FINSLER METRICS WITH PROPERTIES OF THE KOBAYASHI METRIC ON CONVEX DOMAINS

FINSLER METRICS WITH PROPERTIES OF THE KOBAYASHI METRIC ON CONVEX DOMAINS


Introduction
The Riemann mapping theorem says that all simply connected domains in C, different from C are biholomorphically equivalent .It is a well known fact that this theorem does not hold for domains in T' for n > 1, and the classification of bounded domains up to biholomorphism has been an important problem in several complex variables .One approach to understanding the structure of bounded domains is to study biholomorphically invariant métrics such as the Kobayashi or Carathéodory metrics [K]] [BD] [L3] [Pa] .In [L1] and [L2], Lempert showed that these metrics are extremely well-behaved in the special case when the domain is strictly linearly convex and has smooth boundary: In this case, the two metrics coincide, and the infinitesimal form FK of the Kobayashi metric falls into a special class of smooth Finsler metrics with constant holomorphic sectional curvature K = -4.Since the notion of a strictly linearly convex domain is not a biholomorphically invariant concept, it is natural to ask how far Lempert's results can be extended to a more general (biholomorphically invariant) complex manifolds .
One approach to this problem is to study FK from a more invariant point of view.The first step is to characterize the properties of an abstract Finsler metric F on an abstract complex manifold M' which are necessary for Lempert's results to hold.A second, and more difficult, step is to determine when the Kobayashi metric of a bounded domain in (U" has these properties .In [F], Faran analyzed the local structure of (complex) Finsler manifolds and obtained a set of local invariants by applying Cartan's method of equivalente .He proved that vanishing of certain local invariants forces F to coincide with the Kobayashi metric of the underlying manifold M provided that F is a complete metric with K = -4.However, from the complex process of constructing these local invariants it is not easy to see how these invariants naturally arise from the properties of Kobayashi metrics obtained from Lempert's work.Thus, one would like to formulate a somewhat more direct description of the local structure; that is intuitively more appealing .In this paper, we give such description from the point of view of the calculus of variations by examining the local properties of the Kobayashi metric on strictly linearly convex domains, and derive equivalent conditions to the vanishing of the Faran's invariants from a simple property of Kobayashi metric (Property 1.3) .
In order to describe the local structure of the Kobayashi metric, we give brief review of Lempert's work .We define the infinitesimal Kobayashi metric FK on a complex manifold M as follows : For each v E Tx M, x E M, let f be a holomorphic map from the unit disk 0 C C into M such that f(0) = x and f(0) = Af v for Af > 0. The magnitude FK (v)  of v with respect to the infinitesimal Kobayashi metric FK is defined to be the infimum of where the infimum is taken over all such f.If f actually attains the infimum (Le.FK(v) = ár ), then f is called extremall.It can be easily seen that the metric FK is invariant under the action of the group of biholomorphisms of M. Lempert showed that, if M = D C C V`is a bounded strictly linearly convex domain with smooth boundary, then FK is a smooth complex Finsler metric [L1], [L2], Le.FK is smooth outside the zero section of TD and satisfies the following conditions: (1 .1)FK(v) > 0 f'or v , : 0, FK(z .v)= Iz1FK(v) for z E C, and (1 .2) FK(vi + V2) .<FK(v1) + FK(v2) for vl, v2 E T.D, x E D, where equality in (1.2) holds only when vl and v2 are colinear .Moreover, he proved the following theorem: Theorem (Lempert) .Suppose that D is a bounded strictly linearly convex domain with smooth boundary .
(1) There is a unique extremal map corresponding to each v E TD .
(2) All the extremal maps are proper isometric imbeddings, and can be smoothly extended to the closed unit disk 0 .
(3) The extremal disks f(,~i) passing through a pointt x E D form a complex foliation of D -{x} .(4) Extremal disks are (the only) one-dimensional holomorphic retracts of D .
