As an application, we give some solutions of the complex Einstein equations in a weakly gravitational space

Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis 31 (2015), 139–151

www.emis.de/journals ISSN 1786-0091

EINSTEIN EQUATIONS OF G-NATURAL COMPLEX FINSLER METRICS

ANNAM ´ARIA SZ ´ASZ

Dedicated to Professor Lajos Tam´assy on the occasion of his 90th birthday

Abstract. In this paper, we endow the holomorphic tangent bundle with a generalized Sasaki type liftGof the fundamental metric tensor of a complex Finsler space. In order to build the Einstein equations on the holomorphic tangent bundle, we determine the Levi-Civita complex linear connection corresponding to this metric. As an application, we give some solutions of the complex Einstein equations in a weakly gravitational space.

1. Preliminaries

In the papers [3, 4] are studied complex Einstein equations for the weakly gravitational field and for complex version of Schwartzschild metric. This study is based on the idea to write the complex Einstein equations for these metrics relative to the Chern-Finsler connection, which is metrical but with torsion. For such theory to be consistent some restrictions are required, called conservation laws, because the connection used is with torsion.

An alternative to this theory is the one expressed in the following. We extend the metric structure of the weakly gravitational field to one on the holomorphic tangent bundle T⁰M of a complex manifold M, and we then consider the Levi-Civita connection of this lifted metric, which is metrical and torsion-free. Therefore the complex Einstein equations with respect to the Levi-Civita connection have the classical from. In particular, if the space is vacuum the complex Einstein equations reduce to the vanishing of the complex Ricci tensor. Basically, the idea seems to be simple, but the first problem is how to get such a lift and how they can be general. Then is the issue of writing

2010Mathematics Subject Classification. 53B40, 53C60.

Key words and phrases. Holomorphic tangent bundle, Chern-Finsler connection, complex Einstein equations.

This paper is supported by the Sectoral Operational Programme Human Resources De- velopment (SOP HRD), ID134378 financed from the European Social Fund and by the Romanian Government.

139

the curvature tensors and Ricci tensors onT⁰M. For this we turn again to the well-known adapted frames of Chern-Finsler connection, and express all in this complex adapted frames. A similar idea has been applied to the real case by M. Anastasiei and H. Shimada, [5]. Finally, we propose to solve these complex Einstein equations, at least in same particular cases of weakly gravitational metric.

Let M be a complex manifold of complex dimension n. We consider z ∈ M, and so z = (z¹, . . . , zⁿ) are complex coordinate in a local chart. Since z^k = x^k +√

−1x^k+n, k = 1, . . . , n, the complex coordinates induce the real coordinates {x¹, x², . . . , x²ⁿ} on M. Let T_RM be the real tangent bundle. Its complexified tangent bundle T_CM splits into the sum of holomorphic tangent bundleT⁰M and its conjugate T⁰⁰M, under the action of the natural complex structure J on M. The holomorphic tangent bundle T⁰M is itself a complex manifold, and the coordinates in a local chart will be denoted by u= (z^k, η^a), k, a= 1, . . . , n. withη^a =y^a+√

−1y^a+n, a= 1, . . . , n. Trough this paper the indices i, j, k, . . . and a, b, c, . . . run over {1, . . . , n}, where the second denotes geometric objects in local fibers of the holomorphic tangent bundle. This notation of the indices is important for the clarity of the notions in geometry of T⁰M manifold.

Consider the sections of the complexified tangent bundleT_CM. LetV T⁰M ⊂ T⁰(T⁰M) be the vertical bundle, locally spanned by

n ∂

∂η^a

o

a=1,n, and V T⁰⁰M its conjugate. The idea of the complex non-linear connection, briefly c.n.c. is an instrument in ‘linearization’ of the geometry of T⁰M manifold. A c.n.c. is a supplementary subbundle to V T⁰M in T⁰(T⁰M), i.e. T⁰(T⁰M) = HT⁰M ⊕ V T⁰M. The horizontal distribution H_uT⁰M is locally spanned by

