2 Berwald connection

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 non- Hermitian 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− com- plex 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,nbe the complex coordinates in a local chart. The complexified of the real tangent bundle TCM splits into the sum of holomorphic tangent bundle T⁰M and its conjugate T⁰⁰M . The bundle T⁰M 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 z^0k= z^0k(z) and η^0k= ^∂z_∂z^0klη^l.

A R− complex Finsler space is a pair (M, F ), where F is a continuous function F : T⁰M −→ R+ satisfying the conditions:

D

° Balkan Society of Geometers, Geometry Balkan Press 2014.c

i) F := L² is smooth on ]T⁰M := T⁰M \{0};

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

∂ηⁱ∂η^j ; g_i¯j = ∂²L

∂ηⁱ∂ ¯η^j ; g¯i^¯j= ∂²L

∂ ¯ηⁱ∂ ¯η^j. Some consequences of the homogeneity condition iii), [10], are

∂L

∂ηⁱηⁱ+ ∂L

∂ηⁱηⁱ = 2L ; gijηⁱ+ g_j¯iηⁱ= ∂L

∂η^j; (1.2)

2L = gijηⁱη^j+ 2g_ijηⁱη^j+ g_ijηⁱη^j;

∂gik

∂η^jη^j+∂gik

∂η^j η^j = 0 ; ∂g_ik

∂η^j η^j+∂g_ik

∂η^jη^j = 0.

with their complex conjugates.

We say that a function f on ]T⁰M is R− homogenous of degree p in the fibre variables η iff _∂η^∂fiηⁱ+_∂η^∂fiηⁱ = 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 the regularity condition: gij = _∂η^∂i²∂η^Lj is nondegenerated, i.e. det (gij) 6= 0 at any point u ∈ ]T⁰M , and defines a positive definite quadratic form for all z ∈ M , [11].

Consider the sections of the complexified tangent bundle of T⁰M. Let V T⁰M ⊂ T⁰(T⁰M ) be the vertical bundle, locally spanned by {_∂η^∂k}. V T⁰⁰M is its complex con- jugate. A complex nonlinear connection, briefly (c.n.c.), is a supplementary complex subbundle to V T⁰M in T⁰(T⁰M ), i.e. T⁰(T⁰M ) = HT⁰M ⊕ V T⁰M. The horizontal distribution HuT⁰M is locally spanned by {_δz^δk = _∂z^∂k − N_k^j_∂η^∂j}, where N_k^j(z, η) are the coefficients of the (c.n.c.), i.e. they transform by a certain rule

(1.3) N_j⁰ⁱ∂z^0j

∂z^k =∂z⁰ⁱ

∂z^jN_k^j− ∂²z⁰ⁱ

∂z^j∂z^kη^j.

The pair {δk := _δz^δk, ˙∂k := _∂η^∂k} will be called the adapted frame of the (c.n.c.) which obey to the change rules δk = ^∂z_∂z^0jkδ⁰_j and ˙∂k = ^∂z_∂z^0jk ˙∂_j⁰. By conjugation every- where we have obtained an adapted frame {δ¯k, ˙∂¯k} on T_u⁰⁰(T⁰M ). The dual adapted bases are {dz^k, δη^k} and {d¯z^k, δ ¯η^k}.

We consider c (t) a curve on complex manifold M and (z^k(t) , η^k(t) =^dz_dt^k) its ex- tension on T⁰M. The Euler-Lagrange equations with respect to a complex Lagrangian L are

(1.4) ∂L

∂zⁱ − d dt

µ∂L

∂ηⁱ

¶

= 0,

where L is considered along the curve c on T⁰M. Following the same arguments form concerning the complex geodesic curves, in [11] is obtained that:

(1.5) g_ijd²z^j

dt² + ∂²L

∂z^j∂ηⁱη^j− ∂L

∂zⁱ = 0,

which is equivalent with

(1.6) d²z^h

dt² + 2G^h¡

z^h(t) , η^h(t)¢

= 0, where

G^h(z, η) = 1 2g^ih

µ ∂²L

∂z^j∂ηⁱη^j− ∂L

∂zⁱ

¶ (1.7)

= 1

4g^ih µ∂gri

∂z^j +∂gji

∂z^r −∂grj

∂zⁱ

¶

η^rη^j−1 2g^ih∂g¯l¯s

∂zⁱη¯^lη¯^s +g^ih

µ∂gi¯s

∂z^j −∂gj¯s

∂zⁱ

¶ η^jη¯^s.

