 =
eφ ,
l = y /F̄ , l1 = − e
λ
y1
λ
y1
1 1+1/λ
1 1/λ
√
y
1
1
−φ
2
−φ y
m =− λ+1
e , m = −√
e
,
y2
y2
λ+1
1 1+2/λ
y
λ2
K=
· ν3 ·
· e−2φ ,
λ+1
y2
(δi K)mi = Kb ,
δi K = ∂i K − Gr ∂˙r K,
i
∂i K = −2φi K,
λ(λ + 2)
1
∂˙i K =
· ν3 · e−2φ · 2 ·
λ+1
y
y1
y2
2/λ
,
9
On Matsumoto’s statement of Berwald’s theorem on projective flatness
25
1 1+2/λ
y
1
λ(λ
+
2)
−2φ
· ν3 · e
·
· 2,
∂˙2 K = −
2
λ+1
y
y
δ1 K = λKφ1 ,
λ
δ2 K = −
φ2 K,
λ+1
1 1/λ √
y1
φ2
−φ y
,
Kb = −λKe
λ + 1φ1 2 −
y2
y
(λ + 1)3/2
Ks = (δi K)li ,
1 1/λ y
y1
φ2
Ks = λKe−φ
φ
.
−
1 2
y2
y
λ+1
Projective flatness holds iff IKs = −3Kb . Since φi are functions of x1 and
x2 only, this identity implies both λ = 1 and λ = −1/2, a contradiction of
K 6= 0, φ1 , φ2 6= 0. This is an example of a 2-dimensional Bewarld space with
I 2 = 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 the Handbook of Finsler Geometry, M. Matsumoto claims
that a Berwald 2-space with I 2 = 9/2 and F = γ 2 /β, where γ and β are independent 1-forms, must be projectively flat (with rectilinear extremals). This
is not Berwald’s theorem. It has been misunderstood by Prof. Matsumoto. Indeed, taking γ = eφ(x) ·(y 2 ), β = y 1 , with φ(x) = −α1 x1 +(λ+1)α2 x2 +ν3 x1 x2 ,
λ = 1, then I 2 = 9/2, from above, and noting F leads to a positive definite
gij on a suitable positively conical region of Ṫ M 2 , 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
Z
d
ds = 0.
dγ γ
Furthermore, from [1], vol. II, page 801, we have
(29)
∂x1 ∂˙x2 F − ∂x2 ∂˙x1 F = 0
as a necessary and sufficient condition for projectively flat with rectilinear
coordinates. Using (28) and (29), one easily arrives at
(30)
Z∂x1 Z − ∂x2 Z = 0.
The solution Z = constant yields R = 0, but there are plenty of non-constant
solutions Z(x, y). By integration we arrive at
(31)
x1 + x2 Z = M (Z),
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Peter L. Antonelli, Solange F. Rutz and Carlos E. Hirakawa
10
where M (Z) is an arbitrary smooth function of Z. Berwald also showed that
3
M 00 (Z)
y2
(32)
R=
·
.
(M 0 (Z) − x2 )3
y 1 + Zy 2
Therefore, those Fs for which M 00 (Z) > 0, M 0 (Z) − x2 > 0 will be candidates
for projective flat with rectilinear coordinates. The system equations are now
written as
 2 1
1
2 dx1
dx
d
x
dx

1
2

,

 dt2 + dt = Φ(x , x ) dt
dt
(33)

