Connections on almost complex Finsler manifolds and Kobayashi hyperbolicity

J. Korean Math. Soc. 44 (2007), No. 1, pp. 237–247

CONNECTIONS ON ALMOST COMPLEX FINSLER MANIFOLDS AND KOBAYASHI HYPERBOLICITY

Dae Yeon Won and Nany Lee

Reprinted from the

Journal of the Korean Mathematical Society Vol. 44, No. 1, January 2007

c

°2007 The Korean Mathematical Society

CONNECTIONS ON ALMOST COMPLEX FINSLER MANIFOLDS AND KOBAYASHI HYPERBOLICITY

Dae Yeon Won and Nany Lee

Abstract. In this paper, we establish a necessary condition in terms of curvature for the Kobayashi hyperbolicity of a class of almost complex Finsler manifolds. For an almost complex Finsler manifold with the con- dition (R), so-called Rizza manifold, we show that there exists a unique connection compatible with the metric and the almost complex structure which has the horizontal torsion in a special form. With this connection, we define a holomorphic sectional curvature. Then we show that this holomorphic sectional curvature of an almost complex submanifold is not greater than that of the ambient manifold. This fact, in turn, implies that a Rizza manifold is hyperbolic if its holomorphic sectional curvature is bounded above by −1.

1. Introduction

It is known that if a complex manifold admits a Hermitian metric with holomorphic sectional curvature bounded above by −1 , it is hyperbolic in the sense that its Kobayashi pseudo-distance is a distance (see, e.g., [5]). In [6], S.

Kobayashi extended his concept of hyperbolicity to almost complex manifolds and obtained the hyperbolicity criterion for almost Hermitian manifolds. In [9], authors studied various connections on almost complex Finsler manifolds in pursuit of application to global properties of the manifolds. Here we fur- ther explore the usage of such connections and extend Kobayashi’s results to so-called Rizza manifolds, almost complex Finsler manifolds with the compat- ibility condition (R) on the Finsler metric and the almost complex structure.

In §2, we set up the notations for Rizza manifolds and cook up a unitary frame bundle. For a Rizza manifold (M, J, L), we consider the pull-back bundle

˜

π : p ^∗ T M → g T M of the tangent bundle π : T M → M by the projection p : g T M → M. Here g T M = T M \ {zero section of π : T M → M }. Let ˆ π : FM → g T M be a Finsler bundle: an associated frame bundle of ˜ π : p ^∗ T M →

Received December 14, 2005.

2000 Mathematics Subject Classification. Primary 53C60; Secondary 32Q45, 32Q60, 53B40.

Key words and phrases. Finsler metric, Rizza manifold, Kobayashi hyperbolicity, almost complex Finsler manifold.

The first author was partially supported by Duksung Women’s University, 2005.

c

°2007 The Korean Mathematical Society

237

T M . We define a generalized Finsler structure G g ij satisfying G ij = G pq J _i ^p J _j ^q from L. Then we construct a special subbundle FU(M ) of the Finsler bundle FM over g T M : FU(M ) is roughly the intersection of the complex frame bundle FC(M ) defined by the almost complex structure J and the orthogonal frame bundle FO(M ) defined by the generalized metric G ij . It turns out that FU(M ) → g T M is a principal bundle over g T M with the structure group U(n).

In §3, we cook up various connections and define the torsion and the curva- ture. A connection on FU(M ) induces a connection ∇ on ˜ π : p ^∗ T M → g T M satisfying ∇G = 0 and ∇J = 0. To choose the most natural connection among such connections, we impose an extra condition on the torsion of the connection

∇. In order to define the torsion, we introduce a linear connection on T ( g T M ) ^C . The complexification p ^∗ T M ^C of p ^∗ T M can be decomposed into p ^∗ T M ^1,0 and p ^∗ T M ^0,1 by the almost complex structure J. Now the connection ∇ on p ^∗ T M can be extended complex linearly to p ^∗ T M ^C . If we have a non-linear connec- tion, i.e., T ( g T M ) = H + V , then we have T ( g T M ) ^C = H ^C + V ^C . The connection

∇ on p ^∗ T M ^C produces a linear connection on T ( g T M ) ^C and so we can define a torsion of ∇ as that of a linear connection on T ( g T M ) ^C . And hence the torsion has horizontal and vertical components. We then prove a Theorem 3.1 on the existence of a special connection. This is a generalization of the canonical con- nection of a almost Hermitian manifold introduced by Kobayashi [6, 7]. With the canonical connection ∇ , we define the holomorphic sectional curvature K F (v) along v ∈ T x M as usual.

