- Email: [email protected]
Lift of the Finsler foliation to its normal bundle
Differential Geometry and its Applications 24 (2006) 209–214 www.elsevier.com/locate/difgeo
Lift of the Finsler foliation to its normal bundle A. Miernowski, W. Mozgawa ∗ Institute of Mathematics, Maria Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland Received 19 July 2004; received in revised form 7 January 2005 Available online 21 October 2005 Communicated by O. Kowalski
Abstract E. Ghys in [E. Ghys, Appendix E: Riemannian foliations: Examples and problems, in: P. Molino (Ed.), Riemannian Foliations, Birkhäuser, Boston, 1988, pp. 297–314. [3]] has posed a question (still unsolved) if any Finslerian foliation is a Riemannian one? In this paper we prove that the natural lift of a Finslerian foliation to its normal bundle is a Riemannian foliation for some Riemannian transversal metric. The methods we used here are closely related to those used by M. Abate and G. Patrizio in [M. Abate, G. Patrizio, Finsler Metrics—A Global Approach, Springer-Verlag, Berlin, 1994]. © 2005 Elsevier B.V. All rights reserved. MSC: 53C12 Keywords: Finsler foliation; Natural lift; Good connection
1. Basic facts In this chapter we define a Finslerian foliation in an analogous manner as for Riemannian foliation (cf. [4,5]). Let (W, F ) be a Finsler manifold, where F : T W → R is a Finsler metric (for necessary definitions, see [1,2]). Definition 1.1. A diffeomorphism f : W → W is said to be a Finsler isometry if F f (p), f∗ (v) = F (p, v) for each p ∈ W and v ∈ Tp W . 2 v Example. Let W = R2 \ {(0, 0)} and F (p, v) = 1e p 2 − easy to see that each homothety is a Finsler isometry.
p,v , p2
where p ∈ W , v ∈ Tp W and 0 < e < 1. Then it is
Definition 1.2. A foliated cocycle {Ui , fi , γij } on a manifold M is said to be a Finslerian foliation F if * Corresponding author.
E-mail addresses: [email protected] (A. Miernowski), [email protected] (W. Mozgawa). 0926-2245/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.difgeo.2005.09.008
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(a) {Ui } is an open covering of M, (b) fi : Ui → W is a submersion, where (W, F ) is a Finslerian manifold, (c) γij is a local Finslerian isometry of (W, F ) such that for each x ∈ Ui ∩ Uj fi (x) = (γij ◦ fj )(x). The Finslerian manifold (W, F ) will be called the transversal manifold of foliation F . The local submersions {fi } define by pull-back a Finsler metric in the normal bundle Q = Q(M) of the foliation F , which we shall denote again by F . Note that for any basic vector Y for the foliation and any vector field X tangent to F we have X(F (Y )) = 0. Denote by (x i , y α ), i = 1, . . . , p, α = 1, . . . , q, p + q = dim M a local foliated chart on M and let ( ∂x∂ i , ∂y∂ α ) be a ¯
local frame of T M. If we denote by ( ∂y∂ α ) the corresponding local frame of Q then we can induce a chart (x i , y α , uα¯ ) ¯
on Q, where uα¯ ∂y∂ α is a transversal vector at a point (x i , y α ). Note that in this coordinate system the metric F in Q does not depend on (x i ). Lemma 1.1. A distribution Lin( ∂x∂ 1 , . . . , ∂x∂ p ) defines a foliation FQ on Q called the natural lift of F to Q.
