Geodesics on the indicarix of a complex Finsler manifold Gheorghe Munteanu Abstract ˜ is studied as a In this note the geometry of the indicatrix (I, L) hypersurface of a complex Finsler space (M, L). The induced ChernFinsler and Berwald connections are defined and studied. The induced Berwald connection coincides with the intrinsic Berwald connection of the indicatrix bundle. We consider a special projection of a geodesic curve from (M, L) complex Finsler space, called the induced complex geodesic, and a ˜ obtained by using the complex geodesic curve on the indicatrix (I, L) variational problem for their horizontal lift to TC (T 0 M ). Is determined the circumstances in which the induced geodesic coincides with the complex geodesic on the indicatrix. If (M, L) is a weakly K¨ahler Finsler space then one condition for these curves coincide is that the weakly K¨ ahler character conveys to the indicatrix via the induced Chern-Finsler connection.
M.S.C. 2000: 53B40. Key words: complex Finsler spaces, indicatrix, geodesics.
1
Preliminaries and settings
Let us begin our study with a short survey of complex Finsler geometry and set up the basic notions and terminology. For more, see [A-P, Mu]. Let M be a complex manifold, (z k ) 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 in its turn a complex manifold, the local coordinates in a chart will be denoted by (z k , η k ). 1
A complex Finsler space is a pair (M, F), where F : T 0 M → R+ is a continuous function satisfying the conditions: 0 M := T 0 M \{0}; ] i) L := F2 is smooth on T ii) F(z, η) ≥ 0, the equality holds if and only if η = 0; iii) F(z, λη) = |λ|F(z, η) for ∀λ ∈ C, condition; the homogeneity 2 iv) the Hermitian matrix gi¯j (z, η) , with gi¯j = ∂η∂i ∂Lη¯j the fundamental metric tensor, is positive definite. Equivalently, it mean that the indicatrix Iz = {η / gi¯j (z, η)η i η¯j = 1} is strongly pseudoconvex for any z ∈ M . Consequently, from iii) we have: ∂gi¯j k ∂gi¯j k ∂L k ∂L k η = η ¯ = L ; η = η¯ = 0 ∂η k ∂ η¯k ∂η k ∂ η¯k
(1.1)
and L = gi¯j η i η¯j . Roughly speaking, the geometry of a complex Finsler space consists in study of the geometric objects of complex manifold T 0 M endowed with a Hermitian metric structure defined by gi¯j . In this sense, the first step is the study of sections of the complexified tangent bundle of T 0 M which is decomposed into the sum TC (T 0 M ) = T 0 (T 0 M ) ⊕ T 00 (T 0 M ). Let V (T 0 M ) ⊂ T 0 (T 0 M ) be the vertical bundle, locally spanned by { ∂η∂ k } and let V (T 00 M ) be its conjugate. At this point the idea of complex nonlinear connection, briefly (c.n.c.), is one fundamental in ’linearizing’ of this geometry. A (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 ) = H(T 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.), which obey a certain rule of change at the charts changes such that δzδk = 0j ∂ ∂z 0j δ performs. Obviously, we also have that ∂η∂ k = ∂z . The pair ∂z k δz 0j ∂z k ∂η 0j ∂ δ ˙ {∂k := ∂ηk , δk := δzk } will be called the adapted frame of the (c.n.c.). By conjugation everywhere an adapted frame {∂˙k¯ , δk¯ } is obtain on Tu00 (T 0 M ). The dual adapted bases are {dz k , δη k } and {d¯ z k , δ η¯k }. Let us consider the Sasaki type lift of the metric tensor gi¯j , G = gi¯j dz i ⊗ d¯ z j + gi¯j δη i ⊗ δ η¯j .
(1.2)
Certainly, one main problem in this geometry is to determine a (c.n.c.) related only to the fundamental function of the complex Finsler space (M, L).