In §4, we prove one of the main ingredient: non-increasing property of the curvature for submanifolds. On an almost complex submanifold M ⁰ of a Rizza manifold (M, J, L) , we will produce a canonical connection and its curvature.

And then we show that the holomorphic sectional curvature of M ⁰ does not exceed that of M (see, e.g., [2] for the Hermitian case). This is crucial in proving the Theorem 5.2.

In §5, we do some analysis on the holomorphic map f : D(0, 1) → M.

The canonical connection defined in §3 enables us to do such computation.

As a consequence, we generalize a theorem of Kobayashi on the hyperbolicity criterion for almost Hermitian manifolds by Kobayashi [6] to Rizza manifolds:

Theorem 5.2 (Main Theorem). If (M, J, L) is a Rizza manifold whose holo- morphic sectional curvature K F is bounded above by −1, then M is hyperbolic.

Acknowledgement. S. Kobayashi pointed out to us the possibility of extend- ing the choice of connections of almost Hermitian manifolds to almost complex Finsler manifolds. Nany Lee wishes to express her deep gratitude to late S. S.

Chern for his generous support during her stay in Tianjin in August, 2004.

2. Preliminaries

Let M be a 2n-dimensional manifold with an almost complex structure J

and a Finsler metric L. We call the triple (M, J, L) a Rizza manifold if the

following condition, the Rizza condition, is satisfied:

(R) L(x, φ θ (y)) = L(x, y) for all x ∈ M , y ∈ T x M and θ ∈ R ,

where φ θ (y) = (cos θ)y + (sin θ)Jy. The Rizza condition guarantees that the tangent space at any point of a Rizza manifold is a complex Banach space.

Let (x ¹ , . . . , x ²ⁿ ) be a local coordinate system of M and (x ¹ , . . ., x ²ⁿ , y ¹ , . . ., y ²ⁿ ) be a local coordinate system of T M induced by (x ¹ , . . . , x ²ⁿ ).

Now consider the pull-back bundle ˜ π : p ^∗ T M → g T M of the tangent bun- dle π : T M → M by the projection p : g T M → M. Here g T M = T M \ {zero section of π : T M → M } is the slit tangent bundle.

p ^∗ T M −−−−→ T M ^{p ˜}

˜ π

  y

  y ^π T M g −−−−→

p M

Let g ij = ¹ _{2 ∂y} ^∂L

∂y

. If g ij satisfies g ij (x, y) = g pq (x, y)J _i ^p (x)J _j ^q (x) , then g ij

is a-priori a Riemannian metric(see [3]). This is why we are interested in the condition (R) for almost complex Finsler manifolds.

Now define G ij (x, y) = ¹ ₂ {g ij (x, y) + g pq (x, y)J _i ^p (x)J _j ^q (x)}. Then G ij is a generalized Finsler structure and satisfies G ij = G pq J _i ^p J _j ^q . This G ij defines a Riemannian structure G on ˜ π : p ^∗ T M → g T M by

G _(x,y) (U, V ) = G ij (x, y)u ⁱ v ^j for U = u ^{i ∂} _∂x

¯ ¯

(x,y) and V = v ^{i ∂} _∂x

¯ ¯

(x,y) . This G is an almost Hermitian structure on p ^∗ T M , i.e., G(U, V ) = G(JU, JV ).

Let ˆ π : FM → g T M be an associated frame bundle of ˜ π : p ^∗ T M → g T M as in [10]. Now consider a subbundle FU(M ) of FM defined in the following way.

Let R ²ⁿ be equipped with a Euclidean inner product and T x M ' ˜ π ⁻¹ (x, y) with the Hermitian structure G _(x,y) . u ∈ FM with ˆ π(u) = (x, y) is in FU(M ) if u , as a linear map u : R ²ⁿ → T x M , is orthogonal and satisfies J ◦ u = u ◦ J o . Here J o is the canonical complex structure of R ²ⁿ .