dx2
d2 x2
dx2 dx2


1
2

+
= Φ(x , x )
,
dt2
dt
dt
dt
where we have used the parameter transformation s = A − B exp(−t), > 0,
the Douglas tensor and
1
(34)
φ(x1 , x2 ) = 0
.
M (Z) − x2
We know D = 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 , ẋ1 = y 1 and ẋ2 = y 2 .
Proof. We know x + yZ = M (Z), with M 00 (Z) 6= 0. Clearly,
(35)
yZxx = M 00 (Z) · (Zx )2 + M 0 (Z) · Zxx ,
so that
M 00 (Z) · (Zx )2
.
y − M 0 (Z)
But, clearly, 1 + yZx = M 0 (Z)Zx , so
1
Zx = 0
.
M (Z) − y
Therefore,
Zxx =
(36)
Zxx = −M 00 (Z) · (Zx )3
and, since any local extrema must satisfy
(Zx )x = 0 = (Zx )y
we would have Zx = 0 at any such point (xo , yo ). This contradicts Zx =
1/(M 0 (Z) − y) never zero. 11
On Matsumoto’s statement of Berwald’s theorem on projective flatness
27
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 = xo and solve to get
yo = − ab xo − cb so that Φ(xo , yo ) = 0. If b = 0, take x = xo = −c/a. If
a = b = 0, then Φ = c = Zx and (35) implies
M 00 (Z) · c2 = 0
so c = 0 (since M 00 (Z) 6= 0). Therefore, in all cases of Φ linear we see that
Zx = 0 at some point, which is impossible.
(B) Suppose Φ = aij xi xj + bi xi + c (with x1 = x, x2 = y and summation
convention). Clearly, Φx = 0 = Φy has always at least one solution (unique, if
det(aij ) 6= 0). Hence, Zxx = 0 and (36) imply Zx = 0 at this point. Proposition 3. The Berwald I 2 = 9/2 geodesics are
 2 1 dx2 dx1
d x

1
2


 ds2 = Φ(x , x ) ds ds ,
(37)
2 dx2

d2 x2

1 , x2 ) dx


Φ(x
=
.
ds2
ds 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
Zxx = −M 00 (Z)(Zx )3
and since
00
3
K = M (Z) · (Zx ) ·
ẏ
ẋ + Z ẏ
we have
K = −Zxx
ẏ
ẋ + Z ẏ
3
,
3
.
Taking ∂x of K we get
Kx =
ẏ
ẋ + Z ẏ
3 3Zxx Zx ẏ
· −Zxxx +
.
ẋ + Z ẏ
Therefore, Kx = 0 at (xo , yo ) = (x, y) ⇔
Zxxx =
3Zxx Zx
,
(ẋ/ẏ + Z(x, y))
or, from (36),
Zxxx = −
3M 00 (Z)(Zx )4
.
ẋ/ẏ + Z(x, y)
28
Peter L. Antonelli, Solange F. Rutz and Carlos E. Hirakawa
12
Note
∂ẋ K =
1
ẋ
∂ẋ/ẏ K, ∂ẏ K = − 2 · ∂ẋ/ẏ K ⇒ ∂ẋ K, ∂ẏ K never zero.
ẏ
(ẏ)
The left-hand side is just ∂x ∂x ∂x Z(x, y) and is independent of ẋ, ẏ or ẋ/ẏ, since
Z(x, y) is such, by definition. But, since M 00 (Z) 6= 0 and Zx 6= 0 everywhere
in (x, y)-space, the derivative of the right-hand side with respect to ẋ/ẏ is
3M 00 (Z)(Zx )4
(ẋ/ẏ + Z(x, y))2
and is never zero – contradiction. Hence, Kx is never zero in (x, y)-space and
(likewise, Ky 6= 0 ∀(x, y)). We have proved
Theorem E. K(x, y, ẋ, ẏ) as a function on T̃ M 2 , the slit tangent bundle,
cannot have local extremals on M 2 .
The only 2-dimensional Riemannian space with rectilinear extremals and
K > 0 is the sphere, all points of which are local extremals since K = 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 precursors 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 evolutionary theories of Woese and Margulis and because we treated both within
a single logical framework, in spite of the fact that the communes of protocells in Woese’s theory predate the evolved Eukaryote symbiocosm, sensible
comparison was achieved. We are now going to recount the detailed mathematical material.
Using the Volterra-Hamilton systems as a logical method we have compared (see [12] for recent advances in endosymbiosis) the evolutionary theories
13
On Matsumoto’s statement of Berwald’s theorem on projective flatness
29
of Carl Woese and Lynn Margulis and found the former suffers from robust instability while the later is robustly stable in its production processes [13], [18].
N i = density of i-th bacterial population, assumed to satisfy classical
logistic dynamics
dN i
= λN i (1 − α(i) N i ),
dt
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
= k(i) N i .
dt
We require that, for either model, our evolved system has the form
j
k
d2 xi
i dx dx
+
G
= 0,
jk
ds2
ds ds
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 functional, ds = F (x, dx) > 0. Moreover, F is to be positively homogeneous of
degree 1 in dx = (dx1 , . . . , dxn ), that is, for any positive constant c,
dx
dx
F x, c
= c F x,
,
dt
dt
so that ds/dt, the rate of production in the symbiocosm, depends on individual bacterial rates dxi /dt through the cost F (x, dx/dt). The arc-length
Rt
s = t0 F (x, dx/dt)dt represents the total production of the symbiocosm in
the interval (t2 − t1) along a given curve x(t) = (x1 (t), . . . , xn (t)). The homogeneity of F means that, if all the individual rates dxi /dt are magnified by a
factor of c, then ds/dt is so magnified. This forces s to be independent of the
time measure.
We have to introduce the expression Hs = (1/2)F 2 (x, dx/dt), which
yields Euler-Lagrange equations. Note that Hs is therefore positively homogeneous of degree 2 in dx/dt, thus, multiplying each dxi /dt by a positive
constant c implies that Hs is multiplied by c2 . Furthermore, Hs defines two
classes of systems, namely, the Riemannian class, where Hs is quadratic in
dx/dt, and the Finsler (non-Riemannian) class, where Hs is not quadratic,
but is homogeneous of degree 2 in dx/dt. Here are two examples, one for each
30
Peter L. Antonelli, Solange F. Rutz and Carlos E. Hirakawa
14
class. For the former, quadratic case, we may have
"
2
n 2 #
1 2αi xi
dx1
dx
Hs = e
+ ··· +
,
2
dt
dt
while for the latter, non-quadratic but homogeneous case, we have
(∗)
Hs =
1 2φ(x) (dx2 /dt)2+2/λ
,
e
2
(dx1 /dt)2/λ
where we have taken n = 2, λ is a positive constant and φ(x) is an arbitrary
polynomial on x1 and x2 . 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) = −α1 x1 + (l +
1)α2 x2 + ν3 x1 x2 , and λ > 0, αi > 0 and ν3 non-zero, the Euler-Lagrange
equations are