2
One almost classical now is the Chern-Finsler (c.n.c.) ([A-P, Mu]): CF ¯ Njk = g mk
∂glm¯ l η. ∂z j
(1.3)
The next step is to specify the action of a derivative law D on the sections of TC (T 0 M ). A Hermitian connection D, of (1, 0)−type, which satisfies in addition DJX Y = JDX Y, for all horizontal vectors X and J the natural complex structure of the manifold, is the so called Chern-Finsler linear connection, in brief C-F, which is locally given by the following set of coefficients (see the [Mu] notations): CF Lijk =
li
g δk (gjl ) ;
CF i = Cjk
CF
¯
CF
¯ı g li ∂˙k (gjl ) ; L¯¯ıjk = 0 ; C¯jk = 0,
(1.4)
i i ˙ ∂i , D∂˙k δj = Cjk δi etc. Of where Dδk δj = Lijk δi , Dδk ∂˙j = Lijk ∂˙j , D∂˙k ∂˙j = Cjk ¯ course, DX Y = DX¯ Y is performed. On TC (T 0 M ) the following 1−form is well defined:
ω = ω 0 + ω 00 := ηk dz k + η¯k d¯ zk ,
(1.5)
∂L i j i k where ηk := gk¯j η¯j = ∂η k . Also, from (1.1) it follows that Cjk η = Cjk η = 0. In [A-P], a complex Finsler space is said to be weakly K¨ahler iff gi¯l (Lijk − Likj )η k η¯l = 0, is K¨ahler iff Lijk η j = Likj η j and strongly K¨ahler iff Lijk = Likj . Further on, if an indices has a superscript bar then we understand that ¯ it refers to the conjugate object, i.e. η¯j := η j .
2
The intrinsic geometry of complex indicatrix
The study of the indicatrix of a real Finsler space is one of interest, first, because it is a compact and strictly convex set surrounding the origin ([B-C-S], p. 84). It is the motive that, for instance, the indicatrix plays a special role in the definition of the volume on a Finsler space. The geometry of the indicatrix as a hypersurface of the total space is studied in [AZ], p. 147, and it is proved that it plays a special role in obtaining necessary and sufficient conditions for an isotropic Finsler manifold be of constant sectional curvature. 3
A smooth compact and connected manifold with the properties of a indicatrix lead R. Bryant ([Br]) to call such a manifold with generalized Finsler structure. In this last study the K¨ahler structure of a generalized Finsler space with constant positive flag curvature is taken into account. The unit tangent bundle of a Riemannian or Hermitian manifold is a more general position of an indicatrix bundle and here there are some interesting results, see [Bl, Na, Bo], etc. Let us consider dimC M = n + 1 and π : T 0 M → M be its holomorphic bundle, (z k , η k )k=1,n+1 complex coordinates on the manifold T 0 M. We consider Iz = {η / gi¯j (z, η)η i η¯j = 1} the indicatrix at z of a complex Finsler space (M, L) and πI : I → M the indicatrix bundle (or, cf. [Bl] p.142, the holomorphic spheric bundle), I = ∪z∈M Iz . I ⊂ T 0 M is a holomorphic subbundle and as a complex manifold I is a compact and strictly connected in 0 hypersurface of T 0 M. Let i : I → T 0 M be the inclusion map and i∗ : TC I → TC (T 0 M ) the extension of the tangent inclusion map to the complexified bundles. Further on we will study the geometry of the hypersurface I of T 0 M complex manifold. If we consider (˜ z k , θα )α=1,n a parametric representation of the indicatrix hypersurface, then we have the following local representation: z˜k = z k and η k = Bαk (z)θα , with rank(Bαk ) = n.
(2.1)
On T 0 I we have ∂ ∂ ∂ ∂ ∂ j = k + Bαk θα j ; = Bαk k , k α ∂ z˜ ∂z ∂η ∂θ ∂η j where Bαk =
j ∂Bα . ∂z k
(2.2)
The dual frames are connected by
k α dz k = d˜ z k and dη k = Bαj θ d˜ z j + Bαk dθα .
(2.3)
The vertical distribution V I, spanned by {∂˙α := ∂θ∂α }, is a subdistribution of V (T 0 M ). On the indicatrix I we have L(˜ z k , η k (θ)) = 1 and by derivation with respect to ∂˙α it results that: ∂gi¯j k i j B η η¯ + gi¯j Bαi η¯j = 0. ∂η k α ∂g
¯
In account of (1.1) homogeneity conditions ∂ηikj η i = 0 it follows that gi¯j Bαi η¯j = 0, which say that the Liouville vector N = η k ∂η∂ k is normal to 4
the vertical distribution V I spanned by the tangential vectors ∂˙α to the hypersurface I. Moreover, N is one unitary vector since ηk η k = 1. Let us consider the frame R = {∂˙α = Bαk ∂η∂ k , N = η k ∂η∂ k } along V I and R−1 = {Bkα , ηk } the inverse matrices of this frame, that is: Bkα Bβk = δβα ; Bkα η k = 0 ; Bαk ηk = 0 ; Bαk Bjα + η k ηj = δjk ; ηk η k = 1.