Let {² 1 , . . . , ² 2n } be the standard basis of R ²ⁿ and let {e 1 , . . ., e n , Je 1 , . . ., Je n } be a local orthonormal frame field of p ^∗ T M → g T M over an open set O ⊆ g T M . To each u ∈ FU(M ) , we assign an element (x, y, α) ∈ O × U(n) , where ˆ π(u) = (x, y) and α = (α ^j _i ) defined in the following way. If we put u(² a ) = A ^b _a e b + B _a ^b Je b for a = 1, 2, . . . , n, then

u(² n+a ) = u ◦ J o (² a ) = J ◦ u(² a ) = −B _a ^b e b + A ^b _a Je b . Furthermore, by the orthogonality of u , we have

δ _a ^c = G(u(² a ), u(² c )) = (AA ^t + BB ^t ) ^c _a ,

0 = G(u(² a ), u(² c )) = (BA ^t − AB ^t ) ^c _a ,

where A = (A ^b _a ) and B = (B _a ^b ). Now define α = A + √

−1B.

Then

α¯ α ^t = AA ^t + BB ^t + √

−1(BA ^t − AB ^t ) = Id , i.e.,

µ A B

−B A

¶

∈ O(2n). Thus α ∈ U(n). And the map u 7→ (x, y, α) gives a local trivialization of FU(M ). In summary, we have

Proposition 2.1. FU(M ) → g T M is a principal bundle over g T M with the structure group U(n).

3. Connections on the unitary frame bundles

The principal bundle FU(M ) in Proposition 2.1 admits a connection. Its connection form ˜ ω is a u(n) -valued 1-form on FU(M ). The connection form

˜

ω induces an affine connection ∇ on p ^∗ T M → g T M making both G and J parallel: ∇G = 0 and ∇J = 0.

Now the connection form for ∇ is a locally defined u(n)-valued 1-form ω on T M . Fix an orthonormal frame field σ(x, y) = g ©

e 1 , . . . , e n , Je 1 , . . . , Je n

ª on p ^∗ T M → g T M . This σ is a local section of FU(M ) → g T M and let ω = σ ^∗ ω. ˜ Then ω is a u(n) -valued 1-form on g T M of the form

(ω _a ^b ) =

µ P Q

−Q P

¶ .

We then have a covariant derivative on p ^∗ T M → g T M defined by

∇e a = P _a ^b e b + Q ^b _a Je b ,

∇Je a = −Q ^b _a e b + P _a ^b Je b .

Note that the complexification (p ^∗ T M ) ^C of p ^∗ T M can be decomposed into p ^∗ T M ^1,0 and p ^∗ T M ^0,1 by the almost complex structure J , where p ^∗ T M ^1,0 = {X − √

−1JX | X ∈ p ^∗ T M } and p ^∗ T M ^0,1 = {X + √

−1JX | X ∈ p ^∗ T M }.

Now the connection ∇ on p ^∗ T M can be extended complex linearly to p ^∗ T M ^C . In terms of the local basis {e a = ¹ ₂ (e a − √

−1Je a )} ⁿ _a=1 for the complex vector space of type (1, 0) and {¯e a = ¹ ₂ (e a + √

−1Je a )} ⁿ _a=1 for that of type (0, 1) , we have

∇e a = ¹ ₂ ∇(e a − √

−1Je a ) = ψ _a ^b e b and ∇¯e a = ∇e a = ¯ ψ ^b _a ¯e b , where ψ _a ^b = P _a ^b + √

−1Q ^b _a .

Next we consider a non-linear connection of T g T M . Let N _i ^j be a fixed non- linear connection of T g T M . Then T g T M = H ⊕ V , where H is the horizontal subspace of T g T M with the basis

n δ

δx

= _∂x ^∂

− N _i ^j _∂y ^∂

o _n

i=1 and V is the vertical subspace of T g T M with the basis n

∂

∂y

o _n

1=1 . With the identifications χ ^H ( _∂x ^∂

) =

δ

δx

from p ^∗ T M onto H and χ ^V ( _∂x ^∂

) = _∂y ^∂

from p ^∗ T M onto V , we have the

almost complex structure J on H and on V in such a way that J ◦χ ^H = χ ^H ◦J and J ◦ χ ^V = χ ^V ◦ J.

