Table Of ContentFinsleroid-regular space. Landsberg-to-Berwald
implication
G.S. Asanov
Division of Theoretical Physics, Moscow State University
8
0 119992 Moscow, Russia
0
(e-mail: [email protected])
2
n
a
J
0
3
]
G
D
.
h
t
a
m
[
1
v
8
0
6
4
.
1
0
8
0
:
v
i
X
r
a
Abstract
By performing required evaluations, we show that in the Finsleroid-regular space
the Landsberg-space condition just degenerates to the Berwald-space condition (at any
dimension number N 2). Simple and clear expository representations are obtained.
≥
Due comparisons with the Finsleroid-Finsler space are indicated.
Keywords: Finsler metrics, spray coefficients, curvature tensors.
1
1. Description of new conclusions
The Finsler geometry theory can mentally be divided into two great parts: the
development of the general theory to search for tensorial and geometrical implications of
ageneralFinsler metricfunction“F”andtheinvestigation ofpossible results ofspecifying
the F in an attractive particular way [1,2]. Obviously, it is the latter way that one is to
follow when hoping to develop genius and handy applications.
Below, we deal with the Finsler space notion specified by the condition that the basic
Finslerian metric function, to be denoted by K(x,y), is constructed functionally from the
set g(x),b (x),a (x),y , where g(x) is a scalar, b (x) is an involved vector field, and
i ij i
{ }
a (x) is a (positive-definite) Riemannian metric tensor; y stands for the tangent vectors
ij
supported by a point x of the underlined manifold. Denoting by
c = b b (1.1)
Riemannian
|| || ≡ || ||
the respective Riemannian norm value of the input 1-form b = b (x)yi and assuming the
i
range
0 < c < 1, (1.2)
we construct the particular function K(x,y) which occurs being globally regular. The
entailed positive-definite Finsler space will be denoted by PD.
FRg;c
The extrapolation
PD = PD (1.3)
FRg;c→1 FFg
takes place, where PD is the Finsleroid-Finsler space which was constructed and de-
FFg
veloped in [3-6] under the assumption b = 1.
|| ||
This scalar c(x) proves to play the role of the regularization factor. Indeed, in the
spaces PD and PD the metric function K is constructed such that K involves the
FRg;c FFg
square-root variable q(x,y) = √S2 b2 (see (A.4)). Differentiating various tensors of the
−
space PD as well as the space PD gives rise, therefore, to appearance of degrees of
FRg;c FFg
the fraction 1/q.
If b = 1, we have q = 0 when y = b.
|| || ±
As far as 0 < c < 1, we have q = 0 whenever y = 0 (because of the inequality (A.5)),
6 6
so that the fraction 1/q does not produce any singularities on TM 0.
\
We use
REGULARITY DEFINITION. The Finsler space PD under consideration is reg-
FRg;c
ular in the following sense: globally over all the slit tangent bundle TM 0, the Finsler
\
metric function K(x,y) of the space is smooth of the class C∞ regarding both the argu-
ments x and y, and also the entailed Finsler metric tensor g (x,y) is positive-definite:
ij
det(g ) > 0.
ij
The PD–space is smooth of the class C2, and not of the class C3, on all of the
FFg
slit tangent bundle TM 0. The PD–space is smooth of the class C∞ on all of the
\ FFg
b-slit tangent bundle
M := TM 0 b b (1.4)
b
T \ \ \−
(obtained by deleting out in TM 0 all the directions which point along, or oppose, the
\
directions given rise to by the 1-form b).
The Finsleroid-Finsler space PD developed involves an attractive realization of
FFg
the Landsberg condition over the b-slit tangent bundle (see [3-6]). The realization cannot
2
be extrapolated to the b-section of the tangent bundle TM, because on the section the
smoothness of the space PD degenerates to but the C2-level.
FFg
It is impossible to lift realization of the Landsberg condition from the space PD
FFg
to the space PD. Indeed, the following theorem is valid.