(2.4)
˜ z , θ) = L(z, η(θ)) of the complex Finsler The fundamental function L(˜ ¯ space defines a metric tensor on the indicatrix I, gαβ¯ = Bαj Bβk¯gj k¯ , where ¯
¯
¯
¯
Bβk¯ = Bβk . It is easy to check that g βα = g ji Biα B¯jβ is the inverse of gαβ¯ and ¯
¯
¯
¯
g ji = Bαi Bβj¯g βα + η i η j . Also on the indicatrix Iz from ηk η k = 1 it follows that ¯
θα θα = 1, where θα = gαβ¯θβ . ˜ α ∂jα ; ∂˙α = ∂α } On T 0 I we consider the local frame {δ˜k = ∂∂z˜k − N k ∂θ ∂θ k α α α j ˜ ˜ and its dual frame {d˜ z ; δθ = dθ + Nj d˜ z }, where Nkα will be called the ˜jα d˜ coefficients of the induced (c.n.c.) iff δθα = Bkα δη k , that is dθα + N zj = α k k j Bk (dη + Nj dz ), and therefore in view of (2.3) it satisfies ˜ α = B α (B k θβ + N k ). N j k βj j CF Njk
CF
(2.5) ¯
2˜
¯ ∂gγ β¯ γ θ . ∂ z˜j
L βα Let be the (1.3) Chern-Finsler (c.n.c.) and Njα = g βα ∂ z˜∂j ∂θ ¯ = g β Then we have:
CF
Proposition 2.1. The induced (c.n.c.) by the Chern-Finsler (c.n.c.) Njk CF
coincides with Njα . Proof. We make a straight computation by using (2.2) and (2.4): CF
¯
2˜
¯
¯
˜
¯
¯
2
L L ∂L βα ∂ βα k l γ Njα = g βα ∂ z˜∂j ∂θ (Bβk¯ ∂η Bβ¯( ∂z∂j ∂η ¯) ¯ = g ¯) = g ¯ + Bγj θ glk β k k ∂ z˜j ¯
¯
¯
2
¯
2
β ∂ L ∂ L ¯ k l γ mp ¯ k k l γ = g mp Bpα Bm Bpα (δm ¯) = g ¯) ¯ η )( ∂z j ∂η k¯ +Bγj θ glk ¯ Bβ¯ ( ∂z j ∂η k¯ +Bγj θ glk ¯ −ηm ¯
2
2
p γ ∂ L ¯ p k ∂ L l γ = Bpα (g mp − η p ηl Bγj θ ) ¯ + Bγj θ − η η ∂z j ∂η m ∂z j ∂η k¯ CF
CF
p γ ˜jα . = Bpα (Bγj θ + Njk ) =N We take in reductions from the last part the fact Bpα η p = 0 .
We note that {δ˜k } generally are not d−tensor field, i.e. they are not change like vectors on the manifold. Also i∗ (δ˜k ), which for convenience will be often identified with δ˜k on T 0 M , is written as: 5
˜ α Bαj ∂ j = δ˜k = ∂∂z˜k − N k ∂η (2.5), we have:
∂ ∂z k
j ˜ α Bαj ) ∂ j and, by using (2.4) and + (Bαk θα − N k ∂η
δ˜k = δk + Hk0 N and ∂˙k = Bkα ∂˙α − ηk N
(2.6)
j θα + Nkj )ηj . where Hk0 = (Bαk ˜ k = d˜ Further, let us consider the dual induced coframe dz z k and δθα = α α j dθ + Nj d˜ z . The induced frame and coframe on the whole TC I and the induced metric structure are obtain by conjugation everywhere:
˜ i ⊗ d¯ ˜z j + gαβ¯δθα ⊗ δ θ¯β , ˜ = gi¯j dz G
(2.7)
where gi¯j (˜ z , η(θ)) is the metric tensor of the space along the points of the indicatrix. In [Mu1] we emphasize that another (c.n.c.) have a special meaning in study of the geodesics of a complex Finsler space, named the canonical c
CF
CF
CF
(c.n.c.), given by Nji := 21 ∂˙j ( Nki η k ) = 21 (Lijk η k + Nji ), which will play an essential role in construction of the complex Berwald connection and comes from a spray (see [Mu]). Let us study the same problem of its induced canonical (c.n.c.), c
c
CF
CF
˜ α = B α (B k θβ + N k ) = B α {B k θβ + 1 (Lk η i + N k )} N j j j ji k βj k βj 2 CF
CF
k ˜jα . = 12 Bkα (Bβj + Bβi Lkji )θβ + 12 N Indeed, an intrinsic canonical (c.n.c.) on T 0 I is uncalled for a definition because the problem of a complex spray is undesirable on T 0 I. Now, let us proceed to find the induced C-F linear connection. For this we consider the Gauss-Weingarten equations of the hypersurface I with respect to the C-F complex linear connection. Consider for any X ∈ Γ(TC I) and Y ∈ Γ(VC I) the decomposition:
˜ X Y + H(X, Y ) DX Y = D
(2.8)
˜ X Y ∈ Γ(VC I) is the tangential component and H(X, Y ) is the normal where D component. CF
CF
CF
CF
˜ ˜ ∂˙β =L ˜ α ∂˙α and D ˜ ˙ ∂˙β =C˜ α ∂˙α . Since D preserves the Let be D βk βγ δk ∂γ j ˙ ˙ distributions and V I is spanned by ∂α = Bα ∂j and V ⊥ I is generated by N, then we have 6