Consider the complex tangent space H ^1,0 of type (1, 0) of H ^C with the basis

© e ^H _a = ¹ ₂ ¡

χ ^H (e a ) − √

−1Jχ ^H (e a ) ¢ª _n

a=1 and H ^0,1 of type (0, 1) with the basis

© e ¯ ^H _a = ¹ ₂ ¡

χ ^H (e a ) + √

−1Jχ ^H (e a ) ¢ª _n

a=1 . Similarly V ^C = V ^1,0 ⊕V ^0,1 , where V ^1,0 is spanned by ©

e ^V _a = ¹ ₂ ¡

χ ^V (e a ) − √

−1Jχ ^V (e a ) ¢ª _n

a=1 and V ^0,1 is spanned by

© e ¯ ^V _a = ¹ ₂ ¡

χ ^V (e a ) + √

−1Jχ ^V (e a ) ¢ª n

a=1 . Then we define a Hermitian metric on T g T M ^C so that ©

e ^H _a , ¯ e ^H _a , e ^V _a , ¯ e ^V _a ª

is orthonormal and we define the induced linear connection on g T M by

∇e ^H _a = ψ ^b _a e ^H _b , ∇¯ e ^H _a = ¯ ψ ^b _a ¯ e ^H _b ,

∇e ^V _a = ψ ^b _a e ^V _b , ∇¯ e ^V _a = ¯ ψ ^b _a ¯ e ^V _b .

We now define the torsion of the connection ∇ on p ^∗ T M → g T M as that of the induced linear connection on g T M . Let ©

θ ^a , ¯ θ ^a , φ ^a , ¯ φ ^a ª

be the coframe field dual to ©

e ^H _a , ¯ e ^H _a , e ^V _a , ¯ e ^V _a ª

. For the canonical 1-form η = θ ^a ⊗ e ^H _a + φ ^a ⊗ e ^V _a on T M , define the torsion Dη of the connection ∇ by g

Dη =dθ ^a ⊗ e ^H _a + dφ ^a ⊗ e ^V _a − θ ^a ∧ ψ _a ^b e ^H _b − φ ^a ∧ ψ _a ^b e ^V _b

=Θ ^a ⊗ e ^H _a + Φ ^a ⊗ e ^V _a ,

where Θ ^a = dθ ^a + ψ ^a _b ∧ θ ^b and Φ ^a = dφ ^a + ψ ^a _b ∧ φ ^b .

Definition 3.1. Θ ^a = dθ ^a + ψ _b ^a ∧ θ ^b is called the horizontal torsion of the connection ∇ and Φ ^a = dφ ^a + ψ _b ^a ∧ φ ^b is called the vertical torsion of the connection ∇.

Next we define the holomorphic sectional curvature of the connection ∇. Let the curvature of the connection be

Ω ^a _b = dω _b ^a + ω ^a _c ∧ ω _b ^c and let a (1, 1)-component Ω ^a _b ^1,1 of Ω ^a _b be

R _{b c ¯} ^a _d θ ^c ∧ ¯ θ ^d + R _{b c: ¯} ^a _d θ ^c ∧ ¯ φ ^d + R _{b ¯} ^a _c:d θ ¯ ^c ∧ φ ^d + R _{b :c ¯} ^a _d φ ^c ∧ ¯ φ ^d .

Definition 3.2. The holomorphic sectional curvature K F (v) of L along v ∈ T x M is defined by

K F (v) = 2

G(v) ² hΩ(χ, ¯ χ)χ, χi v

= 2

G(v) ² G σ¯ a R _{b c ¯} ^σ _d v ^c ¯ v ^d v ^b v ¯ ^a = 2

G(v) ² R _{b¯ ac ¯ d} v ^c ¯ v ^d v ^b v ¯ ^a ,

where χ : g T M → H ^1,0 is the horizontal radial section defined by χ(v) = v ^a e ^H _a | v

for v ∈ g T M .

Finally we are ready to define a special connection on a Rizza manifold (M, J, L). We call this uniquely defined connection the canonical connection of the Rizza manifold (M, J, L).

Theorem 3.1. For a Rizza manifold (M, J, L), there exists a unique connection

∇ on p ^∗ T M → g T M such that ∇G = 0, ∇J = 0 and Θ ^a (H, ¯ H) = 0 and Θ ^a (H, ¯ V) = 0, where Θ ^a is the horizontal torsion of ∇.