FRg;c
Landsberg-to-Berwald Theorem. In the Finsleroid-regular space PD the
FRg;c
Landsberg-space condition entails the Berwald-space condition.
To verify this theorem, it is sufficient to pay a due attention to the factor (1 c2)
−
which enters the right-hand part of the formula (A.32) which precedes the explicit (and
simple) expression (A.33) for the contraction AiA (see more detail in Note placed in the
i
end of Appendix A).
In Appendix A, the explicit form of the Finsleroid-regular metric function K is
presented, the space PD is rigorously defined, and entailed representations of various
FRg;c
key tensors are given. The knowledge of the associated spray coefficients Gi, as given by
the explicit representation (A.37) derived, can open up various convenient possibilities
to evaluate and study associated geodesic equations, connection coefficients, as well as
curvature tensors. Amazingly, the coefficients (A.37) provide us readily with the Berwald
space of the regular type: the Berwald case would imply g = const and b = 0 (see
i j
∇
(A.35) and (A.36)), and also c = const. In the two-dimensional case, however, the
space degenerates to the locally Minkowskian space (whenever g = 0). In the dimensions
6
N 3 the obtainable PD-Berwald spaces can be “neither Riemannian nor locally
≥ FRg;c
Minkowskian.” TheBerwaldcaseoftheFinsler spaceisattractivebecauseofitssimplicity.
In Appendix B, we investigate the derivatives of the spray coefficients in the partic-
ular case (B.1). Calculations involved are simple, showing the validity of the following
theorem.
Particular Theorem. Given a scalar k = k(x). If the conditions g = const and
b = kr are fulfilled, then
i j ij
∇
A˙ = (1 c2)k m A +m A A A . (1.5)
knj 1 knj 2 k n j
−
(cid:0) (cid:1)
In (1.5), m and m are two scalars, which form is indicated explicitly in (B.20)
1 2
of Appendix B. Remarkably, m , as well as m , doesn’t vanish identically when g = 0.
1 2
6
Therefore, if g = 0 and 0 < c < 1, the right-hand part in (1.5) can vanish identically only
6
in the case k = 0 which is the Berwald case.
We shall use the notation
1
= yjD (1.6)
j
D K
with D standing for the h-covariant (horizontal) Finslerian derivative (see [2]); the tensor
j
A˙ is identical to that used in [2], namely,
knj
A˙ := A . (1.7)
knj knj
D
The condition b = kr , when taken in conjunction with g = const, realizes the
i j ij
Landsberg space, th∇at is, A˙ = 0, in the Finsleroid-Finsler space PD (see [3-6]).
knj FFg
Lifting the condition to the Finsleroid-regular space PD results in the representation
FRg;c
(1.5) which (beautifully?) extends the Landsberg condition A˙ = 0.
knj
3
The occurrence of the regularizing factor (1 c2) in the right-hand parts of the
−
formulae (A.32) and (1.5) presents an astonishingly simple and explicit illustration to the
above Landsberg-to-Berwald Theorem.
Appendix A: Involved PD-notions
FRg;c
Let M be an N-dimensional C∞ differentiable manifold, T M denote the tangent
x
space to M at a point x M, and y T M 0 mean tangent vectors. Suppose we are
x
∈ ∈ \
given on M a Riemannian metric = S(x,y). Denote by = (M, ) the obtained
N
S R S
N-dimensional Riemannian space. Let us also assume that the manifold M admits a
non–vanishing 1-form b = b(x,y), denote by
c = b b (A.1)
Riemannian
|| || ≡ || ||
the respective Riemannian norm value. Assuming
0 < c < 1, (A.2)
we get
S2 b2 > 0 (A.3)
−
and may conveniently use the variable
q := √S2 b2. (A.4)
−
Obviously, the inequality
1 c2
q2 − b2 (A.5)
≥ c2
is valid.