CF
CF
CF
CF
˜ ˜ ∂˙β = D ˜ B j ∂˙j = δ˜k (B j )∂˙j + B j D δ +H 0 N ∂˙j = {B i + B j Li D jk βk β β β β δk δk k k CF
i i +Bβj Hk0 η p Cjp } ∂˙i . But in view of the homogeneity of Finsler metrics η p Cjip = 0, thus we have: CF
CF
˜ α = B α (B i + B j Li ). L jk βk i βk β CF
CF
CF
(2.9) CF
α i ˜ α¯ =C˜ α = 0. Similar we find that C˜βγ = Biα Bβj Bγk Cjk and L β¯ γ βk CF
¯ = G(H(X, ∂˙α ), N) ¯ and For the normal component we have G( D X ∂˙α , N) ˜ furthermore if we take X to be a vector of the adapted frames δk , δ˜k¯ , we easily obtain that CF
j Hαk = Bαk ηj + Bαj Lijk ηi and Hαk¯ = 0. CF
i ηi If X is ∂˙β or ∂˙β¯, then the fundamental form will be Hαβ = Bαj Bβk Cjk and Hαβ¯ = 0. CF
CF
CF
CF
˜ ˜ δ˜j = D δ +H 0 N ˜ i δ˜i , then direct computation of D ˜ ˜ δ˜j =L Further, if D jk δk δk k k CF
(δj + Hj0 N), by CF because of D N CF Hi0 Lijk .
CF
˜ i =Li , using the homogeneity conditions, finally get that L jk jk δj = 0, and the normal part is Hkj = δk (Hj0 ) + Nk0 N(Hj0 ) −
It is obvious that even if the C-F connection is one normal, that is Dδk δj = Lijk δi , Dδk ∂˙j = Lijk ∂˙j , the induced connection is not one normal. Other property of interest which do not preserved by this projection is that of CF
CF
Berwald type connection, namely it is well known that Lijk = ∂˙j Nki , while for the induced connection we have: CF CF CF CF CF CF ˜ α +B α B k (Lj − Lj ), ˜ α = ∂˙β {B α (B j θγ + N j )} = B α (B j + B i Lj ) =L ∂˙β N k
j
γk
k CF ˜α L βk
j
βk
β
ik
βk
j
β
ki
ik
and hence it reduces to if the space is K¨ahler for instance. Now let us introduce another useful linear connection for our study, the complex Berwald connection. We prove in [Mu1] that it place an interesting role in characterization of totally geodesic curves on a holomorphic subspace. c
Let us consider Njk the canonical (c.n.c.) introduced above and we define the 7
complex Berwald connection as being c B
DΓ=
B (Nki , Lijk = c
B B ∂ Nki B¯ı i ¯ı , L = 0, C = 0, C ¯ ¯ jk jk jk = 0). ∂η j
(2.10)
We see that the Berwald connection is in vertical bundle V (T 0 M ) and it is easy to check that B CF CF CF c 1 CF 1 1 CF Nki = (Lik0 + Nki ) and Lijk = (Lijk + Likj ) + ∂˙j (Likm )η m . 2 2 2 B i By using again the G-W decomposition, since Cjk = 0, is obtain easy that B
B
B
B
B
˜ α¯ =C˜ α¯ = ˜ α = Biα (B i +B j Li ) and C˜ α =L the induced Berwald connection is L jk βk βk βk β βk βk B
˜i . 0. Of course, we can not speak about horizontal components L jk But the induced Berwald connection has the property that it is of Berwald type. Indeed, starting from: c
CF
CF
˜ α = 1 ∂˙β {Biα (B i θγ + Li η k )+ N ˜jα } ∂˙β N j γk jk 2 =
1 α i B {(Bβj 2 i
+
Bβk
CF Lijk
CF
CF
+∂˙β (Lijk )η k } +
1 2
˜α ∂˙β N j
CF
˜ α , after reducing the same terms, it and replacing the expression of ∂˙β N k result: CF c CF CF ˜ α = 1 B α {(B i + B k Li +B m ∂˙m (Li )η k } + 1 L ˜α . ∂˙β N j jk kj β βj β βj 2 i 2 On the other hand from (2.9), B
B
CF
CF
CF
˜ α = B α (B i + B k Li ) = B α {B i + 1 B k (Li + Li ) + 1 B k ∂˙j (Li )η m } L i i βj βj β jk βj jk kj km 2 β 2 β CF
CF
CF
i = 12 Biα {(Bβj + Bβk Likj +Bβm ∂˙m (Lijk )η k } + Comparing the last lines, it deduces
1 2
˜α . L βk
Proposition 2.2. The induced Berwald connection coincides with the intrinsic Berwald connection of the indicatrix bundle, that is: c B ˜α = L βj
˜jα ∂N . ∂θβ
8
(2.11)
Similarly, following the general settings from the geometry of hypersurfaces ([Mu1]), a linear connection D induces a normal connection D⊥ and let AN X := A(X) ∈ V I be the shape operator. Then we have: ⊥ DX N = −AN X + DX N.