Proof. Let (x i , y α , uα¯ ) be an another coordinate system in Q. Then the lemma follows directly from the transformation rule
x i = x i (x i , y α ),
y α = y α (y α ),
uα¯ =
∂y α α β¯ (y )u . ∂y β
2 ¯
¯
Let Q(Q) denote the bundle over Q transversal to the foliation FQ . The vectors ∂y∂ α and ∂u∂ α¯ , α = 1, . . . , q, form the natural frame of Q(Q) at the point (x i , y α , uα¯ ) ∈ Q. Thus we can take as coordinates in this bundle the following ¯ ¯ functions (x i , y α , uα¯ , Y α , U α¯ ), where (Y α , U α¯ ) are the coefficients of the vector Y α ∂y∂ α + U α¯ ∂u∂ α¯ . The canonical projection π : Q → M, π(x i , y α , uα¯ ) = (x i , y α ) induces an another projection π∗ : T Q → T M which maps the tangent vectors to FQ in the vectors tangent to F . Thus π∗ induces a mapping π˜ ∗ : Q(Q) → Q given in local coordinates by (x i , y α , uα¯ , Y α , U α¯ ) → (x i , y α , Y α ). Note that V = V (Q) = ker π˜ ∗ is a vertical bundle ¯ spanned on the vectors ∂u∂ α¯ , α = 1, . . . , q, and the local coordinates in this bundle are given by (x i , y α , uα¯ , U α¯ ). Corollary 1.1. We have the following rule of transformations of local coordinates in Q(Q)
x i = x i (x i , y α ),
y α = y α (y α ),
∂y α u = α (y α )uα¯ , ∂y α¯
U α¯ =
∂y α α α¯ (y )U . ∂y α
Lemma 1.2. Let o : M → Q(M) be the zero section of the bundle Q(M). Then the set o(M) is saturated on Q(M) with foliation FQ . Proof. Since the leaves of FQ are locally defined by the equations y α = const., uα¯ = const. and the equations uα¯ = 0 are preserved under the coordinate changes then the zero section is the saturated set. 2
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211
˜ = Q(M) ˜ We put Q = Q(M) \ {o}. In the sequel we denote simply by V the restriction of the vertical bundle V ˜ to Q. 2. Vertical connection ˜ ⊗ V ) related to considered The purpose of this section is to define a linear connection ∇ : X (V ) → X (T ∗ (Q) ˜ foliated structure. Since the bundle Q is foliated we are looking for a Bott connection that is such one that for any vector field X tangent to FQ and any transversal vector field Y we have ∇X Y = p [X, Y˜ ] , where p : T Q → Q(Q) is the canonical projection and p(Y˜ ) = Y . Definition 2.1 (cf. [1]). The vector field i : Q → V given by ∂¯ ∂uα¯ is said to be the radial vector field. i(x i , y α , uα¯ ) = uα¯
It can be checked that this definition is well-posed. ˜ ⊗ V ) be a Bott connection. Then ∇X i = 0 for any vector field tangent to FQ . Lemma 2.1. Let ∇ : X (V ) → X (T ∗ (Q) The proof of this lemma is obvious and follows from the definition of the Bott connection. ¯ ¯ ¯ Consider the local frame { ∂x∂ i , ∂y∂ α , ∂u∂ α¯ } of T Q and recall that the vectors { ∂u∂ α¯ } form the basis of V . With these settings we put ¯ ∂¯ β¯ ∂ = Γ , i α ¯ ∂uα¯ ∂uβ¯ ¯ ∂¯ γ¯ ∂ = Γ , ∇ ∂¯ ¯ α β¯ ∂uγ¯ ∂y α ∂uβ ¯ ∂¯ γ¯ ∂ ∇ ∂¯ = Γα¯ β¯ γ¯ . ¯ β ∂u ∂uα¯ ∂u ∇
