Geometrical Objects on Subbundles

Marcela Popescu and Paul Popescu

Abstract The reductions to a vector subbundle of a pull back vector bundle are studied. They are related to the Finsler splittings (defined earlier by one of the authors) and to geometrical objects, defined here.

Mathematics Subject Classification: 57R22, 53C99 Key words: vector bundles, geometrical objects

1 Reductions of vector bundles

Let ξ = (E, π, M) be a vector bundle (denoted in the sequel v.b.), with the fibre F ≅ IR^{n}, G ⊂ GL_{n}(IR) a Lie subgroup and ξ′ = (E′, π′, M) be a vector subbundle (denoted in the sequel v.sb.), with the fibre F′ ≅ IR^{k} ⊂ IR^{n}. Let us denote as L(ξ) = (L(E), p, M) the principal bundles (p.b.s) of the frames of the v.b. ξ and the induced p.b. Lξ′(ξ) = π^{′∗}L(ξ) = (π^{′∗}L(E) = LE′(E), p_{1}, E′), which is also the p.b.s of the frames of the v.b. ξ′(ξ) = π^{′∗}ξ = (π^{′∗}(E) = E′(E), π_{1}, E′).

Definition 1.1 If the p.b. L(ξ)_{G} is a reduction of the group GL_{n}(IR) of L(ξ) to G, then there is a local trivial bundle ξ_{G}, associated with the p.b. L(ξ)_{G}, defined by the left action of G on F (it is used the left action of GL_{n}(IR) on F restricted to G). We say that the bundle ξ_{G} is the G-reduced bundle of ξ. If H is a subgroup of G and there is a reduction of the group G of L(ξ)_{G} to H, in an analogous way we say that ξ_{H} is a H-reduced bundle of ξ_{G}.

Notice that a reduction of the group G of L(ξ)_{G} to H is also a reduction of the group GL_{n}(IR) of L(ξ) to H.

Example 1.1 Consider the subgroup of the automorphisms which invariate the vector subspace F′ ≅ IR^{k} of IR^{n}:

G_{0} =\{( \begin{matrix}A & C \\ 0 & B \end{matrix} ); A ∈ GL_{k}(IR), B ∈ GL_{n−k}(IR), C ∈ M_{k,n−k}(IR)\}⊂GL_{n}(IR). (1)

The p.b. L(ξ)_{G_{0}} always exists and it consists of all the frames of L(ξ) which extend frames on ξ′; we call in the sequel these frames as frames on ξ, adapted to ξ′. For the same G_{0} as above, we can consider the p.b. Lξ′(ξ)_{G_{0}}, which also consists of frames on Lξ′(ξ) which extend frames on ξ′(ξ′), called as frames on ξ′(ξ), adapted to ξ′(ξ′).

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 105-111 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Example 1.2 Let G_{0} be as above, F′′ ≅ IR^{n−k} a vector subspace of F, so that F = F′ ⊕ F′′ and H_{0} ⊂ G_{0} the subgroup of the elements which invariate the vector subspaces F′ and F′′:

H_{0} = \{( \begin{matrix}A & 0 \\ 0 & B \end{matrix} ); A ∈ GL_{k}(IR), B ∈ GL_{n−k}(IR)\} . (2)

The p.b. L(ξ)_{H_{0}} always exists and it is a reduction of the group G_{0} of L(ξ)_{G_{0}} to H_{0}. It consists of frames of L(ξ)_{G_{0}} which are adapted also to other subbundle ξ′′ = (E′′, π′′, M) of ξ. It follows that at every point x ∈ M we have the direct sum of vector spaces E_{x} = E′_{x} ⊕ E′′_{x}. Such a reduction is also called a Whitney sum of the v.b.s ξ′ and ξ′′ and it is denoted as ξ′ ⊕ ξ′′. This is equivalent with a left splitting S of the inclusion morphism i : ξ′ → ξ, when ξ′′ = ker S. In the case of the p.b. Lξ′(ξ), a reduction of the group G_{0} of Lξ′(ξ)_{G_{0}} to H_{0} is equivalent to a left splitting S of the inclusion i′ = π^{′∗}i : π^{′∗}ξ′ = ξ′(ξ′) → π^{′∗}ξ = ξ′(ξ), as we have called Finsler splitting (see. [4]). In this case ξ′(ξ) has an H_{0}-reduction as Whitney sum ξ′(ξ′) ⊕ ker S.