From G(DX N, ∂˙k¯ ) = −G(A(X), ∂˙k¯ ), where X is taken to be one of the vectors δ˜k , δ˜k¯ , ∂˙β or ∂˙β¯, is obtain that: Aαk = Blα (Nkl − η j Lljk ) ; Aαk¯ = 0 ; Aαβ = −Bkα and Aαβ¯ = 0. Let us remark that both C-F and canonical (c.n.c.) are homogeneous and hence Nkl = η j Lljk . Consequently for the C-F or Berwald connections results Aαk = 0. We compute below the normal component of the Berwald connection. For c
X =δ˜k , we have c
B
D ˜c
δk
c
CF
CF
c
c
N ={δ˜k (η i )+ Nkl ηl η i }∂˙i + η j Lijk ∂˙i = (η j Lijk − Nki )∂˙i + Nkl c
ηl N = Nkl ηl N Thus we prove that c
B
c
AN δ˜k = 0 and D⊥c N = Nkl ηl N. δ˜k
c
c
B
For X =δ˜k¯ , is deduced that AN δ˜k¯ = 0 and D⊥c N = 0. δ˜k
Similar calculus get : B
B
AN ∂˙γ = −∂˙γ and D∂⊥˙ N = − Bγl ηl N ; AN ∂˙γ¯ = 0 and D∂⊥˙ N = 0. γ γ ¯ For the Berwald connection the torsion and curvature tensors are denoted by: c
c
c
c
vT (δ k , δ j ) = Ωijk ∂˙i where vT (δ k¯ , δ j ) = Θij k¯ ∂˙i
c
c
c
c
c
c
Ωijk =δj Nki − δk Nji (2.12)
Θij k¯ =δ k¯ Nji c
c
vT (∂˙k¯ , δ j ) = ρij k¯ ∂˙i
ρij k¯ = ∂˙k¯ Nji
and the nonzero curvatures: c
c
c
i Rjkh Xi = R(δ h , δ k )Xj c P i Xi = R(∂˙h , δ k )Xj
c
Rji kh ¯ )Xj ¯ Xi = R(δ h , δ k c P i¯ Xi = R(δ h , ∂˙k¯ )Xj
jkh
j kh
9
c
where Xi is first δ i and then ∂˙i . The calculus of these curvatures leads to c
B
c
B
B
B
B
B
i = δ h Lijk − δ k Lijh + Lljk Lilh − Lljh Lilk ; Rjkh B
c
(2.13)
B
B
i i i i ˙¯ Li . Rji kh ¯ Ljh ; Pjkh = ∂˙h Ljk ; Pj kh ¯ = − δk ¯ = −∂k jh
Proposition 2.3. The induced torsions and curvature with respect to Berwald connection are given by: ˜ α = B α Ωi ; Θ ˜ α¯ = B α Θi ¯ ; ρ˜α¯ = B k¯¯B α ρi ¯ . i) Ω i i jk jk jk jk jβ β i jk ii) B
c
c
B
c
B
c
B
˜ α = B α {B j Ri + δ h (B j ) Li − δ k (B j ) Li +B j (N 0 Li − N 0 Li )} ; R jk jh h lk k lh βkh i β jkh β β β ¯ j j ˜ α¯ = −Biα {B Ri ¯ + B ηk¯ η l P i¯ } ; R β j kh β β kh j lh B
B
α i i P˜βkγ = Bγh {Bβk + Bβj Lijk )(∂˙h Biα + ηh N(Biα ) + Biα Bβj Pjkh + ηh Biα Bβj Lijk ; α j i h i α ˙ α P˜βαkγ ¯ = −Bi Bβ Pj kh ¯ − Bγ Bβ k ¯ (∂h Bi + ηh N(Bi ).