dy 1

2
1 2


 ds + λ(α1 − ν3 x ) · (y ) = 0,
(∗∗)
 dy 2
ν3 1


+ λ α2 +
x · (y 2 )2 = 0.

ds
λ+1
Note that, if ν3 = 0, then the original double logistic system are obtained.
(Where ds = eλt dt must be employed to transform to realtime t from parameter s.) Moreover, Liapunov stability of this system is completely determined
by the sign of the curvature
1 1+2/λ
λ2
y
K=
ν3
· exp(−2[−α1 x1 + (λ + 1)α2 x2 + ν3 x1 x2 ]).
λ+1
y2
If ν3 > 0, then stability results, while the reverse is true for ν3 < 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 coefficients, 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 the n-dimensional functional, using
15
On Matsumoto’s statement of Berwald’s theorem on projective flatness
31
the additional assumption that our community is simple. This means precisely
that each bacterial species in the conglomerate exchanges chemical information 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 Hs , 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 Gijk as
Giii = αi , Gijk = 0, i 6= j 6= k,
Giij = Giji = αj ,
i=
6 j,
i
Gjj = −αi ,
i=
6 j.
(∗ ∗ ∗)
These coefficients of interaction completely characterize our model of Ancestral
Commune. It is especially significant that the Gijk , with all indexes different,
vanishes. For example, for a 3 species conglomerate, G123 , G213 , G312 all 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(−2αi xi )[(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 curvature K play an important role in stochastic problems
in curved spaces.
The stochastic version of the above theory comparison have been published [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
Giii = αi ,
Gijj = 0,
Giij = Giji = −βi ,
Gjij
=
Gjji
i 6= j, αi > 0,
= βj , i 6= j, βi , βj > 0,
32
Peter L. Antonelli, Solange F. Rutz and Carlos E. Hirakawa
16
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 parameter t between 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
[email protected]
UFPE, Mat. Dept., Recife, PE, Brazil
[email protected]
UERJ, IME, Rio de Janeiro, RJ, Brazil
[email protected]