∂ ∂x i
From the above formulas it follows that β¯
Γi α¯ = 0, ∇
∂¯ ∂uα¯
i=
γ¯ δα¯
β¯
+u
γ¯ Γα¯ β¯
¯ ∂ . ∂uγ¯
The Bott connection ∇ allows us to define a mapping ˜ → X (V ), L : X Q(Q) L(X) = ∇X˜ i, ˜ = X. If we denote by Λ the restriction of the linear mapping L to the bundle V then we can state where p(X) Definition 2.2 (cf. [1]). The connection ∇ is said to be a good vertical connection if Λ : V → V is a bundle isomorphism. γ¯
¯
γ¯
Observe that ∇ is a good vertical connection if and only if the matrix δα¯ + uβ Γα¯ β¯ is nonsingular. If we put H = ker L then we can split the bundle Q(Q) into a direct sum Q(Q) = H ⊕ V . Let {∂α , ∂˙β¯ } be a local frame of
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Q(Q), where ∂α = ¯
∂¯ ˙ ∂y α , ∂β¯
=
∂¯ . ∂uβ¯
Then the coefficients of the mapping L in this basis are
γ¯
L(∂α ) = uβ Γα β¯ ∂˙β¯ , γ¯ ¯ γ¯ γ¯ L(∂˙β¯ ) = δβ¯ + uδ Γβ¯ δ¯ ∂˙γ¯ = Lβ¯ ∂˙γ¯ . β¯
β¯
β¯ ¯
γ¯
It is easy to check that the vectors δα = ∂α − Γα ∂˙β¯ , where Γα = (L−1 )γ¯ uδ Γα δ¯ form a basis of ker L. In the sequel β¯ ¯ ¯ it is convenient to use the basis {δα , ∂˙β¯ } as well as its dual {dy α , ψ β = duβ + Γα dy α }. Using this coframe we can define a local connection forms by
∇ ∂˙β¯ = ωβα¯¯ ⊗ ∂˙α¯ , where ωβα¯¯ = Γγα¯β¯ dy γ + Γγ¯α¯β¯ duγ¯ = Γγα¯β¯ − Γε¯α¯β¯ Γγε¯ dy γ + Γγ¯α¯β¯ ψ γ¯ = Kγα¯ β¯ dy γ + Kγα¯¯ β¯ ψ γ¯ . ¯ Note that Kγα¯ β¯ uβ = Γγα¯ . The formula θ (∂˙α¯ ) = δα defines a linear mapping θ : V → H . This mapping allows us to extend the connection ∇ to the horizontal bundle H by ∇X Z = θ ∇X θ −1 (Z) ,
where Z ∈ X (H ), X ∈ X (T Q). In this way we construct a linear connection in Q(Q) ∇X Y = ∇X ν(Y ) + ∇X Y − ν(Y ) , where Y ∈ X (Q(Q)), X ∈ X (T Q) and ν : Q(Q) → V is the vertical projection. In particular, we have ∇δα = ωαβ ⊗ δβ ,
β¯
ωαβ = ωα¯ .
If ϕ ∈ X (Q∗ (Q) ⊗ Q(Q)) is a 1-form given locally by ¯ ϕ = ϕ α ⊗ δα + ϕ β ⊗ ∂˙β¯
then following [1] we define an exterior differential Dϕ putting ¯ β¯ Dϕ = dϕ α − ϕ γ ∧ ωγα ⊗ δα + dϕ β − ϕ γ¯ ∧ ωγ¯ ⊗ ∂˙β¯ . Simple but long calculations show that the above formula is well-defined. The bundle Q∗ (Q) ⊗ Q(Q) admits a natural section η given by η = dy α ⊗
∂ ∂ ¯ ¯ + duβ ⊗ = dy α ⊗ δα + ψ β ⊗ ∂˙β¯ . ¯ β ∂y α ∂u
It is clear that the form η is well-defined. Definition 2.3. The form ϑ = Dη is called the torsion form of the connection ∇. Locally the form ϑ can be expressed by ¯ ¯ β¯ Dη = −dy α ∧ ωαγ ⊗ δγ + dψ β − ψ α¯ ∧ ωα¯ ⊗ ∂˙β¯ = ϑ γ ⊗ δγ + ϑ β ⊗ ∂˙β¯ , where ϑγ = ¯
1 γ¯ γ¯ γ¯ Kκ α¯ − Kα κ¯ dy α ∧ dy κ − Kκ α¯ dy α ∧ ψ κ¯ , 2 ¯
β¯
β¯
ϑ β = dΓγβ ∧ dy γ − Kκ α¯ ψ α¯ ∧ dy κ − Kκ¯ α¯ ψ α¯ ∧ ψ κ¯ .
