On Asanov's Finsleroid-Finsler metrics as the solutions of a conformal rigidity problem

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by  Csaba Vincze

ON ASANOV’S FINSLEROID-FINSLER METRICS AS THE SOLUTIONS OF
A CONFORMAL RIGIDITY PROBLEM
CS. VINCZE
Abstract. Finsleroid-Finslermetricsformanimportantclassofsingular(y-local)Finslermet-
rics. They were introduced by G. S. Asanov [2] in 2006. As the special case of the general
6
construction Asanov produced singular (y - local) examples of Landsberg spaces of dimension
1
0 at least three that are not of Berwald type. The existence of regular (y - global) Landsberg
2 metricsthatarenotofBerwaldtypeisanopenproblemuptothisday;foradetailedexposition
of the so-called unicorn problem in Finsler geometry see D. Bao [3].
n
a In this paper we are going to characterize the Finsleroid-Finsler metrics as the solutions of
J a conformal rigidity problem. We are looking for (non-Riemannian) Finsler metrics admitting

9 a (non-homothetic) conformal change such that the mixed curvature tensor of the Berwald
2 connection contracted by the derivatives of the logarithmic scale function is invariant. We

provethatthe solutionsofclassatleast 2 onthecomplementofthe zerosectionareconformal
] C
G to Finsleroid-Finsler metrics.
D
.
h
t Introduction
a
m
In the paper we prove the following theorems.
[

1 Theorem A Let M be a manifold of dimension n 3 equipped with the Finslerian metric
v function F and consider a function α: M R satisfy≥ing the regularity condition d α = 0 at a
7 p
→ 6
7 point p M. If the sectional curvature of the indicatrix hypersurface at p is positive and
1 ∈
8 P˜l = Pl
ijk ijk
0
. under the conformal change F˜ = eαF, where Pl is the mixed curvature tensor of the Berwald
1 ijk
0 connection, then F is a locally Riemannian metric function.
6
1
: Theorem B Let M be a manifold of dimension n 3 equipped with the Finslerian metric
iv function F and consider a function α: M R satisfy≥ing the regularity condition dpα = 0 at a
X → 6
point p M. If the sectional curvature of the indicatrix hypersurface at p is positive and
r ∈
a P˜l α = Pl α
ijk l ijk l
under the conformal change F˜ = eαF, where α denotes the partial derivatives of α (depending
l
only on the position), then F is locally conformal to a Finsleroid-Finsler metric or it is a locally
Riemannian metric function.
Looking for results like Theorem A and Theorem B was originally motivated by the so-called
Matsumoto’s problem [12] in 2001: are there conformally related (non-Riemannian) Berwald
manifolds? The problem is closely related to the intrinsic characterization of Wagner manifolds
[17], see also [20]. Wagner manifolds form a special class of generalized Berwald manifolds
admitting alinear connection onthebase manifoldsuch that theparallel transports preserve the
Finslerian norm of tangent vectors. Especially, the compatible linear connection of a Wagner
manifold is semi-symmetric with a special (exact or at least closed) one-form in the usual
decomposition of its torsion. Wagner manifolds can be also defined as conformally Berwald
1991 Mathematics Subject Classification. 53C60, 58B20.
Key words and phrases. Finsler spaces, Conformality, Finsleroid-Finsler metrics.
1
2 CS. VINCZE
Finsler manifolds due to M. Hashiguchi and Y. Ychijyo [10], see also [14], [15] and [19]. The
logarithm of the scale function (the logarithmic scale function) corresponds to the potential of
the one-form in the torsion of the compatible linear connection up to a constant proportional
term. It is clear that the scale function of the conformal change of a Finslerian metric function
toa Berwaldian oneis uniquely determined (up toa constant homothetic term) if andonly if the
conformality between two (non-Riemannian) Berwald manifolds must be trivial (homothetic).