One ofthe key ideas in describing the geometry of D is the construction of the holomorphic retract of D onto the extremal disk f (A) .Lempert proved that the field of holomorphic tangent planes of áD on f(¿9A) can be holomorphically extended to the interior of the disk f (0), and defines a holomorphic field of complex hyperplanes on f (A) that are transversal to f (A) .In other words, there is a well defined (n -1)-dimensional holomorphic vector bundle p : E , f (A) over the extremal disk with fibers defined by the hyperplanes in T'.The union of the hyperplanes contains the domain D, and the holomorphic retract is defined by the restriction to D of the projection map p.
The existente of such holomorphic retracts has further implications.For example, it forces every (locally length minimizing, connected) geodesic curve of FK to be contained in an extremal disk.The properties of the Kobayashi metric that interests us are the following : Corollary .Let f : A -D be a extremal map for v E Tf D .
(1 .3)The extremal disk f (A) coincides with the union of geodesic curves through x tangent to a common complex line in TxM.(1 .4)There is a canonical splitting TD,f(o) = T(f (0)) ® E where E is an (n -1)-dimensional holomorphic subbundle of the restriction TD,f(o) of the tangent bundleTD to f (A) .
We wish to generalize the condition (1 .3) to an abstract complex Finsler metric F defined on an'n-dimensional complex manifold M. Let exp, denote the exponential map from a neighborhood of 0 E T, M ¡rito M defined by the geodesics of F. For each tangent vector v E T~M, the image exp (U,) of a small neighborhood U, of 0 in the complex line 0 v defines a surface in M. A reasonable generalization of the condition (1 .3) is the following: (1.5)For all v E TM, the surface exp (U,) is a complex curve (1- dimensional complex submanifold) in M.
M.-Y .PAlvc This condition was first introduced by Royden [R], and it is, in fact, equivalent to a condition given by Faran [F] .
The purpose of this paper is to study the local structure on of Finsler metrics satisfying condition (1 .5),and further, to show that many properties of the Kobayashi metric of convex domains extend to this more general class of complex manifolds .One of our major results is a construction of a biholomorphically invariant family of complex curves which enjoys many of the properties of extremal disks in convex domains.To see how such complex curves are constructed, recall, from the calculus of variations, that the metric F uniquely determines a vector field X on the cotangent space of M, called the geodesic vector feeld, such that the integral curves of X are mapped into geodesics of F by the projection map 7r : T*M > M. Let Z be the vector field on Tó M = {v E T*MIv =~0} generated by the circle action of unimodular complex numbers defined by multiplication on TO* M. Theorem A. The following conditions are equivalent (Theorem 4 .7): 1.The surface exp (U) C M is a complex curve for all v E TM .
3. The distribution D = e X ® (E Z C T(TO M) is involutive .
If any of the above conditions is satisfied then each complex curve exp (U ) extends uniquely to a maximal, totally geodesic, immersed complex curve E --~M (Theorem 4.9).
The sígnificance of the condition 2 and 3 in the theorem is as follows : The condition 2 provides a computational methods to check whether F satisfies the property (1 .5) .The condition 3 implies that, by Frobenius Theorem, the distribution D defines a 2-dimensional complex foliation .Fo of TóM.The complex curve E is constructed by projectiog each leaf of the foliation Fo by the projection map 7r onto M.
The curves E share many local properties in common with the extremal disks described in the Lempert's theorem.For example, by Theorem A, any real geodesic curve of F is contained in one of the curves E. Furthermore, a generalization of the property (3) in Lempert's theorem holds: the complex curves E passing through a point x form a complex foliation of some neighborhood of x.A less trivial result is the following generalization of property (1.4) : Theorem B. For each complex curve E, there is a canonical splitting TM¡£ = TE ®P-E, where T1 E is an (n -1)-dimensional holomorphic subbundle of TMI£ .(Theorem 4 .9.) The significante of the function r, in the Theorem A is its relation to the holomorphic sectional curvature of F. Note that each complex curve in M is naturally equipped with a Hermitian metric induced by F, and, therefore, has an associated Gaussian curvature .Following the definition by Wong and Royden [W] [R], we define the holomorphic sectional curvature of F at v by the Gaussian curvature of the curve exp (U ) .
