Connections on R-complex non

Connections on R-complex non-Hermitian Finsler
spaces
Gabriela Cˆampean
Abstract. In this paper we extend the results on R- complex Finsler
spaces by studying some properties of R-complex non-Hermitian Finsler
spaces.We introduce two complex linear connections on a R-complex nonHermitian Finsler space, namely Berwald and Rund connections. Various
descriptions of these connections are given related to their corresponding
curvature and torsion tensors.
M.S.C. 2010: 53B40, 53C60.
Key words: R− complex non-Hermitian Finsler space; Berwald connection; Rund
connection.
1
Introduction
Bearing in mind a previous paper with important results on R− complex Hermitian
Finsler spaces [7], we continue the study, making a similar approach to the R− complex non-Hermitian Finsler spaces. Since the complex linear connections are the main
tools in the study of the geometry of the R− complex Finsler spaces, the aim of this
paper is to introduce on a R− complex non-Hermitian Finsler space the notions of
Berwald and Rund connections. The curvature and torsion tensors corresponding
to these connections are obtained from the structure equations. Also, we derive the
Bianchi identities which specify the relations among the covariant derivatives of the
curvature coefficients and which are very useful in our next work.
First step is to make a short introduction in the geometry of R− complex Finsler
spaces, emphasizing properties of R− complex non-Hermitian Finsler spaces.
Let M be a n - dimensional complex manifold and z = (z k )k=1,n be the complex
coordinates in a local chart. The complexified of the real tangent bundle TC M splits
into the sum of holomorphic tangent bundle T 0 M and its conjugate T 00 M . The bundle
T 0 M is itself a complex manifold and the local coordinates in a local chart will be
denoted by u = (z k , η k )k=1,n . These are changed into (z 0k , η 0k )k=1,n by the rules
0k
ηl .
z 0k = z 0k (z) and η 0k = ∂z
∂z l
A R− complex Finsler space is a pair (M, F ), where F is a continuous function
F : T 0 M −→ R+ satisfying the conditions:
Differential Geometry - Dynamical Systems, Vol.16, 2014, pp.
c Balkan Society of Geometers, Geometry Balkan Press 2014.
°
85-91.
86
Gabriela Cˆampean
0 M := T 0 M \{0};
i) F := L2 is smooth on T]
ii) F (z, η) ≥ 0 the equality holds if and only if η = 0;
iii) F (z, λη, z¯, λ¯
η ) = |λ| F (z, η, z¯, η¯), ∀λ ∈ R.
We use the following tensors
(1.1)
gij =
∂2L
∂2L
∂2L
;
g
=
;
g
=
.
¯
¯
i
j
¯
ij
∂η i ∂η j
∂η i ∂ η¯j
∂ η¯i ∂ η¯j
Some consequences of the homogeneity condition iii), [10], are
(1.2)
∂L i ∂L i
η + iη
∂η i
∂η
2L
∂gik j ∂gik j
η +
η
∂η j
∂η j
∂L
;
∂η j
=
2L ; gij η i + gj¯i η i =
=
gij η i η j + 2gij η i η j + gij η i η j ;
=
0;
∂gik j ∂gik j
η +
η = 0.
∂η j
∂η j
with their complex conjugates.
0 M is R− homogenous of degree p in the fibre
We say that a function f on T]
∂f i
∂f i
variables η iff ∂ηi η + ∂ηi η = pf. For example, L is R− homogenous of degree 2 in
the fibre variables.
An R− complex non-Hermitian Finsler space is the pair (M, F ) where F satisfies
2
L
the regularity condition: gij = ∂η∂i ∂η
j is nondegenerated, i.e. det (gij ) 6= 0 at any
0
]
point u ∈ T M , and defines a positive definite quadratic form for all z ∈ M , [11].