The first attempt to solve Matsumoto’s problem was given in [16], where the author investigated
the consequences of the conformal invariance of the mixed curvature tensor of the Berwald
connection (see Theorem A). It is a natural generalization of the original problem1. In what
follows we generalize the basic results of [16] to prove Theorem B which gives the conformal
characterization of Finsleroid-Finsler metrics [2]. The cronology of the basic steps:
1998 - the central symmetric version of the Finsleroid-Finsler metric in G. S. Asanov [1].
•
2003 - the non-symmetric version of the Finsleroid-Finsler metric in [16] as Asanov-type
•
Finslerian metric functions. They satisfy differential equation (46) of the generalized
Matsumoto’sproblem(conformalinvarianceofthemixedcurvaturetensoroftheBerwald
connection) for the Finslerian energy along special directions in the tangent spaces; see
also [19].
2006 - Asanov’s necessary and sufficient conditions for (non-symmetric) Finsleroid-
•
Finsler metrics to be of Landsberg but not of Berwald type;
2016 - non-symmetric Finsleroid-Finsler metrics as the solution of a conformal rigid-
•
ity problem (the invariance of the contracted mixed curvature tensor of the Berwald
connection), see Theorem B and section 6.2. (the converse of the theorem);
for a detailed exposition of the unicorn problem in Finsler geometry see D. Bao [3]. In what
follows we give a characterization of Finsleroid-Finsler metrics as the singular non-Riemannian
solutions of the conformal rigidity problem P˜l α = Pl α , where Pl ’s are the components of
ijk l ijk l ijk
the mixed curvature tensor of the Berwald connection and α ’s are the partial derivatives of the
l
”logarithmic” scale function α depending only on the position. The basic steps of the proof are
Theorem 1, Theorem 2, the solution of a Ricatty-type diffrential equation (sections 4.2, 4.4, 4.5
and 4.6) and Theorem 6.
Acknowledgement
The paper was motivated by the oral communication with Professor David Bao at the 50th
Symposium on Finsler Geometry (21-25. Oct. 2015, Hiroshima, Japan). I would like to thank
him for paying my attention to some correspondences between Asanov’s Unicorn metrics and
Finsler metrics satisfying conformal rigidity properties. I am very grateful for his human and
professional encouragement.
TheworkissupportedbytheUniversity ofDebrecen’sinternalresearchprojectRH/885/2013.
1. Notations and terminology
Let M be a manifold with local coordinates u1,...,un. The induced coordinate system of the
tangent manifold TM consists of the functions
x1,...,xn and y1,...,yn,
where x’s refer to the coordinates of the base point and y’s denote the coordinates of the
directions:
∂
v T M can be written as v = yi(v) , where x(v) = p.
∈ p ∂ui
x(v)
1Bochner’s technic and the theory of geometric vector fields in the tangent spaces of a Finsler manifold give
another way to solve the generalized Matsumoto’s problem in [18].
ON ASANOV’S FINSLEROID-FINSLER METRICS... 3
1.1. Finsler metrics. A Finsler metric is a continuous function F: TM R satisfying the
→
following conditions:
F is smooth on the complement of the zero section (regularity),
•
F(tv) = tF(v) for all t > 0 (positive homogenity),
•
the Hessian
•
∂2E
g =
ij
∂yi∂yj
of the Finslerian energy function E = (1/2)F2 is positive definite at all nonzero elements
v T M (strong convexity). It is called the Riemann-Finsler metric of the Finsler
p
∈
manifold.
In what follows we summerize some basic notations and facts we need to prove our theorems.
As a general reference of Finsler geometry and the forthcoming list of quantities see [3] and [4]:
1
F is a Finsler metric function, E := F2 is the energy function,
• 2
∂F
l = ,
• i ∂yi
∂2E
g = is the Riemann-Finsler metric and its inverse gij = (g ) 1,
• ij ∂yi∂yj ij −
∂
C = yl is the Liouville vector field,
• ∂yl
∂g
C = ij is the so-called first Cartan tensor2, l = glk . The first Cartan tensor is
• ijk ∂yk Cij Cijk
totally symmetric and yk = 0. It is also known that
ijk
C
∂glm
(1) = 2gmk l = 2 lm
∂yi − Cik − Ci
and, consequently,
∂ l ∂ l
(2) Cjk Cik = 2 l m l m , where Ql = l m l m
∂yi − ∂yj CjmCik −CimCjk ijk CjmCik −CimCjk
(cid:0) (cid:1)
is the vv-curvature of the Cartan connection.