Theorem C. If F is a complex Finsler metric satisfying the condition (1 .5)then the holomorphic sectional curvature of F is determined by the function Finally, using the result described above, wc show that, under the condition that F is complete a,nd r, = -4 the complex curves E coincide with the extremal disks.This result was proved earlier by Faran [F] .Note that, from Lempert's result, the Kobayashi metric FK on a strictly linearly convex domain D C e n has constant holomorphic sectional cur- vature -4.(This is a direct consequence of the fact that every extremal map f : A -> D is an isometry with respect to the Poincaré metric and FK such that f (0) is locally defined by exp (U ) .) 4.24 Theorem [F] .Suppose F is a complete complex Finsler metric on a complex manifold, M with constant, holomorphic sectional curvature -4 satisfying th,e property (1 .5).Then F = FK, v)hcre FK is the Kobayashi metric on M.
The paper is organized as follows : In Section 2, we develop basic tools and prove some basic facts about complex Finsler manifolds .Section 3 is an introduction of Legendre foliations and its application to complex Finsler manifolds .In Section 4, we prove the main theorem using the results of Sections 2 and 3 .
Throughout the paper, M denotes an n dimensional complex manifold and F a complex Finsler metric; on M. The f'ollowing notations are used: (1) The indices a, b and c rango from 1 through 2n, and o ., (5, y rango from 1 through 2n -1 .Summation conventions are in forte throughout. (2) (X I , .. ., xn, x n+1 . . . x2n) denote the real coordinates on M obtained from a holomorphic coordinates x, + ixn+ , for v = 1, . .., n. (4) For F E C°°(T*M), Fa, FaL, .. .denote 'g"-, 9zr " and so on.du°( 5) If V is a vector field on a manifold M, e' V : M -M denotes the 1-pararrleter family of dif eomorphisms generated by V. Thus, for Acknowledgments .I would like to thank T. Duchamp for his help and encouragement, and for introducing me to this subject .I also wish to express my thanks to J. Bland for conversations .
In this section, we prove some general facts about complex Finsler metrics .A complex Finsler metric on the cotangent bundle of M is a map F : T*M -R satisfying properties (1 .1)and (1.2) .When M is equipped with a complex Finsler metric, we will call M a complex Finsler manifold .
2 .1The Geodesic Vector Field and Complex Structure .In order to define the exponential map, we introduce the geodesic vector field on T*M .Recall that T*M is naturally equipped with a 1-form defined by the equation (2.2) and that the 2-form d( is a symplectic 2-form on T*M (Le. a smooth closed 2-form on T*M satisfying the non-degeneracy condition (do)2,, =0 ).The geodesic vector field X on T* M is uniquely determined by the condition In particular, tlle identity X F = 0 holds.In terms of coordinates, we Nave (2.4) cach x E M, t ~--~etvx is an integral curve of V starting at x (Le.e°vx = x and d I e tv x = Ve .) .

Complex Finsler Metrics
To describe how X is related to the complex Structure, consider coordinate expression of the complex Structure on T*M .The complex Structure J on M is expressed as -i) (2.5) J 5 = ,In' r~ where (Jb) _ (1 0 , (2 .6)and I is the (n x n)-identity matrix .The natural complex structure on T*M is then given by a b a _a __ ba IgXa It can be directly checked that the complex structure defined by this is independent of the choice of coordinates .By abuse of notation, we will denote this complex structure on T*M by J. From the dcfinition above, it is clear that J o 7r * = 7r* o J.
Note that condition (1 .1)provides a compatibility condition of F and the complex structure, which can be expressed as follows: Let Y be the radial vector field on Tó M generated by the, action of IR 1:>y rnultiplication of é, t E I1Z.It can be easily checked that Y and Z satisfy the relation Thus, if F satisfies the condition (1 .1),we have F(e`v) = F(v) a,nd F(e'v) = e'F(v), and therefore the following identities hold: (2 .8)ZF = 0, YF = F.