It is well known that the reduction of the structural group G of a p.b. P to a subgroup H ⊂ G is equivalent to the existence of a global section in a fibre bundle associated with P, which have the fibre G/H, defined by the natural action of G on G/H [1, pg.57, Propzition 5.6]. A direct computation leads to the following:

Proposition 1.1 There is a canonical identification

G_{0}/H_{0} ≅ M_{k} = \{( \begin{matrix}0 & P \\ 0 & I_{n−k} \end{matrix} ); P ∈ M_{k,n−k}(IR)\} , (3)

the classes being at left, such that the left action ⊙ of the group G_{0} on M_{k} is the adjunction, and

( \begin{matrix}E & G \\ 0 & F \end{matrix} ) ⊙ ( \begin{matrix}0 & P \\ 0 & I_{n−k} \end{matrix} ) = ( \begin{matrix}0 & (E · P + G) · F^{−1} \\ 0 & I_{n−k} \end{matrix} ) . (4)

Given the v.sb. ξ′, the G_{0}-reductions of the group GL_{n}(IR) of the v.b.s L(ξ) and the p.b. Lξ′(ξ) are uniquely defined. It follows that considering the bundles with the fibres GL_{n}(IR)/G_{0}, associated with the p.b.s of frames L(ξ) and Lξ′(ξ), the sections in these bundles, which correspond to the reductions of GL_{n}(IR) to G_{0}, are uniquely determinated by ξ′. In the case of the Example 1.2, the H_{0}-reductions of the group G_{0} of the p.b.s L(ξ)_{G_{0}} and Lξ′(ξ)_{G_{0}} are equivalent with sections in the bundles F_{1} and F_{2} which are associated with these p.b.s and have as fibres G_{0}/H_{0}.

2 Reductions of the group G_{m,n}^{r}

We use in this section some ideas from [2, Cap. IV, Secţiunea 7], but in a more general setting.

Let ξ = (E, p, M) be a v.b., having the fibre F ≅ IR^{n}, dim M = m and G ⊂ GL_{n}(IR) a Lie subgroup. We suppose that the structural group GL_{n}(IR) of L(ξ) is reducible to G. Denote as G_{m,n}^{1} the subgroup of GL(m + n, IR) which consists of the

matrices of the form ( \begin{matrix}A_{j}^{i} & 0 \\ 0 & B_{b}^{a} \end{matrix} ) , (A_{j}^{i}) ∈ GL_{m}(IR), (B_{b}^{a}) ∈ G. Given r ∈ IN^{∗}, the

r-prolongation of the group G_{n,m}^{1}, denoted as G_{n,m}^{r}, is the set of the elements which have the form

a = (A_{j_{1}}^{i}, A_{j_{1}j_{2}}^{i}, …, A_{j_{1}j_{2}···j_{r}}^{i}; B_{b}^{a}, B_{bj_{1}}^{a}, …, B_{bj_{1}···j_{r−1}}^{a}), (5)

where the components are symmetric in the indices j_{k} ∈ \overline{1, m}, and (A_{j}^{i}) ∈ GL_{m}(IR), (B_{b}^{a}) ∈ G and if we fix j_{1}, j_{2}, … j_{p}, then B_{bj_{1}···j_{p}}^{a} ∈ g, where g is the Lie algebra of the Lie group G. The composition law of two elements of the form (5) can be done looking at the cmponents as multi-linear maps. Thus, let a : (A, B) = ((A_{1}(·), A_{2}(·, ·), …, A_{r}(·, …, ·), B_{0}(·), , B_{1}(·, ·), …, B_{r}(·, ·, …, ·))), and b : (C, D) be two elements in G_{n,m}^{r}. We denote ba : (A′, B′), where the expresion of this composition law, using coordinates, can be found in [2, pag. 70]. Notice that if H ⊂ G is a subgroup, then H_{m,n}^{r} ⊂ G_{m,n}^{r} is a subgroup.

