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2 December, 006 Zhejiang University On Complex Finsler Manifolds Shen Yibing Page of

3 Contents Introduction Hermitian Metrics Holomorphic Curvature Proof of the Main Theorem 5 Complex Randers metrics 6 Problems Page of 7 Main References

4 . Introduction Professor S.S.Chern said that it is very sorry that the ancient Chinese can not discover the complex number. Page of

5 . Introduction Professor S.S.Chern said that it is very sorry that the ancient Chinese can not discover the complex number. Complex Finsler manifolds are complex manifolds with complex Finsler metrics, which are more general than Hermitian metrics. Page of

6 . Introduction Professor S.S.Chern said that it is very sorry that the ancient Chinese can not discover the complex number. Complex Finsler manifolds are complex manifolds with complex Finsler metrics, which are more general than Hermitian metrics. In [] S.Kobayashi gave two good reasons for considering complex Finsler structures in a complex manifold. One is that every hyperbolic complex manifold M carries a natural complex Finsler metric in a broad sense. The second reason is as differential geometric tool in the study of complex vector bundles. Page of

7 . Introduction Professor S.S.Chern said that it is very sorry that the ancient Chinese can not discover the complex number. Complex Finsler manifolds are complex manifolds with complex Finsler metrics, which are more general than Hermitian metrics. In [] S.Kobayashi gave two good reasons for considering complex Finsler structures in a complex manifold. One is that every hyperbolic complex manifold M carries a natural complex Finsler metric in a broad sense. The second reason is as differential geometric tool in the study of complex vector bundles. There are many very famous classical metrics on the Teichmüller and the moduli spaces, among which there are three complex Finsler metrics: Teichmüller metric; Caratheodory metric; and Kobayashi metric. Page of

8 . Introduction Professor S.S.Chern said that it is very sorry that the ancient Chinese can not discover the complex number. Complex Finsler manifolds are complex manifolds with complex Finsler metrics, which are more general than Hermitian metrics. In [] S.Kobayashi gave two good reasons for considering complex Finsler structures in a complex manifold. One is that every hyperbolic complex manifold M carries a natural complex Finsler metric in a broad sense. The second reason is as differential geometric tool in the study of complex vector bundles. There are many very famous classical metrics on the Teichmüller and the moduli spaces, among which there are three complex Finsler metrics: Teichmüller metric; Caratheodory metric; and Kobayashi metric. Recently, J.-G. Cao and Pit-Mann Wong ([]) studied Finsler geometry of projective vector bundle and proposed the following question: Suppose that M is a Kähler manifold and E is a holomorphic vector bundle over M. Is E Kähler? They gave some partial results and showed some equivalent conditions for E to be Kähler. Page of

9 Let E be a holomorphic vector bundle of rank r over a complex manifold M of complex dimension n with the natural projection π. We denote a point of E by (z, v), where z represents a point of M and v is a vector in the fibre E z = π (z) of E over z M. Let o : M E be the zero section of E and set E o = E \ {o}. Page 5 of

10 Let E be a holomorphic vector bundle of rank r over a complex manifold M of complex dimension n with the natural projection π. We denote a point of E by (z, v), where z represents a point of M and v is a vector in the fibre E z = π (z) of E over z M. Let o : M E be the zero section of E and set E o = E \ {o}. Definition. A complex Finsler metric on E is a real function G : E R which satisfies the following conditions: () G(z, v) 0, where the equality holds if and only if v = 0; () G C (E o ), that is, G is smooth in E o ; () G(z, λv) = λ G(z, v) for all (z, v) E, λ C \ {0}. Page 5 of

11 Let E be a holomorphic vector bundle of rank r over a complex manifold M of complex dimension n with the natural projection π. We denote a point of E by (z, v), where z represents a point of M and v is a vector in the fibre E z = π (z) of E over z M. Let o : M E be the zero section of E and set E o = E \ {o}. Definition. A complex Finsler metric on E is a real function G : E R which satisfies the following conditions: () G(z, v) 0, where the equality holds if and only if v = 0; () G C (E o ), that is, G is smooth in E o ; () G(z, λv) = λ G(z, v) for all (z, v) E, λ C \ {0}. Theorem(Shen-Du,[6]). Let (M, G) be a complex Finsler manifold of dimension n and T M its holomorphic tangent bundle. Then the Hermitian metric h T M = G i j(z, v)dz i d z j + G i j(z, v)δv i δ v j on T M is Kählerian if and only if (M, G) is a Kähler manifold with zero holomorphic sectional curvature, where G i j = G v i v, i, j, n. j Moreover, we shall consider complex Randers metrics. Page 5 of