1.2. Geodesic spray coefficients:
1 ∂Fl ∂F 1 ∂2E ∂E
(3) Gl = glm yk m F , i.e. Gl = glm yk .
2 ∂xk − ∂xm 2 ∂ym∂xk − ∂xm
(cid:18) (cid:19) (cid:18) (cid:19)
1.3. Horizontal sections:
δ ∂ ∂ ∂Gl
= Gl , where Gl = .
δxi ∂xi − i∂yl i ∂yi
2The Cartan tensor quantities are often defined as
F ∂g
ij
A =
ijk 2 ∂yk
see e.g. [3] and [4]. The symbol follows [7] and [8].
ijk
C
4 CS. VINCZE
1.4. The second Cartan tensor: (Landsberg tensor3)
1 δg 1 ∂g
Pl = glm jm Gkg Gk g = glm jm 2Gk Gkg Gk g ,
ij 2 δxi − ij km − im jk 2 ∂xi − iCjkm − ij km − im jk
(cid:18) (cid:19) (cid:18) (cid:19)
where Gl = ∂Gli.
ij ∂yj
1.5. The mixed curvature of the Berwald connection:
∂Gl
Pl = Gl , where Gl = ij.
ijk − ijk ijk ∂yk
1.6. An identity:
F
(4) Pl = l gklPm
ij −2 m ijk
Proof. Since
∂E
Fl = ,
• m ∂ym
∂E 1 ∂E
Gm = yk ,
• ∂ym 2 ∂xk
1 ∂2E ∂E
g Gm = yk ,
• mi 2 ∂yi∂xk − ∂xi
(cid:18) (cid:19)
∂ ∂E ∂E
Gm g Gm =
mi
• ∂yi ∂ym − ∂xi
(cid:18) (cid:19)
we have
∂E
Fl Pm = Gm =
− m ijk ∂ym ijk
∂ ∂E ∂ ∂ ∂E
Gm g Gm = Gm g Gm g Gm =
∂yk ∂ym ij − mk ij ∂yk ∂yj ∂ym i − mj i − mk ij
(cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:19)
∂ ∂ ∂ ∂E
Gm g Gm g Gm g Gm =
∂yk ∂yj ∂yi ∂ym − mi − mj i − mk ij
(cid:18) (cid:18) (cid:18) (cid:19) (cid:19) (cid:19)
∂ ∂ ∂E ∂
g Gm g Gm = g 2 Gm g Gm g Gm =
∂yk ∂yj ∂xi − mj i − mk ij ∂xi jk − Cjmk i − mj ik − mk ij
(cid:18) (cid:18) (cid:19) (cid:19)
F
2P = 2g Pl Pl = l gklPm
ijk kl ij ⇒ ij −2 m ijk
(cid:3)
as was to be proved
2. Conformality
Definition 1. Let F˜, F: TM R be Finslermetrics. They are conformallyrelated if F˜(x,y) =
eα(x)F(x,y), where α: M R→is a function (depending only on the position).
→
As an easy consequence of the conformality we have:
E˜ = e2αE, g˜ = e2αg ˜k = k.
ij ij ⇒ Cij Cij
For the sake of simplicity let us use the following abbreviation:
∂α
α = , m = 1,...,n.