Consider the sections of the complexified tangent bundle of T 0 M. Let V T 0 M ⊂
0
T (T 0 M ) be the vertical bundle, locally spanned by { ∂η∂ k }. V T 00 M is its complex conjugate. A complex nonlinear connection, briefly (c.n.c.), is a supplementary complex
subbundle to V T 0 M in T 0 (T 0 M ), i.e. T 0 (T 0 M ) = HT 0 M ⊕ V T 0 M. The horizontal
distribution Hu T 0 M is locally spanned by { δzδk = ∂z∂k − Nkj ∂η∂ j }, where Nkj (z, η) are
the coefficients of the (c.n.c.), i.e. they transform by a certain rule
(1.3)
The pair {δk :=
Nj0i
δ
, ∂˙k
δz k
∂z 0j
∂z 0i j
∂ 2 z 0i
=
Nk − j k η j .
k
j
∂z
∂z
∂z ∂z
:=
∂
}
∂η k
will be called the adapted frame of the (c.n.c.)
0j
0j
which obey to the change rules δk = ∂z
δ 0 and ∂˙k = ∂z
∂˙ 0 . By conjugation every∂z k j
∂z k j
00
˙
where we have obtained an adapted frame {δk¯ , ∂k¯ } on Tu (T 0 M ). The dual adapted
bases are {dz k , δη k } and {d¯
z k , δ η¯k }.
k
We consider c (t) a curve on complex manifold M and (z k (t) , η k (t) = dz
dt ) its extension on T 0 M. The Euler-Lagrange equations with respect to a complex Lagrangian
L are
µ
¶
d ∂L
∂L
−
= 0,
(1.4)
∂z i
dt ∂η i
where L is considered along the curve c on T 0 M. Following the same arguments form
concerning the complex geodesic curves, in [11] is obtained that:
(1.5)
gij
∂2L j
∂L
d2 z j
+
η − i = 0,
dt2
∂z j ∂η i
∂z
Connections on R-complex non-Hermitian Finsler spaces
87
which is equivalent with
¡
¢
d2 z h
+ 2Gh z h (t) , η h (t) = 0,
2
dt
(1.6)
where
Gh (z, η)
(1.7)
=
=
µ 2
¶
∂ L j
1 ih
∂L
g
η
−
2
∂z j ∂η i
∂z i
µ
¶
∂g¯
1 ih ∂gri
∂gji
∂grj
1
g
+
−
η r η j − g ih l¯is η¯l η¯s
j
r
i
4
∂z
∂z
∂z
2
∂z
µ
¶
∂g
∂g
j s¯
i¯
s
+g ih
−
η j η¯s .
∂z j
∂z i
The functions Gh are the coefficients of a complex spray on T 0 M and following the
general theory of a complex spray [10] it results that
Nkh =
(1.8)
∂Gh
,
∂η k
is a complex nonlinear connection on T 0 M , which will be called canonical (c.n.c.).
Next, we work only with the canonical (c.n.c.) and thus hereinafter δj is with respect
to (1.8). Taking into account the homogeneity condition of L, we may state the
following properties
(∂˙j Gi )η j + (∂˙r¯Gi )¯
η r = 2Gi ; (∂˙j Nki )η j + (∂˙r¯Nki )¯
η r = Nki ,
(1.9)
which means that Gi and Nki are R−homogeneous of degree 2, respectively 1, with
respect with η, like in R− complex Hermitian Finsler spaces case.
In [11] it is proved that there exists a unique complex linear connection D which
is torsions free (hT (hX, hX) = 0, vT (vX, vX) = 0), metrical compatible (DG = 0)
and G (DX¯ Y, Z) = G (DX¯ Z, Y ) , ∀X, Y, Z ∈ T 0 (T 0 M ) , where
z i ⊗ d¯
z j + g¯i¯j δ η¯i ⊗ δ η¯j .
G = gij dz i ⊗ dz j + gij δη i ⊗ δη j + g¯i¯j d¯
³
´
i
, Cjik¯ , where
Locally it is denoted by DΓ := Nji , Lijk , Lij k¯ , Cjk
(1.10)
(1.11) Lijk
=
Lij k¯
=
2
1 im
1
i
g {δj (gkm ) + δk (gjm ) − δm (gjk )} ; Cjk
= g im ∂˙j (gkm ),
2
2
1 im
1 im ˙
i
g δk¯ (gjm ) ; Cj k¯ = g ∂k¯ (gjm ).