The functions G^hare the coefficients of a complex spray on T⁰M and following the general theory of a complex spray [10] it results that

(1.8) N_k^h=∂G^h

∂η^k,

is a complex nonlinear connection on T⁰M , 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

(1.9) ( ˙∂jGⁱ)η^j+ ( ˙∂r¯Gⁱ)¯η^r= 2Gⁱ; ( ˙∂jN_kⁱ)η^j+ ( ˙∂¯rN_kⁱ)¯η^r= N_kⁱ,

which means that Gⁱ and N_kⁱ 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 ∈ T⁰(T⁰M ) , where

(1.10) G = gijdzⁱ⊗ dz^j+ gijδηⁱ⊗ δη^j+ g¯i^¯jd¯zⁱ⊗ d¯z^j+ g¯i^¯jδ ¯ηⁱ⊗ δ ¯η^j. Locally it is denoted by DΓ :=

³

N_jⁱ, Lⁱ_jk, Lⁱ_j¯k, C_jkⁱ , C_j¯ⁱ_k

´ , where

Lⁱ_jk = 1

2g^im{δj(gkm) + δk(gjm) − δm(gjk)} ; C_jkⁱ =1

2g^im˙∂j(gkm), (1.11)

Lⁱ_j¯k = 1

2g^imδ¯k(g_jm) ; C_j¯ⁱ_k =1

2g^im˙∂¯k(g_jm).

2 Berwald connection

First, we associate to the canonical (c.n.c.) a complex linear connection of Berwald type

BΓ :=

³

N_jⁱ, B_jkⁱ := ˙∂kN_jⁱ, B_j¯ⁱ_k := ˙∂¯kN_jⁱ, 0, 0

´

with its connection form

(2.1) ω_jⁱ(z, η) = Bⁱ_jkdz^k+ B_j¯ⁱ_kd¯z^k. Using (1.9), we deduce that

( ˙∂jB_hkⁱ )η^j+ ( ˙∂¯rBⁱ_hk)¯η^r= 0, i.e. Bⁱ_hk are R−homogeneous of degree 0 and

N_kⁱ = B_jkⁱ η^j+ B_j¯ⁱ_kB_{k ¯}ⁱ_mη¯^m Note that Berwald connection is not metrical compatible.

The connection form of BΓ satisfy the following structure equations

(2.2) d(dzⁱ) − dz^k∧ ωⁱ_k= hΩⁱ; d(δηⁱ) − δη^k∧ ωⁱ_k= vΩⁱ; dωⁱ_j− ω_j^k∧ ω_kⁱ = Ωⁱ_j and their conjugates, where d is exterior differential with respect to the canonical (c.n.c.). Since

d(δηⁱ) = dN_jⁱ∧ dz^j= 1

2K_jkⁱ dz^k∧ dz^j+ Θⁱ_j¯kd¯z^k∧ dz^j +B_jkⁱ δη^k∧ dz^j+ Bⁱ_j¯kδ ¯η^k∧ dz^j

and B_jkⁱ = B_kjⁱ , the torsion and curvature forms are hΩⁱ = −Bⁱ_j¯kdz^j∧ d¯z^k ;

vΩⁱ = −1

2K_jkⁱ dz^j∧ dz^k− Θⁱ_j¯kdz^j∧ d¯z^k− B_j¯ⁱ_kdz^j∧ δ ¯η^k− Bⁱ_j¯kδη^j∧ d¯z^k; Ωⁱ_j = −1

2K_jkhⁱ dz^k∧ dz^h−1

2K_j¯ⁱ_k¯hd¯z^k∧ dz^h+ K_jhkⁱ dz^k∧ dz^h

−Bⁱ_jkhdz^k∧ δη^h− B_j¯ⁱ_k¯hd¯z^k∧ δ ¯η^h− B_jhkⁱ dz^k∧ δη^h+ B_jhkⁱ δη^k∧ dz^h, where