Consider the p.b. OGξ^{r} on the base E, with the group G_{n,m}^{r}, defined by the structural functions

φ_{UU′}(u) = (\frac{∂x^{i′}}{∂x^{i}} (x), …, \frac{∂^{r}x^{i′}}{∂x^{i_{1}} ⋯ ∂x^{i_{r}}} (x), g_{a}^{a′}(x), …, \frac{∂^{r−1}g_{a}^{a′}}{∂x^{j_{1}} ⋯ ∂x^{j_{r−1}}} (x)), (6)

where π(u) = x, and {g_{a}^{a′}(x)} are structural functions of the vertical bundle V ξ,

constant on the fibres, defined on an open cover of E of the form {U = π^{−1}(V), V ⊂ M, V open}. These structural functions proceed from some structural function on M, so the definition is coherent and is equivalent to that used in [2]. We shall use the condition in this form in order to construct some reductions. In the case G = GL_{n}(IR) we get the definitions used in [2].

Let ξ′ be a v.sb. of the v.b. ξ. We denote OGξ′(ξ)^{r} = i^{∗}OGξ^{r}, where i : E′ → E is the inclusion. From now to the end of the section we study the reductions of the group G_{0m,n}^{r} of the p.b. OG_{0}ξ′(ξ)^{r} to the subgroup H_{0m,n}^{r}, where G_{0} and H_{0} are given by the formulas (1) and (2). We consider first the case r = 1.

Proposition 2.1 Let ξ′ be a v.sb. of the v.b. ξ. Then a Finsler splitting S of the inclusion i : ξ′ → ξ induces a H_{0}-reduction of the v.b. ξ′(ξ) to a bundle ξ′(ξ′) ⊕ η, where η = ker S is isomorphic with ξ′(ξ′′), ξ′′ = ξ/ξ′, such that the bundle ξ′(ξ′) ⊕ η is isomorphic to the bundle ξ′(ξ′ ⊕ ξ′′). Conversely, every H_{0}-reduction of ξ′(ξ) as ξ′(ξ′) ⊕ η defines a Finsler splitting S of the inclusion i, such that η = ker S.

Proof. The second statement follows from Example 1.2. In order to prove the first statement, it suffices to prove that η is isomorphic with ξ′(ξ′′). Considering local coordinates, adapted to the v.b.s structures: on M, E′, E′′ and E, it can be shown that η is isomorphic with ξ′(ξ′′). The same reason shows that ξ′(ξ′) ⊕ η is isomorphic with ξ′(ξ′) ⊕ ξ′(ξ′′) ≅ ξ′(ξ′ ⊕ ξ′′). Q.e.d.

Theorem 2.1 Let ξ′ be a v.sb. of the v.b. ξ and r ≥ 1

1) Every Finsler splitting of the inclusion i : ξ′ → ξ defines a canonical reduction of the group G_{0m,n}^{r} of OG_{0}ξ′(ξ)^{r} to H_{0m,n}^{r}, the reduced p.b. being OH_{0}ξ′(ξ)^{r}.

2) Every reduction of the group G_{0m,n}^{r} of OG_{0}ξ′(ξ)^{r} to H_{0m,n}^{r} is OH_{0}ξ′(ξ)^{r} and it is induced by a Finsler splitting, as above.

Proof. 1) Taking on E′(E) a vectorial atlas, which has the structural functions from H_{0}, we obtain structural functions on the p.b. OG_{0}ξ′(ξ)^{r}, which thake values in the subgroup H_{0m,n}^{r}. 2)Considering a reduction as in hytothesis and some structural functions on OH_{0}ξ′(ξ)^{r}, as in the definition, it follows some structural functions on the p.b.s OH_{0}ξ′(ξ)^{r′}, with 1 ≤ t′ ≤ r. Taking r′ = 1 and using the second part and the proof of Propsition 2.1, we obtain a Finsler splitting S. Q.e.d.

3 Geometrical objects

Let ξ be a v.b., G ⊂ GL_{n}(IR) a Lie subgroup and ξ′ a v.sb. of ξ.