2
2
Berwald connection
First, we associate to the canonical (c.n.c.) a complex linear connection of Berwald
type
³
´
i
BΓ := Nji , Bjk
:= ∂˙k Nji , Bji k¯ := ∂˙k¯ Nji , 0, 0
88
Gabriela Cˆampean
with its connection form
i
ωji (z, η) = Bjk
dz k + Bji k¯ d¯
zk .
(2.1)
Using (1.9), we deduce that
i
i
(∂˙j Bhk
)η j + (∂˙r¯Bhk
)¯
η r = 0,
i
i.e. Bhk
are R−homogeneous of degree 0 and
i
Nki = Bjk
η j + Bji k¯ Bki m
¯m
¯η
Note that Berwald connection is not metrical compatible.
The connection form of BΓ satisfy the following structure equations
d(dz i ) − dz k ∧ ωki = hΩi ; d(δη i ) − δη k ∧ ωki = vΩi ; dωji − ωjk ∧ ωki = Ωij
(2.2)
and their conjugates, where d is exterior differential with respect to the canonical
(c.n.c.). Since
d(δη i )
1 i
K dz k ∧ dz j + Θij k¯ d¯
z k ∧ dz j
2 jk
i
+Bjk
δη k ∧ dz j + Bji k¯ δ η¯k ∧ dz j
= dNji ∧ dz j =
i
i
and Bjk
= Bkj
, the torsion and curvature forms are
hΩi
vΩi
Ωij
= −Bji k¯ dz j ∧ d¯
zk ;
1 i
= − Kjk
dz j ∧ dz k − Θij k¯ dz j ∧ d¯
z k − Bji k¯ dz j ∧ δ η¯k − Bji k¯ δη j ∧ d¯
zk ;
2
1 i
1
i
= − Kjkh
dz k ∧ dz h − Kjik¯h¯ d¯
dz k ∧ dz h
z k ∧ dz h + Kjhk
2
2
i
i
i
dz k ∧ δη h − Bji k¯h¯ d¯
z k ∧ δ η¯h − Bjhk
−Bjkh
dz k ∧ δη h + Bjhk
δη k ∧ dz h ,
where
i
Kjk
:= δk Nji − δj Nki ; Θij k¯ := δk¯ Nji ; and
i
i
i
l
i
l
i
Kjkh
:= δh Bjk
− δk Bjh
+ Bjk
Blh
− Bjh
Blk
;
i
l
i
l
i
i
i
Kj k¯h¯ := δh¯ Bj k¯ − δk¯ Bj h¯ + Bj k¯ Blh¯ − Bj h¯ Blk¯ ;
i
l
i
l
i
i
Kjikh
¯ Bjh + Bj k
¯ := δh Bj k
¯ − δk
¯ Blh − Bjh Blk
¯
¯
¯
¯
are hh-, hh- and hh- curvature tensors, respectively;
B i := ∂˙h B i ; B i ¯ ¯ = ∂˙¯ B i ¯ ; B i ¯ := ∂˙h B i ¯
jkh
jk
j kh
h
jk
j kh
jk
¯ v - and h¯
are hv-, h¯
v - curvature tensors, respectively.