Note that the coordinate cxpressions of Y and Z are (2.9)Y = u° 09 , Z = -Ja U'' a'9 a .
In particular, JX and [X, JX] are tangent to the submanifold SFM C ToM .
Proof: : To prove the lemma, recall that we have the identities Cx( = X -id( = -F dF, X F = 0 and C(Z) = 0. Also, recall from identities (2.8) and (2.12) that Z F = 0 and JX = [X, Z] .Using these identities and the fact that Gx is a derivation, compute as follows: Using these identities again, we complete the proof of the lemma: 2.15 The Exponential Map .The exponential map is defined similarly as in the case of Hermitian manifolds .To define it, we introduce the dual complex Finsler metric F : TM , R, satisfying conditions (1.1) and (1.2), and define geodesics as curves with locally length minimizing property with respect to F. We briefly review some concepts of the calculus of variations .For more details about the calculus of variations see [GF] and [S] .
In coordinates, we have 7L (2.17 u2n) Rom (2.17) and conditions (1 .1)and (1 .2), it can be shown that ID is a dif eomorphism with the property that xP(tw) = txP(w) for t > 0. The inverso -D = T-1 is usually called the Legendre transformation.For example, if F is the norm induced from tfre Hermitian metric g, <D : TM -+ T*M defined by {4 >(v)}(w) = g(v, ui) for v, w E T.,M .We define F by F o 4 1) .It is clear from the following lemma that F is a complex Finsler metric.
Also note that from the definition of Lie derivative, c (e* cz x (l tz w » e* t7 (Gzx) (,tz w) .Using these identities arid the identity 7r o e -cG = 7r, we compute Therefore, y(t) = T(e tz v) satisfies ara ordinary differential equation y' --iy with initial conditiori y(0) _ xP(v).From the uniqueness of solution, we can conclude that xP(e ¿Z v) = e -,Z P(v) ; and frorn this, it follows that <D(eciv) = e-tid,(v) .
A curve y : [t1, t2] -M is called a geodesic if it is a critical curve for the furictional Jc (2 .19)/'c2 F (dt dt, y(tr), y(t2) fieed ., Let v E I;, M arrd yv : (-c, e) ---> M be the geodesic such that y (0) = x and ~c (0) = v.As in tlic case of Hermitian metrics, it can be shown that y is solution of a second order ordinary dif erential equation (Le. the Euler-Lagrange equation [GF]), and hence it is uniquely determiried .It can be also shown tliat there is ara opera neighborhood N C T,,M of 0 such that y (1) is defined for all v E N. Herice, the exponential map exp x : N -M at x is defined by exp (v) = y(1).From the uniqueness of solutioris of ordinary differential equations, ( 2.20) exp x (tv) =-Y (t) .
It is a standard fact from the Calculus of Variations that the pro,jection of the integral curves of X are geodesics on M. Hence, it follows that the curve ^y(t) _ 7r(e tx w) is a geodesic for all w E T.M such that -y( 0) = x and 1 (0) _ 7r .X , _ xP(w) .Therefore, from the identity (2 .20),we obt ain (2.21) exp (tP(w)))=-y(t) _ 7 r(etx w), or equivalently, exp (tv) = 7r (etx<D(v))   for v = T (w) .
2.22 Lemma .For all v E TOM the exponencial map can be expressed in terms of the geodesic vector field X as follows: (2 .23)exp (tes2 .v)= {7r o etx o e-sz} (`D(v)) for s E IR and, t, > 0 smallll .

Legendre Foliations
In che following, we give a brief introduction to Legendre foliations.Although the tlreory of Legendre foliations is not required in the statement of the rnain theorem (Theorem 4.2), it is necded in its proof.For more details about Legendre foliations, we refc-;r the reader to [P] .