K_jkⁱ := δkN_jⁱ− δjN_kⁱ ; Θⁱ_j¯k:= δ¯kN_jⁱ; and K_jkhⁱ := δhB_jkⁱ − δkB_jhⁱ + B^l_jkB_lhⁱ − B_jh^l B_lkⁱ ; K_j¯ⁱ_k¯h:= δ¯hB_j¯ⁱ_k− δ¯kB_j¯ⁱ_h+ B^l_j¯kB_l¯ⁱ_h− B_j¯^l_hB_l¯ⁱ_k ; K_j¯ⁱ_kh:= δhB_j¯ⁱ_k− δ¯kB_jhⁱ + B^l_j¯kB_lhⁱ − B_jh^l B_l¯ⁱ_k

are hh-, ¯h¯h- and h¯h- curvature tensors, respectively;

B_jkhⁱ := ˙∂hB_jkⁱ ; B_j¯ⁱ_k¯h= ˙∂¯hB_j¯ⁱ_k; B_j¯ⁱ_kh:= ˙∂hB_j¯ⁱ_k are hv-, ¯h¯v- and h¯v- curvature tensors, respectively.

Taking the exterior differential of the third structure equation from (2.2), it results (2.3) −Ω^l_j∧ ωⁱ_l+ ω^l_j∧ Ωⁱ_l= dΩⁱ_j,

which leads to sixteen Bianchi identities:P

r,k,h{Kⁱ

jkh^b|r

− Bⁱ_jrlK_hk^l } = 0;

Kⁱ

jkh^b| ¯r

= B_{j ¯}ⁱ_rlK_hk^l − Ahk{Kⁱ

j ¯rk^b|h

+ K_{j ¯}ⁱ_mkB_rh^m_¯^¯ − K_jlkⁱ B_h¯^l_r− B_{j ¯}ⁱ_mkΘ^m_rh¯^¯ + Bⁱ_jlkΘ^l_h¯r};

Kⁱ

j¯k¯h^b|r

= B_{j ¯}ⁱ_mrK_¯k¯^m¯_h− Ahk{Kⁱ

j¯hr^b|¯k

+ K_j¯ⁱ_hlB_r¯^l_k− K_j¯ⁱ_{k ¯m}B_¯hr^m¯ + B_{j ¯}ⁱ_m¯hΘ^m_kr¯^¯ − Bⁱ_j¯klΘ^l_r¯h};

P

¯ r,¯k,¯h{Kⁱ

j¯k¯h^b| ¯r

− B_{j ¯}ⁱ_m¯kK_¯r¯^m¯_h} = 0;

Akr{Bⁱ

jkh^b|r

} − K_jkr^{i b}|_h= 0;

A_kr{Bⁱ

j¯hk^b|r

+ B_jrlⁱ B_k¯^l_h+ B_{j ¯}ⁱ_mkB_¯hr^m¯} − K_jkr^{i b}|¯h= 0;

A¯h¯r{Bⁱ

j ¯rk^b|¯h

+ B_{j ¯}ⁱ_rlB^l_k¯h+ Bⁱ_{j ¯m¯h}B_rk^m_¯^¯} + K_j¯ⁱ_h¯r^b|_k = 0;

A¯k¯r{Bⁱ

j¯h¯k^b| ¯r

} − K_j¯ⁱ_k¯r^b|¯h= 0;

Bⁱ

jkh^b| ¯r

− Bⁱ

j ¯rh^B| k

+ K_{j ¯}ⁱ_rk^b|_h+P

kh{B_jlhⁱ B^l_k¯r− B_{j ¯}ⁱ_mhB^m_¯rk^¯} = 0;

Bⁱ

j¯h¯k^b|r

− Bⁱ

j¯hr^b|¯k

− K_j¯ⁱ_kr^b|¯h+P

¯h¯k{B_{j ¯}ⁱ_m¯hB_¯kr^m¯ − B_j¯ⁱ_hlB_r¯^l_k} = 0;

B_jkh^{i b}|_r− Bⁱ_jkr^b|_h= 0; B_jkh^{i b}|_r¯− Bⁱ_{j ¯rk}^b|_h= 0;