With these computations the Gauss, respectively H−Codazzi equations ˜ are deduces from: of the indicatrix subspace (I, L) ˜ ˜ AH(X,˜vZ) Y − AH(Y,˜vZ) X, v˜U ˜ R(X, G (R(X, Y )vZ, v¯U ) = G( Y )˜ v Z, v˜U ) + G (2.14) and ˜ ((DX H)(Y, vZ) − (DY H)(X, vZ), W ) G (R(X, Y )vZ, W ) = G ˜ +G(H( T˜(X, Y ), Z), W ) (2.15) where W ∈ Γ(VC⊥ T 0 I) and v, v¯, v˜, v˜ are the projectors on the corresponding vertical distributions. Similarly, equating the components from VC T 0 I and VC T 0 I⊥ of the normal curvatures we obtain the following A−Codazzi, respectively Ricci equations: ˜ Y A)(W, X) − (D ˜ X A)(W, Y ), v˜Z − ˜ (D G (R(X, Y )W, v¯Z) = G ˜ AW (T (X, Y ), v˜Z G (2.16)
10
and ⊥ ˜ ˜ G (R(X, Y )W, v¯N ) = G R (X, Y )W, v˜N + ˜ H(Y, AW X) − H(X, AW Y ), v˜N G
(2.17)
for ∀X, Y ∈ Γ(TC T 0 I) and W, N ∈ Γ(VC T 0 I⊥ ). An expansive written for these formulas can be otained by replaceing the vectors with the adapted frames of the Berwald (c.n.c.).
3
˜ Complex geodesics on (I, L)
dz k k k Let us consider σ : t → z (t), η (t) = dt a complex geodesic curve on (M, L) complex Finsler space. According to [A-P], p. 101, it satisfies the system d2 z i CFi dz k + N (z(t), η(t)) = Θ∗i , i = 1, n + 1, (3.1) k dt2 dt CF
where Nki are the coefficients of the Chern-Finsler (c.n.c.), and ∗i
Θ =g
mi ¯
CF ¯ gj ¯l {Lln¯ m¯
−
CF ¯ j n Llm¯ ¯ . ¯ n }η η
(3.2) c
¯ In [Mu] we proved that an equivalent written for Θ∗i is Θ∗i = g mi δ i L. ∗i If Θ = 0 then the space is weakly K¨ahler, therefore we call Θ∗ = CF ¯
CF ¯
j ¯ z n ⊗ δi the weakly K¨ahler form. If in addition g mi gj¯l {Lln¯ m¯ − Llm¯ ¯ n }dz ∧ d¯ the torsion of the C-F linear connection satisfies a supplementary condition a special geodesic is obtained, named c−geodesic complex curve, which is the correspondent via a holomorphic map of a geodesic from the unit disc with the Poincar´e metric. We point out that it exists a sensitive difference between these geodesics. Taking into account the 1−homogeneity of the coefficients c
CF
of C-F (c.n.c.), we readily check that N0i = N0i . Thus, the equations (3.1) will be rewritten equivalently as c d2 z i dz k i + N (z(t), η(t)) = Θ∗i , i = 1, n + 1. k 2 dt dt
11
(3.3)
From
B Lijk
c
j k
η η =
∂ Nki j k η η ∂η j
c
=Nki η k = N0i , it results that (3.3) becomes
B d2 z i dz j dz k i = Θ∗i , i = 1, n + 1, + Ljk (z(t), η(t)) 2 dt dt dt
which implies that if the space is weakly K¨ahler, then a complex geodesic is an horizontal curve with respect to the Berwald connection. On the other hand, since η k = complex geodesic curve we have
dz k dt
c
c
and δ η i = dη i + Nki dz k , along a
c
dz i δ η i (t) = Θ∗i and η i (t) = . dt dt
(3.4)
Now, contracting in (3.4) with Biα and taking into account the request c
c
of induced (c.n.c.) δ θα = Biα δ η i , is obtain an induced curve σ ˜ : t → i α ˜ (˜ z (t), θ (t))on (I, L) space of the complex geodesic curve σ from (M, L), which satisfies the equations: c
δ θα (t) = Θ∗α , dt c
α
α
c
(3.5) c
i