(1) δ

δz^k = ∂

∂z^k −N_k^a ∂

∂η^a,

whereN_k^a(z, η) are the coefficients of the c.n.c. The pair{δ_k:= _δz^δk, ∂˙_a:= _∂η^∂a} will be called the adapted frame of the c.n.c., which obey the transformation laws δk = ^∂z_∂z⁰k^jδ⁰_j and ˙∂a = δ_a^kδ_j^b∂z_∂z⁰k^j∂˙_b⁰, where δ^k_a is the Kronecker symbol. By conjugation everywhere we obtain an adapted frame{δk¯,∂˙_¯a}onT_u⁰⁰(T⁰M). The dual adapted frame are {dz^k,dη^a} and {d¯z^k,d¯η^a} its conjugate, where

(2) δη^a = dη^a+N_k^a(z, η)dz^k.

Let N be a c.n.c. on T⁰M. An h−metric onT⁰M is a d−tensor field hG = g_j¯k(z, η)dz^j ⊗d¯z^k, with g_j¯k(z, η) = g_k¯j(z, η), detkg_j¯k(z, η)k 6= 0. A v−metric onT⁰M is a d−tensor field vG =h_a¯b(z, η)δη^a⊗δη¯^b, with h_a¯b(z, η) =h_b¯a(z, η), detkh_a¯b(z, η)k 6= 0. From here we obtain that an (h, v)−metric on T⁰M is tensor field G=hG+vG. So, this metric will be written in the next form:

(3) G(z, η) =g_jk¯(z, η)dz^j⊗d¯z^k+h_a¯b(z, η)δη^a⊗δη¯^b.

A distinguished complex linear connection, briefly d-c.l.c., D on T⁰M is called compatible with the metricG if DG = 0.

Definition 1. An n−dimensional complex Finsler space is a pair (M, F), where F :T⁰M →R⁺ is a continuous function satisfying the conditions:

a) L:=F² is smooth on T]⁰M :=T⁰M \ {0};

b) F(z, η)≥0, the equality holds if and only if η= 0;

c) F(z, λη) =|λ|F(z, η) for ∀λ∈C; d) the Hermitian matrix

(4) g_j¯k=δ^a_jδ^b_k ∂²L

∂η^a∂η¯^b, is positive-definite onT]⁰M.

Then, g_j¯k is called the fundamental metric tensor of the complex Finsler space. Consequently, fromc) we have _∂η^∂Laη^a= _∂¯^∂L_ηaη¯^a=L, ^∂g_∂η^ja^¯kη^a= ^∂g_∂¯η^ja^k¯η¯^a = 0, and L=δ_a^jδ^k_bg_j¯kη^aη¯^b.

From [1, 7] we know there exists a unique metric Hermitian connection D, of type (1,0)−type, which satisfies in addition D_{J X}Y = J D_XY, for all X horizontal vectors, called the Chern-Finsler connection, in brief C-F, which have a special meaning in complex Finsler geometry. The C-F connection DΓ = (N_j^a, Lⁱ_jk, C_bc^a,0,0) is locally given by the following coefficients:

(5) N_j^a =δ_k^aδ_b^lg^mk¯ ∂g_lm¯

∂z^j η^b =δ_k^aδ_b^lL^k_ljη^b; L^h_jk =g^¯lhδ_kg_j¯l; C_bc^a =δ_h^aδ_b^jg^¯lh∂˙_cg_j¯l, where the non-vanishing expressions of the C-F connection areD_δkδ_j =L^h_jkδ_h, D∂˙b

∂˙_a=C_ab^d∂˙_d and its conjugates.

A particular situation of thed−tensor g_jk¯ from (4) is:

Definition 2. Ifg_j¯kdepends only on the variablez, then we say that the space (M, F) is purely Hermitian.

The metric tensorg_j¯kfrom (4) determines a metric structureGSonTC(T⁰M), called the Sasaki lift of g_jk¯, [7], p.96:

(6) G_S =g_j¯kdz^j ⊗d¯z^k+δ^j_aδ_b^kg_jk¯(z, η)δη^a⊗δη¯^b.

We introduce a generalization of the lift (6), which defines also an (h, v)−metric onTC(T⁰M):

(7) G(z, η) =g_jk¯(z, η)dz^j⊗d¯z^k+h_a¯b(z, η)δη^a⊗δη¯^b,

where g_j¯k is the fundamental metric tensor of the Finsler space (M, F), and h_a¯b is an arbitraryd−tensor of (0,2)−type.