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3. Finsler foliation In this section we consider the Finsler transversal metric F : Q → R+ ∪ {0} introduced in Section 1. Let G = F 2 ¯ and let (x i , y α , uβ ) be the local coordinates in Q. We put Gα =
∂G , ∂y α
Gβ¯ =
∂G ∂uα¯
then reasoning in the same way as in [1,2] we have ¯
Gα¯ β¯ uα¯ = Gβ¯ ,
Gα¯ β¯ uα¯ uβ = 2G,
Gα¯ β¯ γ¯ uγ¯ = 0. ¯
It is easy to see that for any vertical vectors v = v α¯ ∂˙α¯ , w = w β ∂˙β¯ the formula ¯
v, w = Gα¯ β¯ v α¯ w β
defines a Riemannian metric in the vertical bundle. In the similar way as in [1] we can generalize to the case of Finslerian foliation Theorem 1.4.2 given there. Theorem 3.1. Let F : Q → R+ ∪ {0} be the transversal Finslerian metric and , be the Riemannian metric on V ˜ ⊗ V ) such that induced by F . Then there exists exactly one Bott vertical connection ∇ : X (V ) → (T ∗ (Q) (i) ∇ is a good connection; ˜ then (ii) if v, w ∈ X (V ), X ∈ T Q Xv, w = ∇X v, w + v, ∇X w; (iii) ϑ(v, w) = 0 for any v, w ∈ V ; (iv) ϑ(v, w) ∈ V for any v, w ∈ H . Proof. The proof is very similar to that in [1] and we obtain the following formulas for the coefficients of connections 1 1 ¯ η¯ Kγ α¯ = Gη¯ μ¯ ∂γ (Gμ¯ α¯ ) + ∂α (Gμ¯ γ¯ ) − ∂μ (Gγ¯ α¯ ) − Gη¯ β Γγκ¯ ∂˙κ¯ (Gα¯ β¯ ) + Γακ¯ ∂˙κ¯ (Gβ¯ γ¯ ) − Γβκ¯ ∂˙κ¯ (Gγ¯ α¯ ) , 2 2 1 β¯ δ¯ ˙ δ¯ Kγ¯ α¯ = G ∂γ¯ (Gα¯ β¯ ), 2 1 η¯ β¯ η¯ η¯ Γγ = G ∂α Gβ¯ γ¯ uα¯ + ∂γ Gα¯ β¯ uα¯ − ∂β Gγ¯ α¯ uα¯ − Kκ¯ γ¯ Gαβ ∂γ (Gβ¯ uγ¯ ) − ∂β (G), 2 β¯
Γi α¯ = 0.
2
Observe that the isomorphism θ does not depend of the coordinates along the leaves. It follows that the Riemannian metric in Q(Q) defined by w1 , w2 = θ −1 (w1 ), θ −1 (w2 ) for all w1 , w2 ∈ H, v, w = 0
for all v ∈ V , w ∈ H
is a transversal Riemannian metric for the lifted foliation FQ . Theorem 3.2. The lift FQ of the Finsler foliation F to Q is a Riemannian foliation. References [1] M. Abate, G. Patrizio, Finsler Metrics—A Global Approach, Springer-Verlag, Berlin, 1994. [2] D. Bao, S.-S. Chern, Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, Berlin, 2000.
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[3] E. Ghys, Appendix E: Riemannian foliations: Examples and problems, in: P. Molino (Ed.), Riemannian Foliations, Birkhäuser, Boston, 1988, pp. 297–314. [4] P. Molino, Riemannian Foliations, Progress in Mathematics, vol. 73, Birkhäuser, Boston, 1988. [5] P. Tondeur, Foliations on Riemannian Manifolds, Universitext, Springer-Verlag, Berlin, 1988.
Further reading [6] [7] [8] [9]
J.C. Álvarez Paiva, C.E. Durán, Isometric submersions of Finsler manifolds, Proc. Amer. Math. Soc. 129 (2001) 2409–2417. S. Deng, Z. Hou, The group of isometries of Finsler space, Pacific J. Math. 207 (2002) 149–155. A. Mba, J.W. Kamga, Prolongement tangent de feuilletage, Demonstratio Math. 26 (1993) 203–206. M. Popescu, P. Popescu, Projectable non-linear connections and foliations, in: Proc. Summer School on Diff. Geometry, Dep. de Mathemática, Universidade de Comibra, September 1999, pp. 159–165. [10] A. Spiro, Chern’s orthonormal frame bundle on a Finsler space, Houston J. Math. 25 (1999) 641–659. [11] R. Wolak, Normal bundles of foliations of order r, Demonstratio Math. 18 (1985) 994–997.