Proof. Let ψ o be the connection form on p ^∗ T M induced by a connection on FU(M ). Then ψ o is a u(n) -valued 1-form. And hence ∇G = 0 and ∇J = 0 , where ∇ is the connection on p ^∗ T M whose connection form is ψ o . Let the (1, 1) -component of the horizontal torsion Θ ^a _o be

(Θ ^a _o ) ^1,1 = A ^a _{b¯ c} θ ^b ∧ ¯ θ ^c + B ^a _{b¯ c} θ ^b ∧ ¯ φ ^c + C ¯ bc ^a θ ¯ ^b ∧ φ ^c + D ^a _{b¯ c} φ ^b ∧ ¯ φ ^c . Now define a new u(n)-valued connection ψ by

ψ _b ^a = ψ o a

b + A ^a _{b¯ c} θ ¯ ^c + B _b¯ ^a _c φ ¯ ^c − ¯ A ^b _{a¯ c} θ ^c − ¯ B _a¯ ^b _c φ ^c .

Clearly G and J are parallel with respect to the new connection ψ. The (1, 1)- component of its horizontal torsion Θ ^a is

(Θ ^a ) ^1,1 = (Θ ^a _o ) ^1,1 − A ^a _{b¯ c} θ ^b ∧ ¯ θ ^c − B _b¯ ^a _c θ ^b ∧ ¯ φ ^c

= C _b¯ ^a _c θ ¯ ^b ∧ φ ^c + D ^a _{b¯ c} φ ^b ∧ ¯ φ ^c .

Then Θ ^a (H, ¯ H) = 0 and Θ ^a (H, ¯ V) = 0. Thus ψ is the desired connection.

Now for the uniqueness of such connections, assume that there exist two such connections ψ 1 and ψ 2 with the corresponding horizontal torsions Θ 1 and Θ ₂ respectively. Since ψ ₁ and ψ ₂ are skew-Hermitian, we may put

ψ 1 a b − ψ 2 a

b = E _bc ^a θ ^c − ¯ E _ac ^b θ ¯ ^c + F _bc ^a φ ^c − ¯ F _ac ^b φ ¯ ^c . Then

(Θ 1 a − Θ 2 a ) ^1,1 = [(ψ 1 a b − ψ 2 a

b ) ∧ θ ^b ] ^1,1

= − ¯ E ^b _ac θ ¯ ^c ∧ θ ^b − ¯ F _ac ^b φ ¯ ^c ∧ θ ^b .

Since Θ ^a _i (H, ¯ H) = 0 and Θ ^a _i (H, ¯ V) = 0 for i = 1, 2, ¯ E ^b _ac = ¯ F _ac ^b = 0 which means ψ 1 a

b = ψ 2 a

b . ¤

Modifying the proof of Theorem 3.1, we further prescribe the horizontal component of the torsion:

Corollary 3.2. For a Rizza manifold (M, J, L), and given P _b¯ ^a _c and Q ^a _{b¯ c} there exists a unique connection ∇ on p ^∗ T M → g T M such that ∇G = 0, ∇J = 0 and

Θ ^a (e ^H _b , ¯ e ^H _c ) = P _b¯ ^a _c and Θ ^a (e ^H _b , ¯ e ^V _c ) = Q ^a _{b¯ c} ,

where Θ ^a is the horizontal torsion of ∇.

We have the following analogue of Theorem 3.1 by imposing the conditions on the vertical torsion.

Theorem 3.3. For a Rizza manifold (M, J, L), there exists a unique connection

∇ on p ^∗ T M → g T M such that ∇G = 0, ∇J = 0 and Φ ^a (V, ¯ H) = 0 and Φ ^a (V, ¯ V) = 0, where Φ ^a is the vertical torsion of ∇.

Corollary 3.4. For a Rizza manifold (M, J, L), and given P _b¯ ^a _c and Q ^a _{b¯ c} there exists a unique connection ∇ on p ^∗ T M → g T M such that ∇G = 0, ∇J = 0 and

Θ ^a (e ^V _b , ¯ e ^H _c ) = P _b¯ ^a _c and Θ ^a (e ^V _b , ¯ e ^V _c ) = Q ^a _{b¯ c} , where Θ ^a is the horizontal torsion of ∇.