2. The Levi-Civita connection on T⁰M

From the standard definition of a complex linear connection on the man- ifold T⁰M, extended on the complexified tangent bundle TC(T⁰M), a com- plex linear connection ∇ can be decomposed in the sum ∇ = ∇⁰ + ∇⁰⁰, where ∇⁰: Γ(T_C(T⁰M)) → Γ(T_C(T⁰M)⊗T⁰(T^0∗M)) and ∇⁰⁰: Γ(T_C(T⁰M)) → Γ(T_C(T⁰M)⊗T⁰⁰(T^0∗M)), which can be decomposed in

∇⁰ =∇^0h+∇^0v and∇⁰⁰ =∇^00h+∇^00v.

So, in the adapted frame of the C-F c.n.c. {δ_k,∂˙_a, δ¯k,∂˙_¯a}, ∇ is defined by the following coefficients:

∇δkδ_j =

1

Lⁱ_jkδ_i+

2

A^d_jk∂˙_d+

3

A^¯ı_jkδ_¯ı+

4

A^d_jk^¯ ∂˙d¯; (8)

∇δk∂˙_a =

1

B_akⁱ δ_i+

2

L^d_ak∂˙_i+

3

B^¯_ak^ı δ_¯ı+

4

B_ak^d¯ ∂˙d¯;

∇δkδ¯j =

1

D¯ⁱjkδ_i+

2

D¯jk^d ∂˙_d+

3

L^¯ı¯jkδ_¯ı+

4

D¯jk^d¯∂˙d¯;

∇δk∂˙_¯a =

1

E_¯akⁱ δ_i+

2

E_¯ak^d ∂˙_d+

3

E^¯_¯ak^ı δ_¯ı+

4

L^d_ak¯^¯ ∂˙d¯;

∇_∂^˙_bδ_j =

1

C_jbⁱ δ_i +

2

F_jb^d∂˙_d+

3

F_jb^¯ıδ_¯ı+

4

F_jb^d¯∂˙d¯;

∇∂˙b

∂˙_a =

1

Gⁱ_abδ_i+

2

C_ab^d∂˙_d+

3

G^¯ı_abδ_¯ı+

4

G^d_ab^¯∂˙d¯;

∇∂˙bδ¯j =

1

H_¯jbⁱ δ_i +

2

H_¯jb^d∂˙_d+

3

C_¯jb^¯ı δ_¯ı+

4

H_¯jb^d¯∂˙d¯;

∇∂˙b

∂˙_¯a =

1

M_¯abⁱ δ_i+

2

M_ab¯^d∂˙_d+

3

M_¯ab^¯ı δ_¯ı+

4

C_ab¯^d¯∂˙d¯

and its conjugates, by∇X^¯Y¯ =∇XY.

Since ∇G= 0 and ∇ is a symmetric connection, direct calculus leads to Theorem 1. The Hermitian manifold (T⁰M, G_H) admits a unique complex linear connection, which is symmetric and metrical in respect toG, defined by (3). This is called the Levi-Civita connection on T⁰M, and its local coefficients are represented in the local adapted frame {δ_k,∂˙_a, δk¯,∂˙_¯a} by the following non- zero expressions:

1

Lⁱ_jk = 1

2g^¯li(δ_kg_j¯l+δ_jg_k¯l);

2

D_¯jk^c =−D^4c_k¯j = 1

2[δ¯jN_k^c−h^dc¯( ˙∂d¯g_k¯j)];

(9)

2

L^c_ak = 1

2[h^dc¯(δ_kh_ad¯) + ˙∂_aN_k^c];

2

E_ak^c_¯ = 1

2h^dc¯[( ˙∂_¯aN_k^e)h_ed¯−( ˙∂d¯N_k^e)h_e¯a];

3

Lⁱ_jk¯ =

1

D_kjⁱ_¯ = 1

2g^¯li(δk¯g_j¯l−δ¯lg_j¯k);

2

F_jb^c = 1

2[h^dc¯(δ_jh_ad¯)−∂˙_aN_j^c];