Definition 3.1 A space of geometrical G−objects of order r is a manifold Θ so that there is a left action of the group G_{n,m}^{r} on Θ. Consider now the fibre bundle with the fibre Θ, associated with the p.b. OGξ^{r}, which correspond to this action. A section in this bundle is a field of geometrical G−objects on the v.b. ξ.

In an analogous way, we can consider the fibre bundle with the fibre Θ , associated with the principal bundle OGξ^{r} A section in this bundle is a field of geometrical G−objects on the v.b. ξ, restricted to the v.sb. ξ′.

In the case G = GL_{n}(IR) we obtain the definitions used in [2] of a space of geometrical objects of order r and of a field of geometrical objects on a v.b..

Example 3.1 Take Θ = IR^{k}, G_{0} given by the formula (1) and the left action ofi G_{0}

on Θ given by ( \begin{matrix}A & C \\ 0 & B \end{matrix} ) v = Av. It is obvious that this action induces a left action

of the group G_{0m,n}^{1} on Θ. It follows a field of geometrical G_{0}-objects of order 1 on the v.b. ξ, which is in fact a section in the v.b. ξ(ξ′) and a field of geometrical G_{0}-objects of order 1 on the v.b. ξ, restricted to the v.sb. ξ′, which is in fact a section in the v.b. ξ′(ξ′).

The second part of the example above, can be extended as follows:

Proposition 3.1 Every field of geometrical objects of order r ≥ 1 on the v.b. ξ′ defines canonicaly a field of geometrical G_{0}-objects of order r on the v.b. ξ, restricted to the v.sb. ξ′.

Example 3.2 A d-connection on the v.b. ξ′ induces a field of geometrical G_{0}-objects of order 2 on the v.b. ξ, restricted to the v.sb. ξ′.

Notice that a field of geometrical G_{0}-objects on the v.b. ξ, restricted to the v.sb. ξ′ is also such H_{0}-objects. So, the fields of geometrical G_{0}-objects from Proposition 3.1 and from Example 3.2 are also H_{0}-objects. A remarcable example of a field of geometrical G_{0}-objects of order k on the v.b. ξ, restricted to the v.sb. ξ′, is given by the following direct consequence of the result [1, pg.57, Propzition 5.6], already stated and used in the first section:

Proposition 3.2 Let G_{0} and H_{0} be given by (1) and (2), and ξ′ be a v.sb. of the v.b. ξ. Then every reduction of the group H_{0m,n}^{r} of the p.b. OG_{0}ξ′(ξ)^{r} to H_{0m,n}^{r} is uniquely defined by a field of geometrical G_{0}-objects on the v.b. ξ, restricted to the v.sb. ξ′, of order r.

Example 3.3 Every Finsler splitting S of the inclusion i : ξ′ → ξ is defined by a field of geometrical G_{0}-objects on the v.b. ξ, restricted to the v.sb. ξ′, of order 1. According he above Proposition 3.2, this G_{0}-object is a section S_{0} in the bundle F_{0}, which is associated with the p.b. OG_{0}ξ′(ξ)^{1} and defined by the left action of G_{0m,n}^{1} on G_{0}/H_{0} (in the form (3), determinated by the left action of G_{0} on G_{0}/H_{0} given by formula 4. This formula can be related to the change rule of the local components of a Finsler splitting, which give also the change rule of the local form of the section S_{0}.

The action (4) can be extended in the general case, but this will be done elsewhere. An example of a field of geometrical H_{0}-objects on the v.b. ξ, restricted to the v.sb. ξ′, of order 2, is given by the following action of H_{0m,n}^{2} on F_{0} = IR^{d}, where d = m^{3} + mk^{2} + m(n − k)^{2} + m^{2}n + k^{2}n + (n − k)^{2}n. Writting F_{0} as (L_{jl}^{i}, L_{βi}^{α}, L_{vi}^{u}, C_{jα}^{i}, C_{ju}^{i}, C_{βγ}^{α}, C_{βu}^{α}, C_{va}^{u}, C_{vw}^{u}), we define the action of H_{0m,n}^{2} on F_{0}, by

an element (A_{j}^{i}, A_{jk}^{i}, ( \begin{matrix}B_{α}^{α′} & 0 \\ 0 & B_{u}^{u′} \end{matrix} ) , ( \begin{matrix}B_{αi}^{α′} & 0 \\ 0 & B_{ui}^{u′} \end{matrix} )), as