12 . Hermitian Metrics Let (z, v) = (z,, z n, v,, v r ) be a local coordinate system for E. A complex Finsler metric G on E is said to be strongly pseudo-convex if the complex Hessian ( ) G (G i j) = v i v j of G is positively definite on E o. In particular, if G(z, v) = h i j(z)v i v j is a Hermitian metric on E, then G(z, v) defines a strongly pseudo-convex Finsler metric on E. α, β, n; i, j, k, r. Page 6 of

13 . Hermitian Metrics Let (z, v) = (z,, z n, v,, v r ) be a local coordinate system for E. A complex Finsler metric G on E is said to be strongly pseudo-convex if the complex Hessian ( ) G (G i j) = v i v j of G is positively definite on E o. In particular, if G(z, v) = h i j(z)v i v j is a Hermitian metric on E, then G(z, v) defines a strongly pseudo-convex Finsler metric on E. α, β, n; i, j, k, r. Introduce the following notations: G,α = G z α, G i = G v i, G j = G v j, G i j = G v i v j, G, β = G z β, G i,ᾱ = G v i z α, G i j, β = G i j z β, etc., Page 6 of

14 . Hermitian Metrics Let (z, v) = (z,, z n, v,, v r ) be a local coordinate system for E. A complex Finsler metric G on E is said to be strongly pseudo-convex if the complex Hessian ( ) G (G i j) = v i v j of G is positively definite on E o. In particular, if G(z, v) = h i j(z)v i v j is a Hermitian metric on E, then G(z, v) defines a strongly pseudo-convex Finsler metric on E. α, β, n; i, j, k, r. Introduce the following notations: G,α = G z α, G i = G v i, G j = G v j, G i j = G v i v j, G, β = G z β, G i,ᾱ = G v i z α, G i j, β = G i j z β, etc., In particular, we can take E = T M, the holomorphic tangent bundle of M, so that r = n. Page 6 of

15 Suppose that a strongly pseudoconvex complex Finsler metric G(z, v) is given on T M. The pair (M, G) is called a complex Finsler manifold. Let M = T M \ {o} denote T M without the zero section. { z, i v }( i, j n) give a local j frame field of the holomorphic tangent bundle T M of M. Page 7 of

16 Suppose that a strongly pseudoconvex complex Finsler metric G(z, v) is given on T M. The pair (M, G) is called a complex Finsler manifold. Let M = T M \ {o} denote T M without the zero section. { z, i v }( i, j n) give a local j frame field of the holomorphic tangent bundle T M of M. Let π : T M M denote the holomorphic tangent bundle of M. Then the differential dπ : T C M T C M of π : M M defines the vertical bundle V over M by V = kerdπ T M, which yields a holomorphic vector bundle of rank n over M. A local frame field of V is given by { v }( j n), and a j natural section ι : M V, called the radial vertical field, is well-defined for (z, v) M by ι(v) = ι(v i ( z i) z) = v i ( v i) v. Associated with G, we now define a Hermite metric on the vertical bundle V by < X, Y > v = G i j(z, v)x i Ȳ j, where (z, v) M and X, Y V v π (z, v). Page 7 of

17 Let D : Γ(V) Γ(T C M V) be the Hermitian connection of the Hermitian vector bundle (V, <, >), where Γ( ) denotes the space of smooth sections. Let denote the covariant differentiation defined by D, and define a bundle map : T M V by (X) = X ι. The horizontal bundle H over M is then defined by H = ker, which is the subbundle of T M consisting of vectors with respect to which ι is parallel. Then it is verified that T M = V H and a natural local frame field { δ δz i }, ( i n) of H is given by δ δz = i z N j i i v, N i j j = G i lg l, j = G i l G v l z. j Thus we get a local frame field { δ δz, i v }, ( i n) of T M. i Let {dz i, δv i } denote the dual frame field of { δ δz, i v }, where i δv i = dv i + N i jdz j. Page 8 of