¯ where δθdt(t) = dθdt + Niα dz and Θ∗α (t) = Biα Θ∗i (t) = Biα g mi δm¯ L along the dt points of σ ˜. Rb i ∂ the tangent vector to σ and l(σ) = F(z i (t), σ˙ i (t))dt Let be σ(t) ˙ = dz i dt ∂z a the length arc, t ∈ [a, b] . i zi Since η i = dz = d˜ and η i = Bαi θα along the induced curve σ ˜ , then dt dt . ∂ d˜ zi ∂ i α its tangent vector will be σ ˜ (t) = dt ∂ z˜i = Bα θ (t) ∂ z˜i and hence, using the homogeneity of the Finsler the length arc of the induced geodesic R b i function i ˜ curve will be l(˜ σ ) = a |Bα | F(˜ z (t), θα (t))dt. We note that from ηi η i = 1 it . follows that σ ˜ (t) is an unitary tangent vector to the indicatrix. ˜ Thus, any complex geodesic curve on (M, L) induces a curve on (I, L) i α with unit tangent vector. Conversely, let be σ ˜ : t → (˜ z (t), θ (t)) a curve α ˜ on (I, L), with θ θα = 1, that is the tangent vector ˜ is unitary. It can to σ dz i i α i be lifted to a curve σ : t → z (t), dt = Bα θ (t) on (M, L). Certainly, its induced curve on the indicatrix by the above procedure is just σ ˜. ˜ ˜ and it The frames {δi } are linear independent and span a distribution HI 0 is not diffeomorphic with T M because δ˜i are not d−tensor fields. We know
12
lh
that T 0 M is diffeomorphic with H(T 0 M ) via the horizontal lift ∂z∂ i → δzδ i . For an appropriate study with that made in [A-P] for the first variation, along the curve σ ˜ we consider the following lift of the tangent vector CF
CF
zi δ d˜ z i ∂ li ˜h d˜ δ → T = = Bαi θα (t) i σ ˜ (t) = i i dt ∂ z˜ dt δz δz .
CF
(3.6)
CF
along the curve σ ˜ . Of course since G( δ i , δ ¯j ) = gi¯j we deduce that k T˜h k= ˜ s a variation of the curve σ 1. Let as consider s ∈ (−ε, ε) and Σ ˜ with fixed ˜i ∂ dΣ i α ˜ points, (˜ z (t, s), θ (t, s)), and for the tangent vector U = ds ∂ z˜i consider their CF
˜i δ Σ . We assume here that the variation of σ ˜ lift to T 0 M denoted by U˜ h = dds δz i h is such k U˜ k= 1, without being a strong restriction. By this, the same construction from Theorem 2.4.1 from [A-P], for the ˜ space iff first variation, get that σ ˜ is a complex geodesic of (I, L) CF
h ∗ h D T˜h +T˜h T˜ = Ξ (T˜ , T˜h )
(3.7)
˜ with respect to where Ξ∗ convey the weakly K¨ahler form on σ ˜ from (I, L), CF
CF
.
¯ ˜ ¯ln¯ m¯ − L ˜ ¯lm¯ σj σ induced C-F linear connection, that is Ξ∗i = g mi gj¯l {L ˜ n along ¯ n }˜ the curve σ ˜. The (3.7) equation of the geodesic say: ! . . CF . . . . . CF CF CF CF CF i i i j k j j ˜ σ ˜ ) δ i = Ξ∗i δ i and because ˜ δ ¯j (σ ˜ )+ L ˜ σ ˜ δ j (σ δi + σ jk ˜ σ d dt
=
CF . σ ˜j
δ dt
CF
δj+
CF . σ ˜j
δ dt
CF
δ ¯j , it follows that : CF . . d .i ˜ (σ ˜ )+ Lijk σ ˜j σ ˜ k = Ξ∗i dt
˜ space. is the equation of a geodesic of (I, L)
(3.8)
.
Now by taking into account that along the σ ˜ curve we have σ ˜ i = Bαi (t)θα (t) CF
CF
˜ i coincides with Li it result: and L jk jk CF d i α (Bα θ )+ Lijk Bβj Bγk θβ θγ = Ξ∗i dt
13
that is, dθα + Biα dt
d i β CFi (B θ + Ljk Bβj Bγk θβ θγ dt β
! = Ξ∗α ,
(3.9)
where Ξ∗α = Biα Ξ∗i . CF
CF
But C-F is a normal connection, i.e. Dδk δj =Lijk δi and Dδk ∂˙j =Lijk ∂˙j , for which we can consider its induced vertical connection (2.9), and hence for (3.9) is obtain dθα + Biα dt
CF d i β k β γ i ˜ αβk Bγk θβ θγ = Ξ∗α . (B )θ − Bβk Bγ θ θ + L dt β
We remark that Bβi depends of t only by means of z(t) and therefore dBβi dz k i = Bβk η k along the curve dz k dt d i i θβ η k (Bβi )θβ − Bβk Bγk θβ θγ = Bβk dt
dBβi dt
=
σ. Accordingly the last bracket became i − Bβk θβ η k = 0, and thus the equation of CF
˜ reduces to: dθα + L ˜ α B k θβ θγ = Ξ∗α , or else the complex geodesic on (I, L) γ βk dt written: zk dθα ˜CF α β d˜ + Lβk θ = Ξ∗α . (3.10) dt dt Further, let see the circumstances in which it coincides to the induces geodesic on (M, L) given by (3.5). CF
CF
CF
CF
c
c
˜ α =L ˜ α +Bjα B k (Lj − Lj ) and dθα =δ θα − N ˜ α d˜ We proved that ∂˙β N zk , k βk β k ki ik which replaced in (3.10), forasmuch c
δ θα + dt CF
CF (Ljki
CF ˜kα )θβ − ∂˙β (N
− c
˜α N k
CF Ljik )η i η k
!