Let P be a srnooth (2m + 1)-dimensional manifold without bouridary with a fixed choice of contact 1-form 17 (Le .17 A (drt)'n 7A 0) .A Legendre foliation is a foliation F of P by m-dimensional integral submanifolds of rl.Two Legendre foliations Fr and F2 are said to be equivalent if there is a dif eomorphism cW : Pr -P2 such that cW*r72 = 17, and cp* .F2 = .Fr where cp* .F2 is the foliation of PI whose leaves are inverse image of leaves in .F2 .
The relation between Legendre foliations and complex Finsler rnanif'olds can be seen from the following fact [P] : The unitt cotangent bundle SrM = {w E T*MIF(w) = 1} is equipped with a natural contact 1-form ri defined by the pulll back of C to S¡M and the Legendre foliation .F defined by ftbers of 7r :SrM1M .
It is shown in [P] that the structure of Legendre foliations defined in this way determines the metric F uniquely: 3 .1 Proposition .If Fr and .'F2are equivalent Legendre foliations on S* M, and , SFZ M2 where Fl and F2 are complex Finsler metrics on Mr and M2 respectively, then Mi and M2 are isometric as Finsler manifolds (i .e .there exists a diffeomorphism cp : Ml -1VI2 such that F,=F2o(p*) .
To describe the local structure of the Legendre foliation on SrM, let L be the tangent bundle to F. One of the basic local invariants of the Legendre foliation is defined by the restriction to L of the symmetric form It is shown in [P] that, if the triangle inequality (1 .2) holds, then 11 is positive definite, and that there is a canonical reduction of the structure group to O(2n -1).Note that, from their definitions, thc; restrictions of vector fields X and Z on SrM are tangent to S¡M, and hence they are invariantly defined vector fields on S* M. If (Z,,) is a local orthonormal frame of L with respect to II, there is a local coframe (B°, r7, ~Q) on SrM such that (ii) IZa~j, y = -F'7fj and Irá = -Irá .Moreover, F is the induced norm of a Hermitian metric if and only if Gap.y = Q%q_í = 0.In this case, the tensors Itap.y and Si are related to the curvature tensor of the Hermitian metric.This can be easily stated in lower dirnensional case: 3 .5Proposition .Let M be a complex curve (i .e .n = 1) .Then F is a, norm induced from a Hermitian metric with Gaussian curvature d9°ir!0 0 00 0 0 0 77 dl;°0 0 7rp (3.4) -77 A ~a + Ga p y 0,9 l~~7 r, if and only if the Legendre foliation on S,',M satisfies the structure equations: (3 .6)dO l =-rlntjl , drl=O'nt 1 , da r =K97n0 .

The Main Theorem
4.1 Equivalent Conditions .In this section, we state the necessary and suffrcient conditions for a complex Finsler manifold to satisfy the property (1 .5) .For each v E TL M, let U be a neiglzborhood of 0 in the complex liase C -v C TM spanned by v, on which the exporrential map exp is defined .The image exp (Us ) defines a surface in M asear x.
4.2 Theorem .The following conditions are equivalent.
Al. exp (U ) is a complex curve for a,ll v E TM with F(v) = 1 .A2. [X, JX] = KZ on S;.M for some smooth function K .A3 .Th,e distribution D spanned by X, JX and Z is an involutive dis- tribution ore S¡ M.
Proof. .We will prove the theorem by showing (i) the equivalente of conditions Al and A2, and (ii) the equivalente of the conditions A2 and A3.Thus, if we denote I', for the surface defined by (t, s) H C , (t, s) = 7r(e tx e -S Z w) for w E S*,M and t > 0 small, condition A1 is equivalent to the condition that I', be a complex curve for all w E S* M. Suppose that the condition A1 holds.Then the surface I', is a complex curve for all w E S¡M .To show that the condition A2 holds, we show [X, JX] = rZ at vw = e' x u) E SUM for sufficiently small e > 0 and v) chosen arbitrarily in S¡M .Let {I'T} denote a family of surfaces parametrized by T defined by the function (t,, s) ~-1 C, (t, s) -7r(e~~-T)XesZCTX?v) .