B_j¯ⁱ_hk^b|_r¯− Bⁱ_j¯h¯r^b|_k = 0; B_{j ¯}ⁱ_r¯k^b|h¯− Bⁱ_j¯h¯k^b|_¯r= 0;

B_j¯ⁱ_hk^b|_r¯− B_{j ¯}ⁱ_rk^b|¯h = 0; B_j¯ⁱ_hk^b|_r− B_j¯ⁱ_hr^b|_k = 0, where ’b

| ’ and ’^b| ’ are horizontal and vertical respectively, covariant derivatives with respect to BΓ andP

and A are symmetric and antisymmetric operators.

3 Rund connection

In this section we consider a new complex linear connection RΓ :=¡

N_jⁱ, Lⁱ_jk, 0, 0, 0¢ ,

where Lⁱ_jk is given in (1.11). By analogy with real case we call this the Rund connec- tion. It is only h - metrical compatible, i.e. g_ij|k= 0, where ’_|’ is horizontal covariant derivatives with respect to RΓ.

The connection RΓ satisfies the following structure equations

(3.1) d(dzⁱ) − dz^k∧ ˜ωⁱ_k= h ˜Ωⁱ; d(δηⁱ) − δη^k∧ ˜ωⁱ_k= v ˜Ωⁱ; d˜ωⁱ_j− ˜ω_j^k∧ ˜ω_kⁱ = ˜Ωⁱ_j, where ˜ω_jⁱ(z, η) = Lⁱ_jkdz^k is connection form of RΓ. The torsion and curvature forms corresponding to RΓ are

h ˜Ωⁱ = 0 ; v ˜Ωⁱ = −1

2K_jkⁱ dz^j∧ dz^k− Θⁱ_j¯kdz^j∧ d¯z^k− (B_jkⁱ − Lⁱ_jk)dz^j∧ δη^k

−Bⁱ_j¯kdz^j∧ δ ¯η^k− B_j¯ⁱ_kδη^j∧ d¯z^k ; Ω˜ⁱ_j = −1

2Rⁱ_jkhdz^k∧ dz^h− Rⁱ_jhkdz^k∧ dz^h− P_jkhⁱ dz^k∧ δη^h− P_jhkⁱ dz^k∧ δη^h,

where

Rⁱ_jkh:= δhLⁱ_jk− δkLⁱ_jh+ L^l_jkLⁱ_lh− L^l_jhLⁱ_lk ;

Rⁱ_jhk:= δ¯hLⁱ_jk are hh- and h¯h- curvature tensors, respectively;

P_jkhⁱ := ˙∂hLⁱ_jk; P_jhkⁱ := ˙∂¯hLⁱ_jkare hv- and h¯v- curvature tensors, respectively.

Taking the exterior differential of the third structure equation from (3.1), it results (3.2) − ˜Ω^l_j∧ ˜ωⁱ_l+ ˜ω^l_j∧ ˜Ωⁱ_l= d ˜Ωⁱ_j,

which leads the following group of Bianchi identities P

r,k,h{Rⁱ_jkh|r− P_jrlⁱ K_hk^l } = 0;

Rⁱ_jkh|¯r− P_{j ¯}ⁱ_rlK_hk^l + Ahk{Rⁱ_{j ¯rk|h}− P_{j ¯}ⁱ_mkΘ^m_rh¯^¯ + P_jklⁱ Θ^l_h¯r} = 0;

Ahk{Rⁱ_j¯hr|¯k− P_j¯ⁱ_klΘ^l_r¯h} − P_{j ¯}ⁱ_mrK_¯k¯^m¯_h= 0;

Akr{P_jkh|rⁱ + P_jklⁱ (L^l_hr− B_hr^l )} − Rⁱ_jkr|h= 0;

Akr{P_j¯ⁱ_hk|r+ P_jrlⁱ B^l_k¯h} − Rⁱ_jkr|¯h= 0;

A¯h¯r{P_{j ¯}ⁱ_rk|¯h} = 0;