4

L^c_ak¯ =

2

Hka¯^c = 1

2h^dc¯[δ¯kh_ad¯−( ˙∂d¯Nk¯^e¯)h_a¯e];

1

Gⁱ_ab = 1

2g^¯li[( ˙∂_bN¯l^d¯)h_ad¯+ ( ˙∂_aN¯l^d¯)h_bd¯];

1

C_kaⁱ =

1

B_akⁱ = 1

2g^¯li[ ˙∂_ag_k¯l+ (δ_kN_¯l^d¯)h_ad¯];

4

H_j^c_¯b =−1

2h^dc¯[( ˙∂d¯N_j^e)h_e¯b+ ( ˙∂¯bN_j^e)h_ed¯];

2

C_ab^c = 1

2h^dc¯( ˙∂bh_ad¯+ ˙∂ah_bd¯);

1

M_¯abⁱ =

3

M_b¯ⁱ_a= 1

2g^¯li[( ˙∂a¯N¯l^d¯)h_bd¯−δ¯lhb¯a];

3

C_jⁱ¯b =

1

E_¯ajⁱ = 1

2g^¯li[ ˙∂¯bg_j¯l−(δ¯lN_j^d)h_d¯b];

4

C_a^c¯b =

2

M¯ba^c = 1

2h^dc¯( ˙∂¯bh_ad¯−∂˙d¯h_a¯b), and its conjugates.

This connection is not h- or v-metrical.

To study the Levi-Civita connection, we may consider a similar connection, which help us to express easier the different properties of the Levi-Civita con- nection. Indeed let De be a d-c.l.c. on T_C(T⁰M):

De_δkδ_j =

1

Lⁱ_jkδ_i; De_δk∂˙_a =

2

L^d_ak∂˙_d; De_δkδ¯j =

3

Lⁱ¯jkδ_¯ı ; De_δk∂˙_a=

4

L^d_ak∂˙_d; (10)

De∂˙aδ_j =

1

C_jaⁱ δ_i; De∂˙b

∂˙_a =

2

C_ab^d∂˙_d; De∂˙aδ¯j =

3

C_¯ja^¯ı δ_i ; De∂˙b

∂˙_a¯ =

4

C_¯ab^d¯∂˙_d and its conjugates, where the local coefficients are expressed in (9).

This d-c.l.c. is metrical with respect to G, i.e.

(11) g_j¯k|m =g_jk¯|d=g_j¯k|m¯ =g_jk¯|d^¯=h_a¯b|m =h_a¯b|d =h_a¯b|m¯ =h_a¯b|d^¯= 0, where with ”_|”, ”|”, ”¯|”, ”¯|” are notated the h−, v−, ¯h− and ¯v−covariant derivatives with respect to D.e

Proposition 1. The non-zero components of the torsion of the d-c.l.c. De are hTe(δk¯, δ_j) = eτ_jⁱ_k¯δ_i; vTe(δ¯k, δ_j) =Θe^d_j¯k∂˙_d; hTe( ˙∂_¯a, δ_j) = Υeⁱ_j¯aδ_i; hTe( ˙∂_a, δ_j) =Qeⁱ_jaδ_i; (12)

vTe( ˙∂¯b,∂˙_a) =χe^d_a¯b∂˙_d; vTe( ˙∂_¯a, δ_j) = ρe^d_j¯a∂˙_d; vTe(δ¯k,∂˙_a) = Σe^d_a¯k∂˙_d; vTe( ˙∂_a, δ_j) =Pe_ja^d∂˙_d; and their conjugates.

After a straightforward computation we obtain the expressions for (12) e

τ_jⁱ_¯k=

3

Lⁱ_j¯k; Θe^d_jk¯ =δk¯N_j^d; Υeⁱ_j¯a =

3

C_j¯ⁱ_a; Qeⁱ_ja=

1

C_jaⁱ ; (13)

e χ^d_a¯b =

4

C_a^d_¯b;ρe^d_j¯a= ˙∂¯aN_j^d; Σe^d_b¯j =

4

L^d_b¯j; Pe_ja^d = ˙∂bN_j^d−L^2d_jb.