With respect to natural local coordinates in the space we have the local repre-
N
R
sentations
aij(x)b (x)b (x) = c(x) (A.6)
i j
q
and
b = b (x)yi, S = a (x)yiyj. (A.7)
i ij
q
The reciprocity aina = δi is assumed, where δi stands for the Kronecker symbol. The
nj j j
covariant index of the vector b will be raised by means of the Riemannian rule bi = aijb ,
i j
which inverse reads b = a bj. We also introduce the tensor
i ij
r (x) := a (x) b (x)b (x) (A.8)
ij ij i j
−
to have the representation
q = r (x)yiyj. (A.9)
ij
q
We choose the Finsler space notion specified by the condition that the Finslerian
metric function K(x,y) be of the functional dependence
K(x,y) = Φ g(x),b (x),a (x),y , (A.10)
i ij
(cid:16) (cid:17)
where g(x) is a scalar (on the background manifold M), subjected to ranging
2 < g(x) < 2, (A.11)
−
4
and apply the convenient notation
1 g(x)
h(x) = 1 (g(x))2, G(x) = . (A.12)
− 4 h(x)
r
We introduce the characteristic quadratic form
1
B(x,y) := b2 +gqb+q2 (b+g q)2 +(b+g q)2 , (A.13)
+ −
≡ 2
h i
where g = (1/2)g+h and g = (1/2)g h, The discriminant D of the quadratic form
+ − {B}
−
B is negative:
D = 4h2 < 0. (A.14)
{B}
−
Therefore, the quadratic form B is positively definite. In the limit g 0, the definition
→
(A.13) degenerates to the quadratic form of the input Riemannian metric tensor: B =
|g=0
b2 +q2 S2. Also, ηB = c2, where η = 1/(1+gc√1 c2). It can readily be verified
≡ |yi=bi −
that on the definition range (A.11) of the g we have η > 0.
Under these conditions, we set forth the following definition.
Definition. The scalar function K(x,y) given by the formulas
K(x,y) = B(x,y)J(x,y) (A.15)
p
and
J(x,y) = e−12G(x)f(x,y), (A.16)
where
G L
f = arctan +arctan , if b 0, (A.17)
− 2 hb ≥
and
G L
f = π arctan +arctan , if b 0, (A.18)
− 2 hb ≤
with
g
L = q + b, (A.19)
2
is called the Finsleroid-regular metric function.
The function L obeys the identity
L2 +h2b2 = B. (A.20)
Definition. The arisen space
PD := ; b (x); g(x); K(x,y) (A.21)
FRg;c {RN i }
is called the Finsleroid-regular space.
Definition. The space entering the above definition is called the associated
N
R
Riemannian space.
5
The associated Riemannian metric tensor a has the meaning
ij
a = g . (A.22)
ij ij g=0
(cid:12)
(cid:12)
Definition. Within any tangent space T M, the Finsleroid-regular metric function
x
K(x,y) produces the regular Finsleroid
PD := y PD : y T M,K(x,y) 1 . (A.23)
FRg;c{x} { ∈ FRg;c{x} ∈ x ≤ }
Definition. The regular Finsleroid Indicatrix I PD T M is the boundary of
Rg;c{x} ⊂ x
the regular Finsleroid, that is,
I PD := y I PD : y T M,K(x,y) = 1 . (A.24)
Rg;c{x} { ∈ Rg;c{x} ∈ x }
Definition. The scalar g(x) is called the Finsleroid charge. The 1-form b = b (x)yi
i
is called the Finsleroid–axis 1-form.
We shall meet the function
ν := q +(1 c2)gb (A.25)
−
for which
ν > 0 when g < 2. (A.26)
| |
Indeed, if gb > 0, then the right-hand part of (A.25) is positive. When gb < 0, we may
note that at any fixed c and b the minimal value of q equals √1 c2 b /c (see (A.5)),
− | |
arriving again at (A.26).