Lift of the Finsler foliation to its normal bundle A. Miernowski, W. Mozgawa ∗ Institute of Mathematics, Maria Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland Received 19 July 2004; received in revised form 7 January 2005 Available online 21 October 2005 Communicated by O. Kowalski
Abstract E. Ghys in [E. Ghys, Appendix E: Riemannian foliations: Examples and problems, in: P. Molino (Ed.), Riemannian Foliations, Birkhäuser, Boston, 1988, pp. 297–314. [3]] has posed a question (still unsolved) if any Finslerian foliation is a Riemannian one? In this paper we prove that the natural lift of a Finslerian foliation to its normal bundle is a Riemannian foliation for some Riemannian transversal metric. The methods we used here are closely related to those used by M. Abate and G. Patrizio in [M. Abate, G. Patrizio, Finsler Metrics—A Global Approach, Springer-Verlag, Berlin, 1994]. © 2005 Elsevier B.V. All rights reserved. MSC: 53C12 Keywords: Finsler foliation; Natural lift; Good connection
1. Basic facts In this chapter we define a Finslerian foliation in an analogous manner as for Riemannian foliation (cf. [4,5]). Let (W, F ) be a Finsler manifold, where F : T W → R is a Finsler metric (for necessary definitions, see [1,2]). Definition 1.1. A diffeomorphism f : W → W is said to be a Finsler isometry if F f (p), f∗ (v) = F (p, v) for each p ∈ W and v ∈ Tp W . 2 v Example. Let W = R2 \ {(0, 0)} and F (p, v) = 1e p 2 − easy to see that each homothety is a Finsler isometry.
p,v , p2
where p ∈ W , v ∈ Tp W and 0 < e < 1. Then it is
Definition 1.2. A foliated cocycle {Ui , fi , γij } on a manifold M is said to be a Finslerian foliation F if * Corresponding author.
E-mail addresses: [email protected] (A. Miernowski), [email protected] (W. Mozgawa). 0926-2245/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.difgeo.2005.09.008
210
A. Miernowski, W. Mozgawa / Differential Geometry and its Applications 24 (2006) 209–214
(a) {Ui } is an open covering of M, (b) fi : Ui → W is a submersion, where (W, F ) is a Finslerian manifold, (c) γij is a local Finslerian isometry of (W, F ) such that for each x ∈ Ui ∩ Uj fi (x) = (γij ◦ fj )(x). The Finslerian manifold (W, F ) will be called the transversal manifold of foliation F . The local submersions {fi } define by pull-back a Finsler metric in the normal bundle Q = Q(M) of the foliation F , which we shall denote again by F . Note that for any basic vector Y for the foliation and any vector field X tangent to F we have X(F (Y )) = 0. Denote by (x i , y α ), i = 1, . . . , p, α = 1, . . . , q, p + q = dim M a local foliated chart on M and let ( ∂x∂ i , ∂y∂ α ) be a ¯
local frame of T M. If we denote by ( ∂y∂ α ) the corresponding local frame of Q then we can induce a chart (x i , y α , uα¯ ) ¯
on Q, where uα¯ ∂y∂ α is a transversal vector at a point (x i , y α ). Note that in this coordinate system the metric F in Q does not depend on (x i ). Lemma 1.1. A distribution Lin( ∂x∂ 1 , . . . , ∂x∂ p ) defines a foliation FQ on Q called the natural lift of F to Q.