The curvature ofDe has twenty components in the form (see p. 44 of [7]):

Reⁱ_jkh =Ahk{δ_h

1

Lⁱ_jk +

1

L^m_jk

1

Lⁱ_mh}; (14)

Re^¯ı_¯jkh =Ahk{δ_h

3

L^¯ı_¯jk +

3

L^m_¯jk^¯

3

L^¯ı_mh¯ };

Reⁱ_j¯kh =δ_h

3

Lⁱ_j¯k−δ¯k 1

Lⁱ_jh+

3

L^m_j¯k 1

Lⁱ_mh−L³ⁱ_m¯k 1

L^m_jh+ (δ_hN¯k^¯e)

3

C_j¯ⁱ_e−(δ¯kN_h^e)

1

C_jeⁱ ; Ωe^d_akh =Ahk{δ_h

2

Lⁱ_ak +

2

L^e_ak

2

L^d_eh}; Ωe^d_¯akh^¯ =Ahk{δ_h

4

L^d_¯ak^¯ +

4

L^e_¯ak^¯

4

L^d_eh¯^¯}; Ωe^d_a¯kh =δ_h

4

L^d_a¯k−δ¯k 2

L^d_ah+

4

L^e_a¯k 2

L^d_eh−L^4d_e¯k 2

L^e_ah+ (δ_hN¯k^¯e)

4

C_a¯^d_e−(δ¯kN_h^e)

2

C_ae^d; Πeⁱ_jkc = ˙∂_c

1

Lⁱ_jk −δ_k

1

C_jcⁱ +

1

L^m_jk

1

C_mcⁱ −L¹ⁱ_mk

1

C_jcⁱ + ( ˙∂_cN_k^e)

1

C_ejⁱ ; Πe^¯_¯^ı_jkc = ˙∂_c

3

L^¯ı_¯jk −δ_k

3

C^¯_¯jc^ı +

3

L_¯^m_jk

3

C^¯_mc^ı −L^¯3ı_mk

3

C_¯jc^¯ı + ( ˙∂_cN_k^e)

3

C_eⁱ_¯j; Πeⁱ_jkc = ˙∂_c

3

Lⁱ_jk −δ_k

1

C_jcⁱ +

3

L^m_jk

1

C_mcⁱ −L³ⁱ_mk

1

C_jc^m+ ( ˙∂_cN_k^e)

1

C_jeⁱ ; Pe_akc^d = ˙∂_c

2

L^d_ak −δ_k

2

C_ac^d +

2

L^e_ak

2

C_ec^d −L^2d_ek

2

C_ac^e + ( ˙∂_cN_k^e)

2

C_ae^d; Pe_akc^d = ˙∂_c

4

L^d_ak −δ_k

4

C_ac^d +

4

L^e_ak

4

C_ec^d −

4

L^d_ek

4

C_ac^e + ( ˙∂_cN_k^e)

4

C_ae^d; Pe_akc^d = ˙∂_c

4

L^d_ak −δ_k

2

C_ac^d +

4

L^e_ak

2

C_ec^d −L^4d_ek

2

C_ac^e + ( ˙∂_cN_k^e)

4

C_ae^d; Θeⁱ_jbh =δ_h

3

C_jbⁱ −∂˙_b

1

Lⁱ_jh+

3

C_jb^m

1

Lⁱ_mh−C³_mbⁱ

1

L^m_jh−( ˙∂_bN_h^e)

1

C_jeⁱ ; Qe^d_abh =δh

4

C_ab^d −∂˙_b

2

L^d_ah+

4

C_ab^e

2

L^d_eh−C⁴_eb^d

2

L^e_ah−( ˙∂_bN_h^e)