• m ∂um
3Following subsection 1.6. the Landsberg tensor is often defined as
. 1 ∂Gl
Aijk=−2Fli∂yji;
see e.g. [3] and [4]. The symbol Pl follows [9] and [11].
ij
ON ASANOV’S FINSLEROID-FINSLER METRICS... 5
Using (3)
G˜l = Gl +ymα yl Eglmα .
m m
−
According to its distinguished role let us introduce the gradient-type vector field
∂
(5) X = Xl , where Xl = glmα g Xl = α
∂yl m ⇒ lm m
and Xlα = Xlg Xm = g(X,X) is just the Riemann-Finsler norm square of the vector field
l lm
X. Under this notation
(6) G˜l = Gl +ymα yl EXl,
m
−
∂E ∂Xl
(7) G˜l = Gl +α yl +ymα δl Xl E .
i i i m i − ∂yi − ∂yi
According to its distinguished role let us introduce the (vector valued) one-form
∂Xl
(8) Xl = .
i ∂yi
We have
2gms l α = 2Xs l or, equivalently,
∂Xl ∂glm − Cis m − Cis
(9) Xl = = α = 2 lmα =
i ∂yi ∂yi m − Ci m 
2gls mα
 − Cis m
and the second line of formula (9) gives the symmetry property

(10) g Xl = 2 mα g Xl = g Xl
lk i − Cik m ⇒ lk i li k
because of the symmetry of the first Cartan tensor. Therefore we also have the following cross-
lifting formula
(11) gklXmg = Xl.
k mj j
Finally
(12) XsXl = 2XsXt l .
s − Cst
Using formula (7)
∂E ∂E ∂Xl
(13) G˜l = Gl +α δl +α δl g Xl Xl Xl E i.
ij ij i j j i − ij − ∂yi j − ∂yj i − ∂yj
According to its distinguished role let us introduce the quantity
∂Xl ∂2Xl
Xl = i = .
ij ∂yj ∂yj∂yi
Proposition 1. (Transformation formula for the Landsberg tensor)
1∂E 1∂E 1 1 1
P˜l = Pl ymα l +EXs l + Xl + Xl + g Xsyl + EXl + EglmXs g .
ij ij − mCij iCjs 2∂yi j 2∂yj i 2 sj i 2 ij 2 im sj
The proof is a long straightforward calculation; see [16] and Hashiguchi [9]. In terms of the
first Cartan tensor
glmXs g = glm∂Xms g = ∂glmXms gsj ∂glmXs g glmXs ∂gsj (1=1)
im sj ∂yi sj ∂yi − ∂yi m sj − m ∂yi
Xl +2 lmXs g 2glmXs g r (1=0)
ij Ci m sj − m srCji
Xl +2 lmXsg 2glmXsg r = Xl +2 l Xs 2Xl r.
ij Ci j sm − r smCji ij Cis j − rCij
6 CS. VINCZE
Therefore
1∂E 1∂E 1
(14) P˜l = Pl ymα l + Xl+ Xl+ g Xsyl+EXl +EXs l +E l Xs EXl r.
ij ij− mCij 2∂yi j 2∂yj i 2 sj i ij iCjs Cis j− rCij
Remark 1. Some further computation shows that
Xl +Xs l + l Xs Xl r = 2Xs v l
ij iCjs Cis j − rCij − ∇sCij
where v denotes the v-covariant derivative with respect to the Cartan connection. Therefore
∇s
1∂E 1∂E 1
P˜l = Pl ymα l + Xl + Xl + g Xsyl 2EXs v l
ij ij − mCij 2∂yi j 2∂yj i 2 sj i − ∇sCij
andtheformulacorrespondstoformula(21)in[16]orformula(3.4C)in[9]; notethat[9]involves
an extra minus sign in the definition of Pl (formula (2.14*), page 35).
ij
3. Special conformal relationships I
From now on we suppose that the Landsberg tensor satisfies the invariance prop-
erty
P˜lα = Plα
ij l ij l
under the conformal change F˜ = eαF. As a direct consequence of formula (14) and the
invariance property we have special expressions for Xl and the contracted quantities Xl α and
ij ij l
XjXl .
ij
Corollary 1. If P˜l = Pl then
ij ij
1∂E 1 ∂E 1
(15) EXl = ymα l Xl Xl g Xsyl EXs l E l Xs +EXl m.
ij mCij − 2∂yi j − 2∂yj i − 2 sj i − iCjs − Cis j mCij
(cid:3)
Proof. Equation (15) is a direct consequence of the transformation formula (14).