Proof. Let (x i , y α , uα¯ ) be an another coordinate system in Q. Then the lemma follows directly from the transformation rule
x i = x i (x i , y α ),
y α = y α (y α ),
uα¯ =
∂y α α β¯ (y )u . ∂y β
2 ¯
¯
Let Q(Q) denote the bundle over Q transversal to the foliation FQ . The vectors ∂y∂ α and ∂u∂ α¯ , α = 1, . . . , q, form the natural frame of Q(Q) at the point (x i , y α , uα¯ ) ∈ Q. Thus we can take as coordinates in this bundle the following ¯ ¯ functions (x i , y α , uα¯ , Y α , U α¯ ), where (Y α , U α¯ ) are the coefficients of the vector Y α ∂y∂ α + U α¯ ∂u∂ α¯ . The canonical projection π : Q → M, π(x i , y α , uα¯ ) = (x i , y α ) induces an another projection π∗ : T Q → T M which maps the tangent vectors to FQ in the vectors tangent to F . Thus π∗ induces a mapping π˜ ∗ : Q(Q) → Q given in local coordinates by (x i , y α , uα¯ , Y α , U α¯ ) → (x i , y α , Y α ). Note that V = V (Q) = ker π˜ ∗ is a vertical bundle ¯ spanned on the vectors ∂u∂ α¯ , α = 1, . . . , q, and the local coordinates in this bundle are given by (x i , y α , uα¯ , U α¯ ). Corollary 1.1. We have the following rule of transformations of local coordinates in Q(Q)
x i = x i (x i , y α ),
y α = y α (y α ),
∂y α u = α (y α )uα¯ , ∂y α¯
U α¯ =
∂y α α α¯ (y )U . ∂y α
Lemma 1.2. Let o : M → Q(M) be the zero section of the bundle Q(M). Then the set o(M) is saturated on Q(M) with foliation FQ . Proof. Since the leaves of FQ are locally defined by the equations y α = const., uα¯ = const. and the equations uα¯ = 0 are preserved under the coordinate changes then the zero section is the saturated set. 2
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211
˜ = Q(M) ˜ We put Q = Q(M) \ {o}. In the sequel we denote simply by V the restriction of the vertical bundle V ˜ to Q. 2. Vertical connection ˜ ⊗ V ) related to considered The purpose of this section is to define a linear connection ∇ : X (V ) → X (T ∗ (Q) ˜ foliated structure. Since the bundle Q is foliated we are looking for a Bott connection that is such one that for any vector field X tangent to FQ and any transversal vector field Y we have ∇X Y = p [X, Y˜ ] , where p : T Q → Q(Q) is the canonical projection and p(Y˜ ) = Y . Definition 2.1 (cf. [1]). The vector field i : Q → V given by ∂¯ ∂uα¯ is said to be the radial vector field. i(x i , y α , uα¯ ) = uα¯
It can be checked that this definition is well-posed. ˜ ⊗ V ) be a Bott connection. Then ∇X i = 0 for any vector field tangent to FQ . Lemma 2.1. Let ∇ : X (V ) → X (T ∗ (Q) The proof of this lemma is obvious and follows from the definition of the Bott connection. ¯ ¯ ¯ Consider the local frame { ∂x∂ i , ∂y∂ α , ∂u∂ α¯ } of T Q and recall that the vectors { ∂u∂ α¯ } form the basis of V . With these settings we put ¯ ∂¯ β¯ ∂ = Γ , i α ¯ ∂uα¯ ∂uβ¯ ¯ ∂¯ γ¯ ∂ = Γ , ∇ ∂¯ ¯ α β¯ ∂uγ¯ ∂y α ∂uβ ¯ ∂¯ γ¯ ∂ ∇ ∂¯ = Γα¯ β¯ γ¯ . ¯ β ∂u ∂uα¯ ∂u ∇