Taking the exterior differential of the third structure equation from (2.2), it results
−Ωlj ∧ ωli + ωjl ∧ Ωil = dΩij ,
(2.3)
which
identities:
P leads toi sixteen iBianchi
l
}
=
0;
K
{K
−
B
b
jrl hk
r,k,h
jkh |r
Connections on R-complex non-Hermitian Finsler spaces
Ki
b
jkh | r¯
l
= Bji r¯l Khk
− Ahk {K i
b
j r¯k |h
89
m
¯
i
l
i
m
¯
i
l
+ Kjimk
¯ Br¯h − Kjlk Bh¯
r − Bj mk
¯ Θr¯h + Bjlk Θh¯
r };
i
l
i
¯
m
¯
i
l
m
¯
i
K i b = Bji mr
B¯m
+ Bji m
¯ Br k
¯ − Kj k
¯m
¯ Θkr
¯ − Bj kl
¯ Θr h
¯ };
¯h
¯ − Ahk {K
b + Kj hl
¯ Kk
¯ hr
¯h
¯
¯
¯
¯
j hr | k
Pj kh |r i
i
m
¯
¯h
¯ {K
¯ Kr¯h
¯ } = 0;
b − Bj m
r¯,k,
¯k
¯h
¯ | r¯
jk
Akr {B i
b
b
jkh |r
i
} − Kjkr
| h = 0;
b
Akr {B i
b
i
m
¯
i
+ Bjrl
Bkl h¯ + Bji mk
¯ = 0;
¯ } − Kjkr | h
¯ Bhr
i
Ah¯
¯ r {B
b
m
¯
i
+ Bji r¯l Bkl h¯ + Bji m
¯ Br¯k } + Kj h¯
¯ r | k = 0;
¯h
i
Ak¯
¯ r {B
b
¯ |r
j hk
b
¯
j r¯k | h
b
¯k
¯ | r¯
jh
Bi
b
− Bi
b
− Bi
jkh | r¯
Bi
} − Kjik¯
¯ = 0;
¯r | h
¯k
¯ |r
jh
b
i
Bjkh
|r
b
Bji hk
¯ | r¯
b
Bji hk
¯ | r¯
b
B
j r¯h | k
+ Kjir¯k | h +
b
P
P
i
l
r
kh {Bjlh Bk¯
i
m
¯
¯k
¯ {Bj m
¯ Bkr
¯
h
¯h
¯ |k
¯
j hr
b
b
b
i
i
− Bjkr
| h = 0; Bjkh
| r¯ − Bji r¯k | h = 0;
b
b
b
i
i
− Bji h¯
¯ − Bj h
¯k
¯ | r¯ = 0;
¯ r | k = 0; Bj r¯k
¯ |h
b
b
b
i
− Bji r¯k | h¯ = 0; Bji hk
¯ |k =
¯ | r − Bj hr
b
− Kjikr
¯ +
¯ |h
m
¯
− Bji mh
¯ Br¯k } = 0;
l
− Bji hl
¯ Br k
¯ } = 0;
b
0, where ’b ’ and ’ | ’ are horizontal
|
P
and vertical respectively, covariant derivatives with respect to BΓ and
and A are
symmetric and antisymmetric operators.
3
Rund connection
In this section we consider a new complex linear connection
¢
¡
RΓ := Nji , Lijk , 0, 0, 0 ,
where Lijk is given in (1.11). By analogy with real case we call this the Rund connection. It is only h - metrical compatible, i.e. gij|k = 0, where ’| ’ is horizontal covariant
derivatives with respect to RΓ.
The connection RΓ satisfies the following structure equations
(3.1)
˜ i ; d(δη i ) − δη k ∧ ω
˜ i ; d˜
˜ ij ,
d(dz i ) − dz k ∧ ω
˜ ki = hΩ
˜ ki = v Ω
ωji − ω
˜ jk ∧ ω
˜ ki = Ω
where ω
˜ ji (z, η) = Lijk dz k is connection form of RΓ. The torsion and curvature forms
corresponding to RΓ are
˜i
hΩ
˜i
vΩ
˜ ij
Ω
= 0;
1 i
i
dz j ∧ dz k − Θij k¯ dz j ∧ d¯
z k − (Bjk
− Lijk )dz j ∧ δη k
= − Kjk
2
−Bji k¯ dz j ∧ δ η¯k − Bji k¯ δη j ∧ d¯
zk ;
1 i
i
i
i
= − Rjkh
dz k ∧ dz h − Rjhk
dz k ∧ dz h − Pjkh
dz k ∧ δη h − Pjhk
dz k ∧ δη h ,
2
90
Gabriela Cˆampean
where
i
Rjkh
:= δh Lijk − δk Lijh + Lljk Lilh − Lljh Lilk ;
i
¯ curvature tensors, respectively;
Rjhk := δh¯ Lijk are hh- and hhi
i
i
i
Pjkh := ∂˙h Ljk ; Pjhk := ∂˙h¯ Ljk are hv- and h¯
v - curvature tensors, respectively.