Remark. For a complex Finsler manifold, the Chern-Finsler connection has a torsion whose horizontal (1, 1) torsion vanishes and whose vertical torsion has no ¯ θ ^a ∧ φ ^b and φ ^a ∧ ¯ φ ^b terms. So both connections in Theorem 3.1 and Theorem 3.3 are generalizations of the Chern-Finsler connection for complex Finsler manifolds. As we will see in §4, the connection in Theorem 3.1 is much more useful than the one in Theorem 3.3. This is why we call the connection in Theorem 3.1 the canonical connection.

4. Curvature of almost complex submanifolds of Rizza manifolds Let (M, J, L) be a Rizza manifold of dimension 2n and let M ⁰ be a 2m- dimensional almost complex submanifold of M, i.e., at each point of x ∈ M ⁰ , the tangent space T _x M ⁰ is invariant under J. A Rizza metric L on M induces a Rizza metric on M ⁰ . Here we show that the holomorphic sectional curvature of M ⁰ is not greater than that of M. This is a generalization of the well-known property in Hermitian geometry(see, e.g., [2]) and is essential in the proof of Lemma 5.1. Hereafter, we denote by R ⁰ _{b¯ ac ¯ d} , K ⁰ _F , etc., the corresponding quantities R _{b¯ ac ¯ d} , K F , on M ⁰ .

Let {e 1 , . . . , e n , Je 1 , . . . , Je n } be a local orthonormal frame field on p ^∗ T M such that {e 1 , . . . , e m , Je 1 , . . . , Je m } are tangent to M ⁰ . If {θ ^a , ¯ θ ^a } is dual to {e ^H _a , ¯ e ^H _a }, then θ ^m+1 = · · · = θ ⁿ = 0 on M ⁰ . Let (ψ _b ^a ) be the canonical connec- tion on M and Θ ^a = dθ ^a + ψ _b ^a ∧ θ ^b be its horizontal torsion.

If we restrict Θ ^r = dθ ^r + P _n

b=1 ψ ^r _b ∧ θ ^b to ] T M ⁰ for r = m + 1, . . . , n, then we have Θ ^r = P _m

b=1 ψ ^r _b ∧ θ ^b . Since Θ ^a (H, ¯ H) = 0 and Θ ^a (H, ¯ V) = 0, ψ _b ^r (1 ≤

b ≤ m, m + 1 ≤ r ≤ n) restricted to ] T M ⁰ are of type (1, 0). And hence we may

put ψ ^r _b = h ^r _bc θ ^c + h ^r _b:c φ ^c on ] T M ⁰ .

Note that ψ ⁰ = (ψ ^a _b ) a,b=1,...,m defines a canonical connection on M ⁰ . Let Ω ⁰ be the curvature of ψ ⁰ . Then

Ω ^0a _b = dψ ^0a _b + X m c=1

ψ ^0a _c ∧ ψ ^0c _b = Ω ^a _b − X n r=m+1

ψ _r ^a ∧ ψ _b ^r = Ω ^a _b − X n r=m+1

ψ _b ^r ∧ ¯ ψ ^r _a

= Ω ^a _b − X

h ^r _bc h ^r _ad θ ^c ∧ ¯ θ ^d − terms containing no θ ^c ∧ ¯ θ ^d and so R ⁰ _{b¯ ac ¯ d} = R _{b¯ ac ¯ d} − P _n

r=m+1 h ^r _bc ¯h ^r _ad . For a unit vector v = P _m

a=1 v ^a e a in T x M ⁰ , we have K ⁰ _F (v) = 2R ⁰ _{b¯ ac ¯ d} v ¯ ^a v ^b v ^c ¯ v ^d = K F (v) − 2

X n r=m+1

h ^r _bc ¯h ^r _ad ¯ v ^a v ^b v ^c v ¯ ^d

= K F (v) − 2 X n r=m+1

¯ ¯

¯ ¯

¯ ¯ X n b,c=1

h ^r _bc v ^b v ^c

¯ ¯

¯ ¯

¯ ¯

2

≤ K F (v).

In summary, we have

Theorem 4.1. Let M ⁰ be an almost complex submanifold of a Rizza manifold (M, J, L). Then

K _F ⁰ (v) ≤ K _F (v) for all v ∈ T _x M ⁰ . for the canonical connection on T M ⁰ .