18 Associated with the decomposition T M = V H, we have the horizontal map Θ : V H given locally by Θ( v ) = δ i δz for i n, and a natural section i χ = Θ ι : M H, called the radial horizontal field, such that χ(v i z i) = vi δ δz i. Page 9 of

19 Associated with the decomposition T M = V H, we have the horizontal map Θ : V H given locally by Θ( v ) = δ i δz for i n, and a natural section i χ = Θ ι : M H, called the radial horizontal field, such that χ(v i z i) = vi δ δz i. Using the horizontal map Θ : V H, we can transfer the Hermitian metric <, > on V to H by setting < X, Y > v =< Θ (X), Θ (Y ) > v, where (z, v) M,X, Y H v π (z, v). Then a Hermitian metric h T M on M canonically associated with G is defined by requiring H to be orthogonal to V, so that Θ : V H and χ : M H are isometric embeddings. ht M is given in local coordinates by h T M = G i j(z, v)dz i d z j + G i j(z, v)δv i δ v j. (.) Page 9 of

20 . Holomorphic Curvature Then the connection form ω = (ωj i ) of the Hermitian connection D of the Hermitian vector bundle ( M, h T M ) is given by ωj i = G ki G j k = Γ i jkdz k + γjkdv i k, (.) where Γ i jk = G li G j l z k, γi jk = G li G j l v k. Page 0 of

21 . Holomorphic Curvature Then the connection form ω = (ωj i ) of the Hermitian connection D of the Hermitian vector bundle ( M, h T M ) is given by ωj i = G ki G j k = Γ i jkdz k + γjkdv i k, (.) where Γ i jk = G li G j l z k, γi jk = G li G j l v k. The curvature form Ω = (Ω i j ) of its curvature R = D D is given by Ω = (Ω i j ) = ( ω j i ), which can be written as Ω i j = κ i jk l dzk d z l +µ i jk l dzk d v l +σ i jk l dvk d z l +τ i jk l dvk d v l, where κ i jk l = Γi jk z l, σ i jk l = γi jk z l, µ i jk l = Γi jk v l, τ i jk l = γi jk v l. Page 0 of

22 Setting κ i jk l = G h jκ h ik l and τ i jk lv i = G h jτ h, we have ik l κ i jk lv i v j = ( G i j, k l + G h p G h j, lg pi, k )v i v j = G, k l + G h p G h, lg p, k, (.) τ i jk lv i = µ i jk lv i = σ i jk l v j = 0. Page of

23 Setting κ i jk l = G h jκ h ik l and τ i jk lv i = G h jτ h, we have ik l κ i jk lv i v j = ( G i j, k l + G h p G h j, lg pi, k )v i v j = G, k l + G h p G h, lg p, k, (.) τ i jk lv i = µ i jk lv i = σ i jk l v j = 0. Corresponding to the decomposition T M = V H, The differential operator d on functions is decomposed as d = d H + d V. We also decompose d H and d V into (,0)-part and (0,)-part as d H = H + H, and d V = V + V, (.) respectively, where we put H f = δf δz dz i, i V f = f v δv i for a i C function f(z, v) on T M. Page of

24 Definition. Let (M, G) be a complex Finsler manifold. The fundamental form associated with G is Φ = G i jdz i d z j which is a real (,)-form on M. (M, G) is called a Finsler- Kähler manifold if d H Φ = 0. It is equivalent to Γ i jk = Γ i kj. Page of

25 Definition. Let (M, G) be a complex Finsler manifold. The fundamental form associated with G is Φ = G i jdz i d z j which is a real (,)-form on M. (M, G) is called a Finsler- Kähler manifold if d H Φ = 0. It is equivalent to Γ i jk = Γ i kj. Definition. Let (M, G) be a complex Finsler manifold. The holomorphic curvature K(z, v) of G along v is given by where K(z, v) = ψ i jv i v j G (z, v), (.) ψ i j = G k ln k i N l j G, i j. Page of

26 Lemma. For the non-linear connection Nj i following formulas: we have the () () N p i v j G p = 0, N p i z j G p = ψ i j, N p i () v G q p j = N p i v G p q, j () N p δg p j k + δg j, k = N p δg p j δz l δz l l δz k + δg j, l δz k. Page of

27 Lemma. For the non-linear connection Nj i following formulas: we have the () () N p i v j G p = 0, N p i z j G p = ψ i j, N p i () v G q p j = N p i v G p q, j () N p δg p j k + δg j, k = N p δg p j δz l δz l l δz k + δg j, l δz k. Proof. A straightforward calculation can prove the lemma. Page of