Taking the exterior differential of the third structure equation from (3.1), it results
˜ il = dΩ
˜ ij ,
˜ lj ∧ ω
˜ li + ω
˜ jl ∧ Ω
−Ω
(3.2)
which leads the following group of Bianchi identities
P
i
l
i
r,k,h {Rjkh|r − Pjrl Khk } = 0;
i
i
l
m
¯
i
l
Rjkh|¯r − Pj r¯l Khk + Ahk {Rji r¯k|h − Pjimk
¯ Θr¯h + Pjkl Θh¯
r } = 0;
i
i
l
i
m
¯
Ahk {Rj hr|
¯ k
¯ − Pj kl
¯ Θr h
¯ } − Pj mr
¯h
¯ = 0;
¯ Kk
i
i
l
i
Akr {Pjkh|r
+ Pjkl
(Llhr − Bhr
)} − Rjkr
|h = 0;
i
i
l
i
Akr {Pj hk|r
+ Pjrl Bkh¯ } − Rjkr |h¯ = 0;
¯
i
Ah¯
¯ r {Pj r¯k|h
¯ } = 0;
i
i
i
i
l
l
i
m
¯
Pjkh|¯
¯ Br¯h = 0;
r − Pj r¯h|k + Rj r¯k |h + Pj r¯l (Bhk − Lhk ) − Pj mk
i
m
¯
m
¯
i
l
i
i
Pj hr|
¯ − Pj mr
¯k
¯ − Lh
¯k
¯ ) + Pj kl
¯ Br h
¯ = 0;
¯ k
¯ + Rj kr
¯ |h
¯ (Bh
i
i
i
i
Pjkh |r − Pjkr |h = 0; Pjkh |r¯ − Pj r¯k |h = 0;
i
i
i
i
Pjihk
¯ = 0; Pj hk
¯ |r − Pj hr
¯ |k = 0,
¯ |r¯ = 0; Pj hk
¯ |r¯ − Pj r¯k |h
where ’ | ’ is vertical covariant derivatives with respect to RΓ.
An example. We consider the function
q
¡
¢4
4
F 2 = L(z, w; η, θ) = e2σ (η + η¯) + θ + θ¯ , with η, θ 6= 0,
on C2 , where σ(z, w) is a real valued function,and we relabeled the usual local coordinates z 1 , z 2 , η 1 , η 2 as z, w, η, θ, respectively. It is R− complex Finsler metric with
the tensor gij invertible. After a direct computation we obtain the local coefficients
of the canonical (c.n.c.)
¡
¢4
θ + θ¯
∂σ 2θ − θ¯ ∂σ
1
N1 = [η +
;
3 ] ∂z +
3 ∂w
3 (η + η¯)
¡
¢3
−2 θ + θ¯ ∂σ 2 (η + η¯) ∂σ
1
N2 =
+
;
2
3
∂w
3 (η + η¯) ∂z
¡
¢
3
2 θ + θ¯ ∂σ
2 (η + η¯) ∂σ
2
− ¡
;
N1 =
¢2
3
∂z
3 θ + θ¯ ∂w
4
N22
=
2η − η¯ ∂σ
(η + η¯) ∂σ
+ [θ + ¡
.
¢3 ]
3 ∂z
3 θ + θ¯ ∂w
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Connections on R-complex non-Hermitian Finsler spaces
91
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Author’s address:
Gabriela Cˆampean
Transilvania Univ., Department of Mathematics and Informatics
Iuliu Maniu 50, Bra¸sov 500091, Romania.
E-mail: [email protected]

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