Corollary 2. If P˜lα = Plα then
ij l ij l
(16) EXl α =
ij l
1∂E 1∂E 1
ymα l Xl Xl g Xsyl EXs l E l Xs +EXl m α
mCij − 2∂yi j − 2∂yj i − 2 sj i − iCjs − Cis j mCij l
(cid:18) (cid:19)
and
(17) EXjXl =
ij
1∂E 1 ∂E 1
Xj ymα l Xl Xl g Xsyl EXs l E l Xs +EXl m .
mCij − 2∂yi j − 2∂yj i − 2 sj i − iCjs − Cis j mCij
(cid:18) (cid:19)
Proof. Equation (16) is a direct consequence of Corollary 1. Since the lowered second Cartan
tensor P is totally symmetric we have that
ijk
Plα = glkP α (=5) XkP = XkP = XkPl g .
ij l ijk l ijk ikj ik lj
Using that g˜ = e2αg , the same computation results in
ij ij
1 1 1
P˜lα = g˜lkP˜ α = XkP˜ = XkP˜ = XkP˜l g˜ = XkP˜l g ,
ij l ijk l e2α ijk e2α ikj e2α ik lj ik lj
i.e. P˜lα = Pl α implies that
ij l ij l
XkPl g = XkP˜l g XkPl = XkP˜l .
ik lj ik lj ⇒ ik ik
(cid:3)
This means, by Corollary 1, that formula (17) holds.
ON ASANOV’S FINSLEROID-FINSLER METRICS... 7
3.1. The first basic step. Equations (16) and (17) will be the key formulas to conclude the
linear dependence of the vector fields
∂ Xl∂F ∂ ∂
Xl C and XsXl (=12) 2XsXt l .
∂yl − F ∂yl s∂yl − Cst∂yl
Since
∂F Xl∂E glmα ∂E ymα
Xl = (=5) m = m
∂yl F ∂yl F ∂yl F
we can also write that
∂ Xl∂F ∂ ymα ∂ ∂ ymα ∂
Xl C = Xl myl = Xl myl .
∂yl − F ∂yl ∂yl − F2 ∂yl ∂yl − 2E ∂yl
Remark 2. Note that the projected vector field
∂ Xl∂F
Xl C
∂yl − F ∂yl
∂
is obviously tangential to the indicatrix hypersurface. The vector field XsXl is also tan-
s∂yl
gential to the indicatrix because of formula (12) and the basic properties of the first Cartan
tensor:
XsXl∂F = XsXsl ∂E (=9) Xs 2Xt l ∂E , where l ∂E = glm ∂E = ym = 0.
s∂yl F ∂yl F − Cst∂yl Cst∂yl Cmst∂yl Cmst
(cid:18) (cid:19)
To conclude the linear dependency we use the substitution of Xs systematically into the
arguments of the difference tensor Bl = G˜l Gl . Especially we prove the following lemma
ijk ijk − ijk
which is the generalization of Lemma 5 in [16] (page 22).
Lemma 1. If P˜lα = Pl α then
ij l ij l
1 1 ymα ∂E
XkXjBl α = Xlα Xsα XkXjα α + m XtXjα (ymα )Xlα +
3 ijk l 2 l i s − k j i 4E j t∂yi − m i l
(cid:18) (cid:19)
(cid:0) (cid:1)
EXjXsXpXrQ
j pisr
where
Bl = G˜l Gl
ijk ijk − ijk
is the difference tensor of the mixed curveture of the Berwald connection and Q = g Ql is
pisr pl isr
the lowered vv-curvature tensor of the Cartan connection.