Under these conditions, we can explicitly extract from the function K the distin-
guished Finslerian tensors, and first of all the covariant tangent vector yˆ = y from
i
{ }
y := (1/2)∂K2/∂yi, obtaining
i
K2
y = (u +gqb ) , (A.27)
i i i
B
where u = a yj. After that, we can find the Finslerian metric tensor g together with
i ij ij
{ }
the contravariant tensor gij defined by the reciprocity conditions g gjk = δk, and the
{ } ij i
angular metric tensor h , by making use of the following conventional Finslerian rules
ij
{ }
in succession:
1 ∂2K2 ∂y 1
i
g := = , h := g y y ,
ij 2 ∂yi∂yj ∂yj ij ij − i jK2
thereafter the Cartan tensor
K ∂g
ij
A := (A.28)
ijk
2 ∂yk
and the contraction
∂ln det(g )
A := gjkA = K mn (A.29)
i ijk
∂yi
(cid:0)p (cid:1)
can readily be evaluated.
It can straightforwardly be verified that
ν K2 N
det(g ) = det(a ) > 0 (A.30)
ij ij
q B
(cid:18) (cid:19)
6
with the function ν given by (A.25) [7], and
Kg 1
A = (q2b bv ), (A.31)
i i i
2qBX −
where the function X is given by
1 B
= N +(1 c2) . (A.32)
X − qν
Contracting yields the formula
g2 1 1
AiA = N +1 (A.33)
i
4 X2 − X
(cid:18) (cid:19)
and evaluating the Cartan tensor results in the lucid representation
1 1
A = X A h +A h +A h N +1 A A A . (A.34)
ijk i jk j ik k ij − − X A Ah i j k
" (cid:18) (cid:19) h #
We use the Riemannian covariant derivative
b := ∂ b b ak , (A.35)
i j i j k ij
∇ −
where
1
ak := akn(∂ a +∂ a ∂ a ) (A.36)
ij j ni i nj n ji
2 −
are the Christoffel symbols given rise to by the associated Riemannian metric .
S
Attentive direct calculations of the induced spray coefficients Gi = γi ynym, where
nm
γi denote the associated Finslerian Christoffel symbols, can be used to arrive at the
nm
following result.
Theorem 1. In the Finsleroid-regular space PD the spray coefficients Gi can
FRg;c
explicitly be written in the form
g
Gi = yjyh b +gqbjf vi gqfi+Ei +ai ynym. (A.37)
j h j nm
ν ∇ −
(cid:16) (cid:17)
We use the notation
vi = yi bbi (A.38)
−
and
∂b ∂b
f = f yn, fi = fi yn, fi = aikf , f = b b n m, (A.39)
j jn n n kn mn m n n m
∇ −∇ ≡ ∂xm − ∂xn
where means the covariant derivative in terms of the associated Riemannian space
∇
R = (M,S) (see (A.35)); ai stands for the Riemannian Christoffel symbols (A.36)
N nm
constructed from the input Riemannian metric tensor a (x); the coefficients Ei involving
ij
the gradients g = ∂g/∂xh of the Finsleroid charge can be taken as
h
2q2 1
Ei = M¯(yg)yi+K (yg)XAi M¯K2g gih, (A.40)
h
gB − 2
7
where (yg) = g yh, X is the function given in (A.32), and the function M¯ is defined by
h
the equality ∂K/∂g = (1/2)M¯K. It can readily be verified that
q2 1 1 1
b Eh = bM¯(yg)+ (yg) M¯ q(bg)+g[b(bg) (yg)] (c2S2 b2) M¯(bg)b2, (A.41)
h
B −2 − ν − −2
" #
(cid:16) (cid:17)
where (bg) = bhg .
h
Contracting (A.37) by b yields
i
g
b Gi ai ymyn = (ys)+gqσ (1 c2)b gqσ+b Ei,
i mn i
− ν − −
(cid:16) (cid:17) (cid:16) (cid:17)
or
g gq2
b Gk ak ymyn = (ys)(1 c2)b σ +b Ek, (A.42)
k mn k
− ν − − ν
(cid:16) (cid:17)
where (ys) = yhyk b and
h k
∇
σ = b fk = bkynf .