Let W(-r) be the tangent vector at 7r(»» defirred by But from Lernma 3.8, [X, JX] does not have a component in the direction of X and Xl, and thus [X, JX] , E L, .Again, from Lernma 3.8, it follows that Sri (X) = 0, and we obtain (4.3) JX = XII and [X, JX] = Si Z< ,.
But, Again, since we proved (Ex [X, JX]), E RX, (D 1FI(Xl), ® L,D, components of the vector Ex [X, JX] in the direction of Xa for a =~0 has to vanish.Therefore, it follows that Si = 0 for a > 1 and [X, JX] _ Si Z, and the condition A2 follows .
(ü) If [X, JX] = rZ holds for some rc E C' (S* M), then the involutivity of D follows from the identities The converse of this is an immediate conséquence of the the identity (3 .10) .
(1) Note that, in the proof of Theorem 4.2, the two equations Sri (X) _ 0 and SQ = 0 for /3 > 1 are equivalent to the single condition [X, JX] = rZ with rc = Si .The condition Sri (X) = 0 can be interpreted as a compatibility condition for F and the complex structure .For example, if F is the induced norm of a Kaehler manifold, this condition is satisfied .In this case, the condition Sp = 0 for ,(j > 1 puts restrictions on the curvature of the Kaehler metric.One special case of this is when M is a Kaehler manifold with constant holomorphic sectional curvature .In this case, it can be verified using results in [P] that Sá = cb í 3 for some constant e. (2) Conditions equivalent to Al-A3 of Theorem 4.2 were introduced by Royden [R] and Faran [F] .
The conditions Al-A3 in Theorem 4 .2can be equivalently stated as conditions on Tó M. 4.7 Theorem .The following conditions are equivalent : B1 .exp(U ) is a complex curve for all v E TOM .132 .[X, JX] = rZ on Tó M for some te E C°°(TO M) .133 .The distribution CX ® 0Z C T(TO M) is involutive .
Moreover, these conditions are equivalent to conditions Al-A3 in Theorem 4 .2 .
Proof: Note that the conditions Al and Bl are clearly equivalent .To prove the theorem, we show (i) that A2 implicas B2, and (ii) that A3 implies B3.The converses of these are trivial to prove.
(i) Suppose that [X, JX] =rc Z holds on SFM for some rc E C°°(SrM).Extend K to Tó M by rc(tw) = tz K(W) for all t > 0 and w E SI*M .
We claim that [X, JX] = rc Z on Tó M. To show this, we show that both W = [X, JX] and W = n Z on Tá M must satisfy the ordinary differential equation GyW = 2W .Note that the integral curves of Y are the radial lirios in Tó M .Thus, if both [X, .IX] and rc Z satisfy the equation, they rnust coincide because the identity [X, JX] = r Z gives the same initial condition at points on SFM.To show that the vector field [X, JX] satisfies the differential equation, recall, frorrr Lemma 2.11, that Gy X = X and GyJX = JX.Using these identities and the fact that Gy is a, derivatiorr, we compute To show that the vector field r, Z satisfies the differential equation, recall that the vector field Y on TóM is generated by the action of II3, by multiplication of e'.Using homogeneity of K, we obtain the following identity : For w E TO M, (ii) The; proof that the condition A3 irnplics the condition B3 imrnediatc ;ly follows frorrr thc; identities in Lcrnrna 3.8 .4 .8Totally Geodesic Complex Curves .We call a complex curve E totallly geodesic if, for any tangent vector v to the complex curve E and geodesic segnrcnt yv : (-e, e) -M such that ^c (0) = v, y, (t) is corrtairred in the complex curve for small t .The main result of this seetiorr is tlrat, under condition (1 .5), the geodesics of F can be uniquely extended to irnmersed complex curves that are totally geodesic submanifolds of M. In fact, tlrese curves are precisely thc ones defirred by the complex curves exp (U)'s in condition (1 .5) .Theorem 4.9.If the condition (1 .5)holds, the complex curve exp (U) can be uniquely extended to a maximal totally geodesic complex curve f : E -M immersed in M.Moreover, there is a canonical (n-1)dimensional holomorphic vector subbundle T'E of f*(TM) transversal to E .