∂ ∂x i
From the above formulas it follows that β¯
Γi α¯ = 0, ∇
∂¯ ∂uα¯
i=
γ¯ δα¯
β¯
+u
γ¯ Γα¯ β¯
¯ ∂ . ∂uγ¯
The Bott connection ∇ allows us to define a mapping ˜ → X (V ), L : X Q(Q) L(X) = ∇X˜ i, ˜ = X. If we denote by Λ the restriction of the linear mapping L to the bundle V then we can state where p(X) Definition 2.2 (cf. [1]). The connection ∇ is said to be a good vertical connection if Λ : V → V is a bundle isomorphism. γ¯
¯
γ¯
Observe that ∇ is a good vertical connection if and only if the matrix δα¯ + uβ Γα¯ β¯ is nonsingular. If we put H = ker L then we can split the bundle Q(Q) into a direct sum Q(Q) = H ⊕ V . Let {∂α , ∂˙β¯ } be a local frame of
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Q(Q), where ∂α = ¯
∂¯ ˙ ∂y α , ∂β¯
=
∂¯ . ∂uβ¯
Then the coefficients of the mapping L in this basis are
γ¯
L(∂α ) = uβ Γα β¯ ∂˙β¯ , γ¯ ¯ γ¯ γ¯ L(∂˙β¯ ) = δβ¯ + uδ Γβ¯ δ¯ ∂˙γ¯ = Lβ¯ ∂˙γ¯ . β¯
β¯
β¯ ¯
γ¯
It is easy to check that the vectors δα = ∂α − Γα ∂˙β¯ , where Γα = (L−1 )γ¯ uδ Γα δ¯ form a basis of ker L. In the sequel β¯ ¯ ¯ it is convenient to use the basis {δα , ∂˙β¯ } as well as its dual {dy α , ψ β = duβ + Γα dy α }. Using this coframe we can define a local connection forms by
∇ ∂˙β¯ = ωβα¯¯ ⊗ ∂˙α¯ , where ωβα¯¯ = Γγα¯β¯ dy γ + Γγ¯α¯β¯ duγ¯ = Γγα¯β¯ − Γε¯α¯β¯ Γγε¯ dy γ + Γγ¯α¯β¯ ψ γ¯ = Kγα¯ β¯ dy γ + Kγα¯¯ β¯ ψ γ¯ . ¯ Note that Kγα¯ β¯ uβ = Γγα¯ . The formula θ (∂˙α¯ ) = δα defines a linear mapping θ : V → H . This mapping allows us to extend the connection ∇ to the horizontal bundle H by ∇X Z = θ ∇X θ −1 (Z) ,
where Z ∈ X (H ), X ∈ X (T Q). In this way we construct a linear connection in Q(Q) ∇X Y = ∇X ν(Y ) + ∇X Y − ν(Y ) , where Y ∈ X (Q(Q)), X ∈ X (T Q) and ν : Q(Q) → V is the vertical projection. In particular, we have ∇δα = ωαβ ⊗ δβ ,
β¯
ωαβ = ωα¯ .
If ϕ ∈ X (Q∗ (Q) ⊗ Q(Q)) is a 1-form given locally by ¯ ϕ = ϕ α ⊗ δα + ϕ β ⊗ ∂˙β¯
then following [1] we define an exterior differential Dϕ putting ¯ β¯ Dϕ = dϕ α − ϕ γ ∧ ωγα ⊗ δα + dϕ β − ϕ γ¯ ∧ ωγ¯ ⊗ ∂˙β¯ . Simple but long calculations show that the above formula is well-defined. The bundle Q∗ (Q) ⊗ Q(Q) admits a natural section η given by η = dy α ⊗
∂ ∂ ¯ ¯ + duβ ⊗ = dy α ⊗ δα + ψ β ⊗ ∂˙β¯ . ¯ β ∂y α ∂u
It is clear that the form η is well-defined. Definition 2.3. The form ϑ = Dη is called the torsion form of the connection ∇. Locally the form ϑ can be expressed by ¯ ¯ β¯ Dη = −dy α ∧ ωαγ ⊗ δγ + dψ β − ψ α¯ ∧ ωα¯ ⊗ ∂˙β¯ = ϑ γ ⊗ δγ + ϑ β ⊗ ∂˙β¯ , where ϑγ = ¯
1 γ¯ γ¯ γ¯ Kκ α¯ − Kα κ¯ dy α ∧ dy κ − Kκ α¯ dy α ∧ ψ κ¯ , 2 ¯
β¯
β¯
ϑ β = dΓγβ ∧ dy γ − Kκ α¯ ψ α¯ ∧ dy κ − Kκ¯ α¯ ψ α¯ ∧ ψ κ¯ .