Proof. Let us introduce the abbreviation
(18) Bl = G˜l Gl (1=3) α δl +α δl g Xl ∂EXl ∂EXl EXl .
ij ij − ij i j j i − ij − ∂yi j − ∂yj i − ij
Since
∂Bl ∂
(19) XkXjBl α = XkXj ijα = Xk XjBl α XkXjBl α
ijk l ∂yk l ∂yk ij l − k ij l
it is enough to compute the terms (cid:0) (cid:1)
∂
Xk XjBl α and XkXjBl α .
∂yk ij l k ij l
By some direct calculations (cid:0) (cid:1)
(20) XjBl α (1=8) Xjα α ∂EXjXlα ymα Xlα EXjXl α
ij l j i − ∂yi j l − m i l − ij l
8 CS. VINCZE
because of
∂E ∂E
Xj = gjmα = ymα and Xjg Xl = α Xl.
∂yj m∂yj m ij i
Using formula (16)
1∂E
EXjXl α = ymα Xj l α XjXlα
ij l m Cij l − 2∂yi j l−
1Xj∂EXlα 1Xjg Xsylα EXsXj l α E l XjXsα +EXl α Xj m (5=),(9)
2 ∂yj i l − 2 sj i l − i Cjs l − Cis j l m l Cij
3 1∂E
ymα Xlα XjXlα E l XjXsα EXsXj l α +EXlα Xj r.
−2 m i l − 2∂yi j l − Cis j l − i Cjs l r l Cij
On the other hand
Xlα Xj r (=9) 1Xlα Xr, XsXj l α (=9) 1XsXlα ,
r l Cij −2 r l i i Cjs l −2 i s l
l XjXsα (1=0) 1g XrXjXs = 1g XrXjXs = 1α XrXs
Cis j l −2 sr i j −2 sj i r −2 s i r
and, consequently,
3 1∂E 1
(21) EXjXl α = ymα Xlα XjXlα + EXlXmα .
ij l −2 m i l − 2∂yi j l 2 i l m
From equations (20) and (21)
1∂E 1 1
(22) XjBl α = Xjα α XjXlα + ylα Xsα EXmXlα .
ij l j i − 2∂yi j l 2 l i s − 2 l i m
Using the previous formula
∂ 1 1∂E 1∂E
(23) Xk XjBl α = Xk Xjα α g XjXlα XjXlα XjXl α +
∂yk ij l k j i − 2 ik j l − 2∂yi k j l − 2∂yi jk l
(cid:18) (cid:19)
(cid:0) (cid:1)
1 1 1 ∂E 1 1
Xk α Xsα + ylα Xs α XmXlα EXmXlα EXmXl α =
2 k i s 2 l ik s − 2∂yk l i m − 2 lk i m − 2 l ik m
(cid:18) (cid:19)
1 1∂E 1∂E
XkXjα α α XjXlα XkXjXlα XjXkXl α +
k j i − 2 i j l − 2∂yi k j l − 2∂yi jk l
1 1 1 1 1
Xlα Xsα + ylα XkXs α ymα XsXlα EXkXmXlα EXmXkXl α =
2 l i s 2 l ik s − 2 m l i s − 2 lk i m − 2 l ik m
1 1∂E 1∂E
α XjXlα XkXjXlα XjXkXl α +
2 i j l − 2∂yi k j l − 2∂yi jk l
1 1 1 1 1
Xlα Xsα + ylα XkXs α ymα XsXlα EXlXkXmα EXkXl Xmα .
2 l i s 2 l ik s − 2 m l i s − 2 i lk m − 2 ik l m
Each term containing Xl means second order partial derivatives of Xl with respect to y’s. To
ij
reduce the order of the partial differentiation in the first, the second and the third indicated
terms we can directly use formula (21); for example (by replacing the free index i with j in (21))
(24) XkXl α (2=1) 1 3ymα Xlα 1 ∂EXkXlα + 1EXlXmα
jk l E −2 m j l − 2∂yj k l 2 j l m
(cid:18) (cid:19)
and we have similar expressions coming from the terms
1 3 1∂E 1
XkXmα = ymα Xtα XkXtα + EXtXmα
lk m E −2 m l t − 2∂yl k t 2 l t m
(cid:18) (cid:19)
and
1 3 1∂E 1
XkXs α = ymα Xlα XkXlα + EXlXmα .