k kn
Let us apply the operator (defined in (1.6)) to the input 1-form b:
D
b = yk∂ b Gkb = ykym b (Gk ak ymyn)b ,
k k k m mn k
D − ∇ − −
that is,
b = (ys) (Gk ak ymyn)b . (A.43)
mn k
D − −
˙
Taking into account (A.42) and denoting b = b, we obtain simply
D
q gq2
b˙ = (ys)+ σ b Ek, (A.44)
k
ν ν −
With the help of the coefficients
∂Gk
Gk = (A.45)
n ∂yn
we can consider also the contracted derivative
1 1
b = yh∂ b Gk b = yh b Gk ak yh b , (A.46)
n h n n k h n n nh k
D − 2 ∇ − 2 −
(cid:18) (cid:19)
or
∂ b Gk ak ymyn
k mn
−
1 !
b = yh b (cid:16) (cid:17) . (A.47)
n h n
D ∇ − 2 ∂yn
This, by virtue of (A.43), can be written as
1 1∂ b
b = yh( b b )+ D . (A.48)
D n 2 ∇h n −∇n h 2 ∂yn
In evaluations, it is convenient to use the derivative value
v ∂ν
ν = k +(1 c2)gb . (A.49)
k q − k ≡ ∂yk
8
If the input 1-form b = yib is exact:
i
f = 0, (A.50)
mn
then the above representation (A.37) reduces to read merely
g
Gi = (ys)vi+Ei +ai ynym, (A.51)
nm
ν
entailing
1 ∂Gi
Gi = gU vi +g (ys)ri +Ei +2ai ym (A.52)
k k ν k k km ≡ ∂yk
with
1 2
U = ν (ys)+ s , (A.53)
k k k
−ν2 ν
where (ys) = yhs and s = yh b ; Ei = ∂Ei/∂yk and ri = aihr . Differentiating
h k h k k k hk
∇
(A.53) yields the coefficients U = ∂U /∂ym given by the formula
km k
1 2 1 2
U = 2 ν ν (ys) (ν s +ν s ) η (ys)+ b , (A.54)
km k m k m m k km m k
ν3 − ν2 − ν2q ν∇
where
1
η = r v v , v = u bb a vj, u = a yj. (A.55)
km km k m k k k kj k kj
− q2 − ≡
From (A.52) we find that the coefficients Gi = ∂Gi /∂ym are given by the repre-
km k
sentation
Gi = gU ri +gU ri +gU vi +Ei +2ai , (A.56)
km k m m k km km km
where Ei = ∂Ei /∂ym. By contracting we obtain
km k
b Gi = (1 c2) gU b +gU b +gU b +b Ei +2b ai (A.57)
i km k m m k km i km i km
−
(cid:16) (cid:17)
and
u Gi = gU v +gU v +gU q2 +u Ei +2u ai . (A.58)
i km k m m k km i km i km
(cid:16) (cid:17)
The following theorem is valid.
Theorem 2. If the Finsleroid-regular space PD is the Berwald space and is not
FRg;c
the locally-Minkowskian space, then g = const.
To verify the theorem, it is worth noting that in the Berwald case the covariant
derivative A of the Cartan tensor vanishes identically (see [1,2]). The implication
n ijk
D
A = 0 = (AiA ) = 0 is obviously valid in any Finsler space. Therefore, in
n ijk n i
D ⇒ D
the Berwald case of dimension N 3 the representations (A.33) and (A.34) just entail
≥
g = const. In the two-dimensional case the representation (A.34) reduces to
A = Iα α α at N = 2 (A.59)
ijk i j k
with the normalized vector α = A /√AhA , and with the main scalar I given by
i i h
I = AhA . (A.60)
h
p