Proof: : Recall from Theorem 4.2 that D = span{X, JX, Z} is an involutive distribution.By the Frobenius theorerrr, this irrrplies that S* M is foliated by 3-dirnerrsional maximal integral subrnanifolds of D .Let (4 .10)E be a leaf of this foliation .-D,then there is a well defined SI C e action on E since Z is tangent to E. Let us denote the quotient space E/SI by E. It is not difñcult to see that E is a complex curve with the complex structure induced from the complex structure of E C T* M, and that there is a holomorphic immersion f such that the following diagram commutes : inclusion E~Srm E J M Recall that the complex curve exp (U ) is the surface defined by From this, it is clear that the complex curve f : E -M is locally defined by exp (U) since e'xc -9ziv E E .From this, it clearly follows that f : E -M is totally geodesic .
To define the transversal holomorphic subbundle T-LE of f* (TM), note that 7rz E is a principal circle bundle .Define the fiber T,LE ofT l EatxEEbyT~E={vEf*(TM)Jw(f*v)=0 for al¡ wEE -,,} .It is clear that Tl E is a holomorphic vector bundle with dimension n-1 .
To show that T~E is transversal to E, suppose v E TLE n TE.Note that xP(w) is tangent to E for all w E Ey since: xP(rv) = 7r* X , and X is tangent to E.Moreover, 'Y(ui) :,¿ 0 because w (xP(w)) = w(7r* X,) = FFa ua = F2 = 1 z~0, where uJ = Ea-l ,a dxa .Since v, f * Y' (w) E T,; E, we have f* v = zkP(w) for some z E (E .But, recall that v E TL E, and therefore, v)(v) = 0.This implies that v = 0 because 71) (V) = w (zXP(w)) = z {w (xP(w))} = z .4.12 The Holomorphic Sectional Curvature .If F satisfies the property (1 .5),there is a natural way to define holomorphic sectional curvature K of F .In this sectio'n, we show that K is determiried by the smooth function K in the condition A2 of Theorem 4 .2 .
Holomorphic sectional curvature K of a complex Firisler rnetric F has been studied by Wong and Royden [W] [R] .To define K(v) for a unit vector v E T, M (Le .F(v) = 1), note that each complex curve U C M tangent to v has a canonical complex Finsler rnetric, defined by the restriction F,1 ,u : TU -> IR.In fact, because U is of complex But, from thc; identities (4 .5), it easily follows that G'r1 = Qr 11 = 0 arrd Si = r,.Hence, we obtain the structure equations (4.16) : From the equations, it is clear that the pulí back to É of rl is a contact form since r) A drt = rt A B' A ' 0 on E.
Proof. .To prove tire theorern, choose v E T,,M and let U C M be a totally geodesic complex curve such that x E U arrd v E T,;U .Also, let f : E , M be the unique extension of U described in Theorem 4.9, and let Éu denote the restriction of the circle bundle 7rr : É -~E to U. Observe that U is also a submanifold of S M/S' since U C E = E/S' C SrM/S' .We claim that the Gaussian curvatura of g orr U is Klu E C_ (U), wlrcre K is regarded as a function orr S¡,M/S' .
The theorern follows from the claim.To sea this, proveed as follows: Note that by commutativity of the diagram (4.10), 7r = f o 7rz .Hence, if w E Eu, then xP(ur) = 7r,X w = f* o (7rr),X, .Therefore, the rrrap SrM , SpM sends Eu into the zenit tarrgerrt bundle S.U of g.Since T is a bundle map ovar U such that T(es'u» = e`T(w), kP maps tu dif eomorphically onto S.U, or equivalently, we llave a bundle rrrap 1 Is,u = <PIs,u : SU -Eu ovar U. Thus, since r, is corrstant along fibers of Eu arrd v E T,,U, we llave r, o <D(v) = K(x) .B,y thc; clairrr, K(x) is the Gaussian curvatura of g at x E U, and the identity K(v) = Koq>(v) follows .