P_jkh|¯ⁱ _r− P_{j ¯}ⁱ_rh|k+ Rⁱ_{j ¯rk}|h+ P_{j ¯}ⁱ_rl(B_hk^l − L^l_hk) − P_{j ¯}ⁱ_mkB_rh^m_¯^¯ = 0;

P_j¯ⁱ_hr|¯k+ R_j¯ⁱ_kr|h¯− P_{j ¯}ⁱ_mr(B_¯h¯^m¯_k− L^m_¯h¯^¯_k) + P_j¯ⁱ_klB_r¯^l_h= 0;

P_jkhⁱ |r− P_jkrⁱ |h= 0; P_jkhⁱ |r¯− P_{j ¯}ⁱ_rk|h= 0;

P_j¯ⁱ_hk|¯r= 0; P_j¯ⁱ_hk|r¯− P_{j ¯}ⁱ_rk|¯h= 0; P_j¯ⁱ_hk|r− P_j¯ⁱ_hr|k= 0, where ’ | ’ is vertical covariant derivatives with respect to RΓ.

An example. We consider the function F²= L(z, w; η, θ) = e^2σ

q

(η + ¯η)⁴+¡ θ + ¯θ¢₄

, with η, θ 6= 0,

on C², where σ(z, w) is a real valued function,and we relabeled the usual local coor- dinates z¹, z², η¹, η²as 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.)

N₁¹ = [η +

¡θ + ¯θ¢₄ 3 (η + ¯η)³]∂σ

∂z +2θ − ¯θ 3

∂σ

∂w; N₂¹ = −2¡

θ + ¯θ¢₃ 3 (η + ¯η)²

∂σ

∂z +2 (η + ¯η) 3

∂σ

∂w; N₁² = 2¡

θ + ¯θ¢ 3

∂σ

∂z −2 (η + ¯η)³ 3¡

θ + ¯θ¢₂∂σ

∂w; N₂² = 2η − ¯η

3

∂σ

∂z + [θ + (η + ¯η)⁴ 3¡

θ + ¯θ¢₃]∂σ

∂w.

References

[1] M. Abate, G. Patrizio, Finsler Metrics - A Global Approach, Lecture Notes in Math., 1591, Springer-Verlag, 1994.

[2] N. Aldea, G. Munteanu, On complex Landsberg and Berwald spaces, Journal of Geometry and Physics, 62 (2012), no. 2, 368-380.

[3] N. Aldea, G. Munteanu, On projective invariants of the complex Finsler spaces, Differential Geom. Appl. 30 (2012), 6, 562–575.

[4] N. Aldea, M. Purcaru, R− complex Finsler spaces with (α, β)-metric, Novi Sad J. Math., 38, 1(2008), 1-9.

[5] A. Bejancu, H. R. Faran, The geometry of pseudo-Finsler submanifolds, Kluwer Acad. Publ., 2000.

[6] D. Bao, S. S. Chern, Z. Shen, An Introduction to Riemannian Finsler Geom., Graduate Texts in Math., 200, Springer-Verlag, 2000.

[7] G. Cˆampean, N. Aldea, Some classes of R− complex Hermitian Finsler spaces, manuscript 2013.

[8] S. Kobayashi, Complex Finsler vector bundles, Contemporary Math. 1996, 145- 153.

[9] M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kai- seisha Press, Saikawa, Otsu, 1986.

[10] G. Munteanu, Complex Spaces in Finsler, Lagrange and Hamilton Geometries, Kluwer Acad. Publ., 141, FTPH, 2004.

[11] G. Munteanu, M. Purcaru, On R− complex Finsler spaces, Balkan J. Geom.

Appl., 14, 1(2009), 52-59.

[12] H. L. Royden, Complex Finsler metrics, Contemporary Math., 49 (1984), 119- 124.

[13] H. Rund, The differential geometry of Finsler spaces, Springer-Verlag, Berlin, 1959.

[14] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.

[15] P. M. Wong, Theory of Complex Finsler Geometry and Geometry of Intrinsic Metrics, Imperial College Press 2011.

Author’s address:

Gabriela Cˆampean

Transilvania Univ., Department of Mathematics and Informatics Iuliu Maniu 50, Bra¸sov 500091, Romania.

E-mail: [email protected]