2

C_je^d; Ξeⁱ_jbc =Acb{∂˙_c

1

C_jbⁱ +C¹_jb^m

1

C_mcⁱ }; Ξe^¯ı_¯jbc =Acb{∂˙_c

3

C_¯jb^¯ı +

3

C_¯jb^m¯

3

C_mc^¯ı_¯ }; Ξeⁱ_jbc = ˙∂_c

3

C_jbⁱ −∂˙_b

1

C_jcⁱ +

3

C_jb^m

1

C_mcⁱ −C³_mbⁱ

1

C_jc^m; Se_abc^d =Acb{∂˙_c

2

C_ab^d +

2

C_ab^e

2

C_ec^d}; Se_¯abc^d¯ =Acb{∂˙_c

4

C_¯ab^d¯ +

4

C_¯ab^e¯

4

C_¯ec^d¯}; Se_a^d¯bc = ˙∂_c

4

C_a^d¯b−∂˙¯b 2

C_ac^d +

4

C_a^e¯b 2

C_ec^d −C⁴_e^d¯b 2

C_ac^e ,

where Ahk means the difference between the terms in the brackets and the terms obtained by replacingk with h.

3. Pure Hermitian metric

The geometrical objects associated withGare generally complicated. Some simplifications appear for particular choices for g_jk¯ and h_a¯b. We studied in a previous paper, [9], the case δ_j^aδ^¯_¯k^bh_a¯b = _a(F¹2)g_j¯k, and G. Munteanu and N.

Aldea studied the case δ^a_jδ^¯b_¯kh_a¯b = g_j¯k, [7, 2]. Here we resume for a detailed analysis the following particular case of the Sasaki type metric:

(15) GH(z, η) =g_j¯k(z)dz^j⊗d¯z^k+h_a¯b(z)δη^a⊗δη¯^b.

The Chern-Finsler c.l.c. of the complex Finsler space (M, F) is reduced to

CF

DΓ = N_j^a=δ^a_iδ_b^lg^{mi ∂g¯} _∂z^lmj^¯η^b, Lⁱ_jk =g^{mi ∂g¯} _∂z^kj^m¯ ,0,0,0

. Thev−and ¯v−covariant derivatives coincides with the partial derivatives with respect to η^a and ¯η^a, respectively. By direct calculation we prove:

Proposition 2. The Levi-Civita connection of the metric (15) is given by the following non-zero coefficients

1

Lⁱ_jk = 1

2g^¯li(∂_kg_j¯l+∂_jg_k¯l);

(16)

2

L^c_ak = 1

2[h^dc¯(∂_kh_ad¯) + ˙∂_aN_k^c];

3

Lⁱ_j¯k=

1

Dⁱ_¯kj = 1

2g^¯li(∂k¯g_j¯l−∂¯lg_j¯k);

4

L^c_a¯k=

2

H¯ka^c = 1

2h^dc¯[∂¯kh_ad¯−( ˙∂d¯N¯k^¯e)h_a¯e];

2

F_jb^c = 1

2[h^dc¯(∂jh_ad¯)−∂˙aN_j^c];

1

M_¯abⁱ =

3

M_b¯ⁱ_a= 1

2g^¯li[( ˙∂_a¯N_¯l^d¯)h_bd¯−∂¯lh_b¯a], where ∂_k= _∂z^∂k.

The curvature of the d-c.l.c.De from (15) is reduced to Reⁱ_jkh =Ahk{∂h

1

Lⁱ_jk+

1

L^m_jk

1

Lⁱ_mh}; (17)

Re^¯ı_¯jkh =Ahk{∂_h

3

L^¯ı_¯jk+

3

L^m_¯jk^¯

3

L^¯ı_mh¯ }; Reⁱ_j¯kh =∂_h

3

Lⁱ_jk¯−∂¯k 1

Lⁱ_jh+

3

L^m_j¯k 1

Lⁱ_mh−L³ⁱ_m¯k 1

L^m_jh; Ωe^d_akh =Ahk{∂_h

2

Lⁱ_ak+

2

L^e_ak

2

L^d_eh}; Ωe^d_¯akh^¯ =Ahk{∂_h

4

L^d_¯ak^¯ +

4

L^¯e_¯ak

4

L^d_eh¯^¯ }; Ωe^d_a¯kh =∂_h

4

L^d_ak¯−∂¯k 2

L^d_ah+

4

L^e_a¯k 2

L^d_eh−L^4d_ek¯ 2

L^e_ah.

Let K be the curvature tensor field of the Levi-Civita connection ∇. We shall denote its components by the same letters as N. Aldea in [2], indexed with two types of indices with the understanding that different indices means