28 . Proof of Main Theorem The sufficiency of the theorem follows directly from Corollary. of []. In the following, we prove the necessity of the theorem. Page of

29 . Proof of Main Theorem The sufficiency of the theorem follows directly from Corollary. of []. In the following, we prove the necessity of the theorem. Let ω be the fundamental -form of the Hermitian metric h T M, that is, ω is defined by ω(x, Y ) = h T M (X, JY ), forx, Y T c (T M). In a local coordinate system for T M, ω can be expressed as ω = G i j(z, v)dz i d z j G i j(z, v)δv i δ v j. Page of

30 . Proof of Main Theorem The sufficiency of the theorem follows directly from Corollary. of []. In the following, we prove the necessity of the theorem. Let ω be the fundamental -form of the Hermitian metric h T M, that is, ω is defined by ω(x, Y ) = h T M (X, JY ), forx, Y T c (T M). In a local coordinate system for T M, ω can be expressed as ω = G i j(z, v)dz i d z j G i j(z, v)δv i δ v j. Taking exterior differentiation of ω, we have dω = ( ){dg i j dz i d z j + dg i j δv i δ v j +G i jd(δv i ) δ v j G i jδv i d(δ v j )} := ( )(I + II + III III) Page of

31 Calculating all of the terms one by one, we obtain I = δg i j δz k dzk dz i d z j + δg i j δ z k d zk dz i d z j + G k l v j dzk d z l δv j + G k l v j dzk d z l δ v j. II = δg i j δz k dzk δv i δ v j + δg i j δ z k d zk δv i δ v j. III = N p k δg p j dz k d z l δ v j + N p G p j δ z l k v i dzk δv i δ v j δg j, k δ z l dz k d z l δ v j G j, k v i dz k δv i δ v j. Page 5 of

32 Calculating all of the terms one by one, we obtain I = δg i j δz k dzk dz i d z j + δg i j δ z k d zk dz i d z j + G k l v j dzk d z l δv j + G k l v j dzk d z l δ v j. II = δg i j δz k dzk δv i δ v j + δg i j δ z k d zk δv i δ v j. III = N p k δg p j dz k d z l δ v j + N p G p j δ z l k v i dzk δv i δ v j δg j, k dz k d z l δ v j G j, k dz k δv i δ v j. δ z l v i Therefore, the coefficient of ( )dz k δv i δ v j in dω is Page 5 of δg i j δz k + N p k G p j v i G j, k v i = 0. Correspondingly, the coefficient of ( )d z k δv i δ v j is also 0.

33 p We know that z ( N k l v G p) = 0, from which it follows that j the coefficient of ( )dz k d z l δ v j is G k l v j + N p k δg p j δ z l δg j, k δ z l = G k l v j + v j ψ k l. Hence, the coefficient of ( )dz k d z l δv j is G k l v j + ψ k l v j. Page 6 of

34 p We know that z ( N k l v G p) = 0, from which it follows that j the coefficient of ( )dz k d z l δ v j is G k l v j + N p k δg p j δ z l δg j, k δ z l = G k l v j + v j ψ k l. Hence, the coefficient of ( )dz k d z l δv j is G k l v j + ψ k l v j. If h T M is a Kähler metric, then we have dω = 0, so that δg i j δz k dzk dz i d z j + δg i j δ z k d zk dz i d z j = 0, (.) Page 6 of G k l G k l + ψ k l = 0, + ψ k l = 0, (.) v j v j v j v j which implies that M is a Finsler-Kähler manifold.

35 By the definition of ψ k l, one can check easily that ψ k l have the same homogeneity as G, that is, ψ k l(z, λv) = λ λψ k l(z, v), λ C. Therefore, we have from which we obtain ψ k l v j vj = ψ k l, (.) Thus, we obtain G k l v j vj = 0, G k l v j vj = 0. (.) ψ k l = 0. (.5) Page 7 of