5. Kobayashi Hyperbolicity of Rizza manifolds

In this section, we establish a necessary condition for the Kobayashi hyper- bolicity of Rizza manifolds. Recall that a mapping f : D(0, 1) → M from a unit disk D(0, 1) into an almost complex manifold M is called holomorphic if J ◦ f ∗ = f ∗ ◦ ˜ J , where ˜ J is the complex structure on D(0, 1). The concepts of Kobayashi pseudo-distance and hyperbolicity for complex manifolds can be extended to our setting as usual. For a description of intrinsic metrics and hyperbolicity in detail, we refer to [5].

Now we do some analysis on the holomorphic maps.

Lemma 5.1. Let w ∈ T p M , w 6= 0 . For a holomorphic map f : D(0, 1) → M with f ∗ (a _∂z ^∂ | 0 ) = w ,

L(w) √

2 ≤ |a| if K F ≤ −1.

Proof. Let S = {z ∈ D(0, 1) : f ∗ is singular at z} and D ⁰ = D(0, 1) \ S and

M ⁰ = f (D ⁰ ). Since f is holomorphic and S is discrete, M ⁰ is an almost complex

submanifold of M. Now we have a basis {f u , f v } of T _{f (z)} M ⁰ for z ∈ D ⁰ , where

f u = f ∗ ( _∂u ^∂ ) and f v = f ∗ ( _∂v ^∂ ). Note that f v = J ◦ f u . Let (u, v, y ¹ , y ² ) be a

local coordinate system of T M ⁰ given by x = f (u, v) and y = y ¹ f u + y ² f v .

For a local orthonormal frame field of p ^∗ T M ⁰ → ] T M ⁰ , consider e(x, y) = f u

p G (x,y) (f u , f u ) and Je(x, y) = f v

p G (x,y) (f u , f u ) .

Now instead of a rectangular coordinate system (y ¹ , y ² ) , let us use a polar coordinate system (r, θ). Then for y ∈ ] T M ⁰ , the Rizza condition (R)

L ² (x, y) = L ² (x, r cos θf u + r sin θf v ) = r ² L ² (x, f u ) implies that ^∂ _∂r

^L

= 0 and ^∂L _∂θ

= 0. Thus

∂

∂y ^k g ij (x, y) = ∂ ³

∂y ^k ∂y ⁱ ∂y ^j µ L ²

2

¶

= X

s+t=3

X 3 t=0

a s,t ∂ ³

∂r ^s ∂θ ^t µ L ²

2

¶

= 0 , i.e., g ij (x, y) is independent of y and so is G ij (x, y). Therefore

q

G _(x,y) (f u , f u ) = q

G _(x,f

₎ (f u , f u ) = L(f u ) is a function of x ∈ f (D(0, r)), say h(x). Then

e(x, y) = f u

h(x) and Je(x, y) = f v

h(x) . Following the notations in §2, we put

e = ¹ ₂ (e − √

−1Je) = _2h ¹ (f u − √

−1f v ) , e ^H = _2h ¹ ¡ _δ

δu − √

−1 _δv ^δ ¢

, e ^V = _2h ¹ ³

∂

∂y

− √

−1 _∂y ^∂

´ , θ = h(du + √

−1dv) , φ = h(δy ¹ + √

−1δy ² ).

Notice that the canonical connection ψ of M ⁰ is of the form αθ − ¯ α¯ θ + βφ − ¯ β ¯ φ.

And its horizontal torsion is

Θ = dθ + ψ ∧ θ = d(log h) ∧ θ + ψ ∧ θ.

Then, by direct computation, Θ(e ^H , ¯ e ^H ) = 0 implies α = d(log h)(¯ e ^H ) =

1

2h

(h u − √

−1h v ) and Θ(e ^H , ¯ e ^V ) = 0 implies β = 0.

Next its curvature is Ω = dα ∧ θ − d¯ α ∧ ¯ θ + α ∧ dθ − ¯ α ∧ d¯ θ. Since dα(¯ e ^H ) = d¯ α(e ^H ) = ∆ log h

4h ² − |∇h| ² 4h ⁴ , αdθ(e ^H , ¯ e ^H ) = −¯ αd¯ θ(e ^H , ¯ e ^H ) = − |∇h| ²

4h ⁴ , we have

Ω(e ^H , ¯ e ^H ) = − ∆ log h 2h ² .