ik s E −2 m i l − 2∂yi k l 2 i l m
(cid:18) (cid:19)
ON ASANOV’S FINSLEROID-FINSLER METRICS... 9
For the last indicated term
EXjXl Xtα (1=7)
ij l t
Xj ymα l 1∂EXl 1∂EXl 1g Xsyl EXs l E l Xs +EXl m Xtα (=9)
mCij − 2∂yi j − 2∂yj i − 2 sj i − iCjs − Cis j mCij l t
(cid:18) (cid:19)
1 1∂E 1 1
ymα Xl Xtα XjXlXtα ymα XlXtα α XsylXtα
m −2 i l t − 2∂yi j l t − 2 m i l t − 2 s i l t−
(cid:18) (cid:19)
1 1
EXs Xl Xtα EXj l XsXtα +EXl Xm Xtα
i −2 s l t − Cis j l t m −2 i l t
(cid:18) (cid:19) (cid:18) (cid:19)
because of
1Xj∂EXl (=5) 1gjmα ∂EXl = 1ymα Xl and ylXt = 0;
2 ∂yj i 2 m∂yj i 2 m i l
see (9) and the basic property yl = 0 of the first Cartan tensor. Therefore
ijl
C
1∂E
EXjXl Xtα = ymα XlXtα XjXlXtα EXj l XsXtα
ij l t − m i l t − 2∂yi j l t − Cis j l t
and we have by substituting the expressions of XkXl α (see (24)), XkXmα , XkXs α and
jk l lk m ik s
EXjXl Xtα in formula (23)
ij l t
∂ 1 1∂E 1 1
Xk XjBl α = (XkXjα )α XjXlXtα + Xlα Xsα ymα XlXtα
∂yk ij l 2 k j i − 2∂yi j l t 2 l i s − 2 m i l t−
(cid:0) (cid:1)
1∂E 2 1
ymα XjXlα + XjXlXmα +
2∂yi −E m j l 2 j l m
(cid:18) (cid:19)
1 3 1∂E 1
ylα ymα Xlα XtXjα + EXlXmα
2E l −2 m i l − 2∂yi j t 2 i l m −
(cid:18) (cid:19)
1 3 1
ymα Xtα Xl + EXlXtXmα
2 −2 m l t i 2 i l t m −
(cid:18) (cid:19)
1 1∂E
ymα XlXtα XjXlXtα EXj l XsXtα .
2 − m i l t − 2∂yi j l t − Cis j l t
(cid:18) (cid:19)
Now the formula has been free from the second order terms containing Xl . On the other hand
ij
XkXjα Bl (1=8) XkXj 2α α g Xlα ∂EXlα ∂EXlα EXl α .
k l ij k i j − ij l − ∂yi j l − ∂yj i l − ij l
(cid:18) (cid:19)
To set the formula free from Xl we use formula (16):
ij
1∂E 1∂E 1
EXl α = ymα l α Xlα Xlα g Xsylα EXs l α E l Xsα +
ij l mCij l − 2∂yi j l − 2∂yj i l − 2 sj i l − iCjs l − Cis j l
EXl mα (1=0) g Xsymα 1∂EXlα 1 ∂EXlα + 1EXsg Xm + 1Eg XmXs+
mCij l − js i m − 2∂yi j l − 2∂yj i l 2 i jm s 2 sm i j
EXl mα
mCij l
and, consequently,
1∂E 1 ∂E
XkXjα Bl = XkXj 2α α g Xlα Xlα Xlα +
k l ij k i j − ij l − 2∂yi j l − 2∂yj i l
(cid:18) (cid:19)
1 1
XkXj g Xsymα EXsg Xm Eg XmXs EXl mα =
k js i m − 2 i jm s − 2 sm i j − mCij l
(cid:18) (cid:19)
1∂E
2XkXjα α g XkXjXlα XkXjXlα +g XkXjXsymα
k i j − ij k l − 2∂yi k j l js k i m−
10 CS. VINCZE
1 1
EXsg XkXjXm Eg XmXkXjXs EXl mXkXjα
2 i jm k s − 2 sm i k j − mCij k l
because of
∂E ∂E
XkXj = 2XkXt j = 2XkXtym = 0.