To prove the clairn, we denote tire Gaussian curvatura of g on U by ic E C'(U), and show K = k.Recall that, f'rorrr Propositiorr 3.5, thc; Legendre foliation on Sg*U iras the structure equations (4.17)  x2,) is taken so that U C M is locally defined by .r,°'= 0 for a =~1, n + 1.To complete the proof; it remains to show that (1) rnaps Eu diffeornorpllically orto S.*U, and (2) " ~= 71 (1) : To show that 0 is a diffeornorphism onto Sg*U, recall that T maps Eu onto SU.Hence, if w E Eu, then the vectors xP(w) and T(Jw) _ -JT(w) f'orm an orthonormal frame of T,;U for some x E U. The following computation shows that the covectors 0(7u) and 0(Jul) = ,Iz/1(w) forro the dual coframe of {xP(w),T(Jul)} : Using the iderltities (2.10), compute; Hence z/>(Elx) C S.*U .Since V) is a bundle map preserving the circle action, it casily follows that 0 is a dif eorrlorpllisrn.
(2) : To prove tlle identity z)*~= TI, recall f'ronl (2.2) that fl = ul dx 1 + un -" dx n+1 .Tllerefóre, fronl (4.19), 't%1 * f = ul dx 1 -}-u "+1.dxn+l .On the other hand, the contact 1-forro 77 on E is defined by the pull-back of 71 _ 1:2n .= to E. But, since dx°= 0 for a =,~1, n+ 1 on E, we have 77 = 7x 1 dx 1 + un+1 dxn+1, Hence the identity 0*~= 17 follows .a The following corollary is a consequence of the proof of Theorem 4.14 : 4.20 Corollary .The Gaussian curvatura of the induced metric .q on E ás r,,É .4.21 Relation to the Kobayashi Metric.In this section, we prove a version of a theorem of Faran which states that vanishing of certain local invariants forres F to be the Kobayashi metric of M, provided that F is complete and satisfies the condition K = -4 (sea Introduction) .In the version of the theorem presented here, the condition of vanishing of invariants is replaced by the equivalent condition (1 .5) .
Recall frorri Lempert's result described in the introduction that, if D C T' is a bounded strictly linearly convex domain with smooth boundary, then cvery extremal disk f : A -> D is an isornetric imbedding (Le.f *FK coincides with the Poincaré norm en ,), and that f (A) is a maximal totally geodesia cornplex curve; in D. Since thc; Poincaré nietrie has Gaussian curvatura -4, the holornorphic; sectional curvaturc of thc; Kobayashi rnctric FK is -4.
Ori the other hand, if F is any cornplex Finsler metric : on a cornplex manifold M, the condition of constant holornorphic; sectional curvatura K = -4 imposes a restriction on the metric F. In faca, we show that, if F is any complete cornplex Finsler metric; with the properties K = -4 and (1 .5),then F rnust coincide with the Kobayashi metric .To show this, we need thc ; following lerrima due to Ahlfors [A] [K] : 4.22 Generalized Schwarz Lemma.Let (N, g) be a, 1-dimensional.
Hermitian manifold such thatt the Gaussian cu,r'vatur'e is bounded, above by a negative consttantt -C .For' any holornorphic mal) f : A , N, the inequality (4 .23)Il .f*vll.~<_ c Ilvll holds for allll v E TA, v)hcre II Ils is the norm on N induced by y and II II denotes the nor'rn defined by the Poincaré rnetric on A.
We call a cornplex Finsler metric: F complete if the geodesic : vector field X is complete; (or equivalently, if every geodesic can be; extended to a geodesic define(¡ en all of IR) .
4 .24Theorem [F] .Suppose F is a complete complex Finsler metric on a complex manifold M with constante holomorphic sectionall curvatura K = -4 satisfyinq the property (1 .5) .The dual metric F coincides with the Kobayashi metric FI< of M.
and the fact that the vector fields Y and Z commute { .Cy (r Z)} = (Yr,) Z = 2(K Z) .