Thus the holomorphic sectional curvature along e ∈ T x M ⁰ is K F (e) = 2R _{1¯ 11¯ 1} = − ∆ log h

h ² ,

which is ≤ −1 by Theorem 4.1.

Put 0 < r < 1. Let D(0, r) be the disk with center 0 and radius r. Con- sider g(z) = _ρ ^h(z)

r

(z) on D ⁰ ∩ D(0, r) , where ρ _r (z) = _r

−|z| ^2r

. Note that dσ ² = ρ r (z)dzd¯ z is the Poincar´e metric on D(0, r) and its (Gaussian) curvature is K = − ^{∆ log ρ} _ρ

r2

= −1. And g(z) is continuous and non-negative. Since h(z) is bounded above on D(0, r) and ρ r (z) → ∞ as |z| → r, lim |z|→r g(z) = 0. Thus g(z) attains its maximum at some τ 0 ∈ D ⁰ ∩ D(0, r). And hence

0 ≥ ∆ log g(τ 0 ) = ∆ log h − ∆ log ρ r

= −K F (e) · h ² (τ 0 ) − ρ r 2 (τ 0 )

≥ h ² (τ 0 ) − ρ r 2 (τ 0 ) , i.e., g(τ 0 ) = _ρ ^h(τ

⁾

r

(τ

) ≤ 1 which means ^h(0)·r ₂ = g(0) ≤ g(τ 0 ) =≤ 1 , for 0 < r < 1.

Letting r → 1 ⁻ , h(0) ≤ 2. Thus L(w) = |a|L ¡ ₁

2 (f u − √

−1f v ) ¢

= |a|h(0)

√ 2 ≤ √ 2|a|.

¤ Recall that the infinitesimal Kobayashi metric K of w ∈ T p M , w 6= 0 , is defined by K(w) = inf {|a|} , where the infimum is taken over the holomorphic maps f : D(0, 1) → M satisfying f (0) = p and f ∗

¡ a _∂z ^∂ | 0

¢ = w. In case, the Kobayashi pseudo-distance d K induced by K is a distance, we call M hyperbolic.

By Lemma 5.1, we have K F (w) ≥ L(w)

√ 2 > 0 for all 0 6= w ∈ T M , which means that d K (p, q) > 0 for p 6= q ∈ M.

In summary, we have the following theorem, which is a generalization of the theorem of Kobayashi [6] on the hyperbolicity criterion for almost Hermitian manifolds to Rizza manifolds.

Theorem 5.2 (Main Theorem). If (M, J, L) is a Rizza manifold whose holo- morphic sectional curvature K F is bounded above by −1, then M is hyperbolic.

References

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[3] Y. Ichijy¯ o, Finsler metrics on almost complex manifolds, Geometry Conference (Parma, 1988), Riv. Mat. Univ. Parma (4) 14 (1988), 1–28.

[4] , Kaehlerian Finsler manifolds, J. Math. Tokushima Univ. 28 (1994), 19–27.

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schaften, Springer-Verlag, Berlin, 1998.

[6] , Almost complex manifolds and hyperbolicity, Results Math. 40 (2001), no. 1-4, 246–256.

[7] , Natural connections in almost complex manifolds, Contem. Math. 332 (2003), 175–178.

[8] S. Krantz, Complex analysis: the geometric viewpoint, Carus Mathematical Mono- graphs, Mathematical Association of America, Washington, DC, 1990.

[9] N. Lee and D. Y. Won, Lichnerowicz connections in almost complex Finsler manifolds, Bull. Korean Math. Soc. 42 (2005), no. 2, 405–413.

[10] M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Japan, 1986.

[11] G. B. Rizza, Strutture di Finsler sulle variet` a quasi complesse, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 33 (1962), 271–275.

[12] , Strutture di Finsler di tipo quasi Hermitiano, Riv. Mat. Univ. Parma (2) 4 (1963), 83–106.

Dae Yeon Won

Department of Mathematics Duksung Women’s University Seoul 132-714, Korea

E-mail address: [email protected] Nany Lee

Department of Mathematics The University of Seoul Seoul 130-743, Korea

E-mail address: [email protected]