k∂yj − Ctk∂yj − Cmtk
Using the symmetry property
EXsg XkXjXm (1=0) EXsg XkXjXm
i jm k s i sm k j
it follows that
1∂E
XkXjα Bl = 2XkXjα α g XkXjXlα XkXjXlα +g XkXjXsymα
k l ij k i j − ij k l − 2∂yi k j l js k i m−
EXsg XkXjXm EXl mXkXjα (1=0)
i jm k s − mCij k l
1∂E
2XkXjα α g XkXjXlα XkXjXlα +g XkXjXsymα
k i j − kj i l − 2∂yi k j l jk s i m−
EXsg XkXj Xm EXl mXkXjα =
i jk m s − mCij k l
1∂E
2XkXjα α Xjα Xlα XkXjXlα +Xjα Xsymα
k i j − i j l − 2∂yi k j l s j i m−
EXsXj α Xm EXl mXkXjα .
i m j s − mCij k l
Finally
∂ 3 3 ∂E 3
Xk XjBl α XkXjα Bl = XkXjα α + ymα XtXjα + Xlα Xsα
∂yk ij l − k l ij −2 k j i 4E m j t∂yi 2 l i s−
(cid:0) (cid:1)
3 3 3
(ymα )2Xlα + EXj l XsXtα + EXsXj α Xm,
4E m i l 2 Cis j l t 4 i m j s
where
3EXsXj α Xm (=9) 3EXt sXj Xmα .
4 i m j s −2 Cit m s j
Therefore
∂ 3 3 ∂E 3
Xk XjBl α XkXjα Bl = XkXjα α + ymα XtXjα + Xlα Xsα
∂yk ij l − k l ij −2 k j i 4E m j t∂yi 2 l i s−
(cid:0) (cid:1)
3 3
(ymα )2Xlα + E Xj l XsXtα Xt sXj Xmα ,
4E m i l 2 Cis j l t − Cit m s j
where (cid:0) (cid:1)
Xj l XsXtα Xt sXj Xmα = 2XpXjXsXrQ
Cis j l t − Cit m s j j pisr
as a straightforward computation shows4. (cid:3)
4
Xj l XsXtα Xt sXj Xmα (=5)Xj l XsXtα Xt sXj Xmg Xr (1=0)
Cis j l t− Cit m s j Cis j l t− Cit m s jr
Xj l XsXtα Xt sXjXmg Xr (1=0)Xj l XsXtα Xt sXjXmg Xr =
Cis j l t− Cit r s jm Cis j l t− Cit r j ms
Xj l XsXtα Xt XjXmXr =XjXs l Xtα Xt Xm (=5)XjXs l Xtα gtrα Xm =
Cis j l t− Cmit r j j Cis l t− Cmit s j Cis l t− rCmit s
XjXs l Xtα α r Xm =XjXs l Xt(cid:0) t Xm α (=9)XjX(cid:1)s l 2X(cid:0)p t t 2Xp m (cid:1)α =
j Cis l t− rCmi s j Cis l −Cmi s t j Cis − Cpl −Cmi − Cps t
−2X(cid:0)pXjXjs CilsCptl−Cmt iC(cid:1)pms αt =−(cid:0)2XpXjXjsQtipsαt(cid:1)=−2XpXjX(cid:0)jsQtip(cid:0)sXrgrt =(cid:1)−2XpX(cid:0)jXjsXrQi(cid:1)p(cid:1)sr =
(cid:0) 2(cid:1)XpXjXjsXrQpisr because of Qipsr =−Qpisr.