= 0, is obtain
d˜ zk = Ξ∗α . dt
CF
CF
CF
˜ α )θβ = Biα B i θβ + B j Li θβ = Biα (B i θβ + N i ) =N ˜ α . Hence the But ∂˙β (N k βk jk βk k k β above equations become c
CF
c
zk δ θα ˜α − N ˜ α ) d˜ + (N = Ξ∗α . k k dt dt
14
CF
c
˜ α and N ˜ α are induced (c.n.c.) and hence each of them satisfies Now, N k k the (2.5) formula. By reducing the same terms, we have: c
CF c zk δ θα α i i d˜ + Bi (Nk − Nk ) = Ξ∗α . dt dt CF
c
˜ N i η k =N i η k (by using the Since along the geodesic curve on (I, L), k k zk homogeneitycondition in their definitions), with η k = d˜ , it result that the dt equations of complex geodesic curve σ ˜ reduces to c
δ θα = Ξ∗α . dt
(3.11)
In conclusion Theorem 3.1. The induced geodesic from (M, L) complex Finsler space co˜ , if incides with the defined complex geodesic on the indicatrix space (I, L) ∗ ∗ and only if the induced weakly K¨ahler form Θ coincides with Ξ . In particular, Colorallary 3.1. If (M, L) complex Finsler space with weakly K¨ahler Finsler ˜ indicametrics then the induced geodesics will be a complex geodesic for (I, L) trix space if and only if the weakly K¨ahler character conveys to the indicatrix via the induced Chern-Finsler induced connection.
References [A-P]
M. Abate, G. Patrizio, Finsler Metrics - A Global Approach, Lecture Notes in Math., 1591, Springer-Verlag, 1994.
[AZ]
H. Akbar-Zadeh, Initiation to global Finsler geometry, NorthHolland Math. Library, 68, Elsevier, 2006.
[Al]
N. Aldea, Contributions to the study of holomorphic curvature, PHD Thesis, Brasov, 2004.
[B-C-S] D. Bao, S.S. Chern, Z. Shen, An Introduction to Riemannian Finsler Geom., Graduate Texts in Math., 200, Springer-Verlag, 2000. 15
[Bl]
D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math, 203, Birkhauser, 2001.
[Bl1]
D.E. Blair, A complex geodesic flow, RIGA 2007, Bull. of Transilvania Univ., 14(49)s (2007), 29-42.
[Bo]
E. Boecks, A Case for curvature: the unit tangent bundle, in Complex, contact and symetric manifolds, Progress in Math. 234 (2005), Birkhauser, 15-25.
[Br]
R. Bryant, Some remarks on Finsler manifolds with constant flag curvature, Houston J. of Math., 28 (2002), 221-262.
[Ma]
M. Matsumoto, Fundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Saikawa, Otsu, 1986.
[Mu]
G. Munteanu, Complex Spaces in Finsler, Lagrange and Hamilton Geometries, Kluwer Acad. Publ., FTPH 141, 2004.
[Mu1]
G. Munteanu, Totally geodesics holomorphic subspaces, Nonlinear Analysis, Real World Appl. 8 (2007), no. 4, 1132–1143.
[Na]
P. T. Nagy, Geodesics on the tangent sphere bundle of a Riemannian manifold, Geom.Dedicata 7 (1978), 233-243.
[Sh]
Z. Shen, Lectures on Finsler Geometry, World Scientific Publishers, 2001.
[Ta]
L. Tamassy, Angle, area and isometries in Finsler spaces, Bull. Transilvania Univ. of Brasov, 1(50), (2008), Series III, 395-404.
Transilvania Univ., Faculty of Mathematics and Informatics Iuliu Maniu 50, Bra¸sov 500091, ROMANIA e-mail:
[email protected]
16