36 By the definition of ψ k l, one can check easily that ψ k l have the same homogeneity as G, that is, Therefore, we have ψ k l(z, λv) = λ λψ k l(z, v), λ C. from which we obtain G k l v j vj = 0, Thus, we obtain Substituting (5.5) into (5.) yields that ψ k l v j vj = ψ k l, (.) G k l v j vj = 0. (.) ψ k l = 0. (.5) G k l G = 0, k l = 0, v j v j which implies that (M, G) is a Hermitian manifold. Then, by (5.), we see that (M, G) is a Kähler manifold. From (.) and (5.5) we know the holomorphic curvature K of M is zero, and the proof of the main Theorem is completed. Page 7 of

37 5. Complex Randers metrics Let M be a complex manifold of complex dimension n, and α = a i j(z)v i v j be a Hermitian metric on M. Suppose that β = b i (z)v i is a holomorphic -form on M. Set F = α + ɛ β β = α + ɛ β, (5.) where ɛ = Definition. The metrics {, for β 0, 0, for β = 0. G := F = α + ɛα β + ɛ β (5.) are called complex Randers metrics. Page 8 of

38 5. Complex Randers metrics Let M be a complex manifold of complex dimension n, and α = a i j(z)v i v j be a Hermitian metric on M. Suppose that β = b i (z)v i is a holomorphic -form on M. Set F = α + ɛ β β = α + ɛ β, (5.) where ɛ = Definition. The metrics {, for β 0, 0, for β = 0. G := F = α + ɛα β + ɛ β (5.) are called complex Randers metrics. It is easy to see that where G i j = F α h i j + ɛf β b ib j + G G ig j, (5.) Page 8 of

39 h i j := a i j α l il j, l i := i α = α i α = a i j v j, l j := jα. Page 9 of

40 h i j := a i j α l il j, l i := i α = α i α = a i j v j, l j := jα. G ji = α a ji + β (α β α + β ) v i v j α F Gγ F γ bi bj α F γ ( βv i bj + βb i v j ), (5.) where det(g i j) = ( ) n F ɛγ α α β det(a i j), (5.5) γ := G + α ( β α ). Page 9 of

41 h i j := a i j α l il j, l i := i α = α i α = a i j v j, l j := jα. G ji = α a ji + β (α β α + β ) v i v j α F Gγ F γ bi bj α F γ ( βv i bj + βb i v j ), (5.) where det(g i j) = ( ) n F ɛγ α α β det(a i j), (5.5) γ := G + α ( β α ). Hence, G is strongly pseudo-convex if G > α ( β(v) α). Page 9 of

42 For the complex Randers metric, the coefficients of the Chern- Finsler connection are ( ) Nj i = α Nj i + ηi b k l k γ z β b k j β z j vk + β b k g ki β z, (5.6) j where η i := βv i + α b i, α N i j := a ki a l k z j vl, g ji := αa ji + (α β α + β ) v i v j α γ γ bi bj α γ ( βv i bj + βb i v j ). From this we can calculus the holomorphic curvature of the complex Randers metric. It is a technical computation which involves a lot of long-winded calculus. So, we will not dwell too much on it. Page 0 of

43 6. Problems Question. Construct complex Randers metrics with constant holomorphic curvature. Page of

44 6. Problems Question. Construct complex Randers metrics with constant holomorphic curvature. Question. Consider the relation between the Kobayashi metric and the complex Randers metric with constant negative holomorphic curvature. Page of

45 6. Problems Question. Construct complex Randers metrics with constant holomorphic curvature. Question. Consider the relation between the Kobayashi metric and the complex Randers metric with constant negative holomorphic curvature. Question. Consider complex projectively flat Randers metrics with constant holomorphic curvature. Page of

46 7. Main References [] Cao,J.-K. and Wang,P.-M., Finsler geometry of projectivized vector bundles, J. Math. Kyoto Univ., (00),8-. [] Abate,M. and Patrizio,G., Kahler Finsler manifolds of constant holomorphic curvature, Internat. J. Math., 8(997), [] Kobayashi,S., Complex Finsler vector bundles, Finsler geometry (Seattle, WA, 995), 5 5, Contemp. Math., 96, AMS, 996. [] Bao,D., Chern,S.S. and Shen,Z., An Introduction to Riemann-Finsler Geometry, GTM 00, Springer, 000. [5] Aldea,N. and Munteanu,G., On complex Finsler spaces with Randers metric, Preprint. [6] Shen,Y.B. and Du,W.P., A note on comlex Finsler manifolds, Chin. Ann. of Math., 7A(006), Page of

47 The End Thank You! Page of