A. Miernowski, W. Mozgawa / Differential Geometry and its Applications 24 (2006) 209–214
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3. Finsler foliation In this section we consider the Finsler transversal metric F : Q → R+ ∪ {0} introduced in Section 1. Let G = F 2 ¯ and let (x i , y α , uβ ) be the local coordinates in Q. We put Gα =
∂G , ∂y α
Gβ¯ =
∂G ∂uα¯
then reasoning in the same way as in [1,2] we have ¯
Gα¯ β¯ uα¯ = Gβ¯ ,
Gα¯ β¯ uα¯ uβ = 2G,
Gα¯ β¯ γ¯ uγ¯ = 0. ¯
It is easy to see that for any vertical vectors v = v α¯ ∂˙α¯ , w = w β ∂˙β¯ the formula ¯
v, w = Gα¯ β¯ v α¯ w β
defines a Riemannian metric in the vertical bundle. In the similar way as in [1] we can generalize to the case of Finslerian foliation Theorem 1.4.2 given there. Theorem 3.1. Let F : Q → R+ ∪ {0} be the transversal Finslerian metric and , be the Riemannian metric on V ˜ ⊗ V ) such that induced by F . Then there exists exactly one Bott vertical connection ∇ : X (V ) → (T ∗ (Q) (i) ∇ is a good connection; ˜ then (ii) if v, w ∈ X (V ), X ∈ T Q Xv, w = ∇X v, w + v, ∇X w; (iii) ϑ(v, w) = 0 for any v, w ∈ V ; (iv) ϑ(v, w) ∈ V for any v, w ∈ H . Proof. The proof is very similar to that in [1] and we obtain the following formulas for the coefficients of connections 1 1 ¯ η¯ Kγ α¯ = Gη¯ μ¯ ∂γ (Gμ¯ α¯ ) + ∂α (Gμ¯ γ¯ ) − ∂μ (Gγ¯ α¯ ) − Gη¯ β Γγκ¯ ∂˙κ¯ (Gα¯ β¯ ) + Γακ¯ ∂˙κ¯ (Gβ¯ γ¯ ) − Γβκ¯ ∂˙κ¯ (Gγ¯ α¯ ) , 2 2 1 β¯ δ¯ ˙ δ¯ Kγ¯ α¯ = G ∂γ¯ (Gα¯ β¯ ), 2 1 η¯ β¯ η¯ η¯ Γγ = G ∂α Gβ¯ γ¯ uα¯ + ∂γ Gα¯ β¯ uα¯ − ∂β Gγ¯ α¯ uα¯ − Kκ¯ γ¯ Gαβ ∂γ (Gβ¯ uγ¯ ) − ∂β (G), 2 β¯
Γi α¯ = 0.
2
Observe that the isomorphism θ does not depend of the coordinates along the leaves. It follows that the Riemannian metric in Q(Q) defined by w1 , w2 = θ −1 (w1 ), θ −1 (w2 ) for all w1 , w2 ∈ H, v, w = 0
for all v ∈ V , w ∈ H
is a transversal Riemannian metric for the lifted foliation FQ . Theorem 3.2. The lift FQ of the Finsler foliation F to Q is a Riemannian foliation. References [1] M. Abate, G. Patrizio, Finsler Metrics—A Global Approach, Springer-Verlag, Berlin, 1994. [2] D. Bao, S.-S. Chern, Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, Berlin, 2000.
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[3] E. Ghys, Appendix E: Riemannian foliations: Examples and problems, in: P. Molino (Ed.), Riemannian Foliations, Birkhäuser, Boston, 1988, pp. 297–314. [4] P. Molino, Riemannian Foliations, Progress in Mathematics, vol. 73, Birkhäuser, Boston, 1988. [5] P. Tondeur, Foliations on Riemannian Manifolds, Universitext, Springer-Verlag, Berlin, 1988.
Further reading [6] [7] [8] [9]
J.C. Álvarez Paiva, C.E. Durán, Isometric submersions of Finsler manifolds, Proc. Amer. Math. Soc. 129 (2001) 2409–2417. S. Deng, Z. Hou, The group of isometries of Finsler space, Pacific J. Math. 207 (2002) 149–155. A. Mba, J.W. Kamga, Prolongement tangent de feuilletage, Demonstratio Math. 26 (1993) 203–206. M. Popescu, P. Popescu, Projectable non-linear connections and foliations, in: Proc. Summer School on Diff. Geometry, Dep. de Mathemática, Universidade de Comibra, September 1999, pp. 159–165. [10] A. Spiro, Chern’s orthonormal frame bundle on a Finsler space, Houston J. Math. 25 (1999) 641–659. [11] R. Wolak, Normal bundles of foliations of order r, Demonstratio Math. 18 (1985) 994–997.