L_{j′l′}^{i′} = (A_{i}^{i′}L_{jl}^{i} − A_{jl}^{i′}) A_{j′}^{j}A_{l′}^{l} , L_{β′i′}^{α′} = (A_{α}^{α′}L_{βi}^{α} − A_{βi}^{α′}) B_{β′}^{β}A_{i′}^{i} ,
L_{v′i′}^{u′} = (B_{u}^{u′}L_{vi}^{u} − B_{vi}^{u′}) B_{v′}^{v}A_{i′}^{i}, C_{j′α′}^{i′} = A_{i}^{i′}A_{j′}^{j}B_{α′}^{α}C_{jα}^{i} ,
C_{ju′}^{i′} = A_{i}^{i′}A_{j′}^{j}B_{u′}^{u}C_{ju}^{i}, C_{β′γ′}^{α′} = B_{α}^{α′}B_{β′}^{β}B_{γ′}^{γ}C_{βγ}^{α} ,

C_{β′u′}^{α′} = B_{α}^{α′}B_{β′}^{β}B_{u′}^{u}C_{βu}^{α}, C_{v′α′}^{u′} = B_{u}^{u′}B_{v′}^{v}B_{α′}^{α}C_{vα}^{u} , C_{v′w′}^{u′} = B_{u}^{u′}B_{v′}^{v}B_{w′}^{w}C_{vw}^{u}.

It follows that there is a local trivial bundle F_{1} with the fibre F_{1}, associated with the p.b. OH_{0}ξ′(ξ)^{2}.

Let S be the Finsler splitting which correspomd to the reduction of the group G_{0} of OG_{0}ξ′(ξ)^{1} to H_{0}, according to Theorem 2.1.

Definition 3.2 A restricted d-connection on ξ (related to the v.sb. ξ′ and the Finsler splitting S) is a section in the above bundle F_{1}.

If follows that a restricted d-connection is uniquely determinated by the local functions on E′(E)

(L_{jl}^{i}, L_{βi}^{α}, L_{vi}^{u}, C_{jα}^{i}, C_{ju}^{i}, C_{βγ}^{α}, C_{βu}^{α}, C_{vα}^{u}, C_{vw}^{u}) (7)

which have as variables (x^{i}, y^{α}). They are given on domains of local maps on E′, which belong to a vectorial atlas on E′, which proceed from one on E′. The coordinates on the fibres change following the rules y^{α′} = g_{α}^{α′}(x^{i})y^{α} +g_{u}^{α′}(x^{i})y^{u}, y^{u′} = g_{u}^{u′}(x^{i})y^{u}.The local functions (7) change according the rules

L_{j′l′}^{i′} = \frac{∂x^{i′}}{∂x^{i}} (\frac{∂x^{j}}{∂x^{j′}} \frac{∂x^{l}}{∂x^{l′}} L_{jl}^{i} + \frac{∂^{2}x^{i}}{∂x^{j′}∂x^{l′}} ) , L_{β′i′}^{α′} = g_{α}^{α′} (g_{β′}^{β} \frac{∂x^{i}}{∂x^{i′}} L_{βi}^{α} + \frac{∂g_{β′}^{α}}{∂x^{i′}} ) ,

L_{v′i′}^{u′} = g_{u}^{u′} (g_{v′}^{v} \frac{∂x^{i}}{∂x^{i′}} L_{vi}^{u} + \frac{∂g_{v′}^{u}}{∂x^{i′}} ) , C_{j′α′}^{i′} = \frac{∂x^{j}}{∂x^{j′}} \frac{∂x^{i′}}{∂x^{i}} g_{α′}^{α}C_{jα}^{i}

C_{j′u′}^{i′} = \frac{∂x^{j}}{∂x^{j′}} \frac{∂x^{i′}}{∂x^{i}} g_{u′}^{u}C_{ju}^{i}, C_{β′γ′}^{α′} = g_{α}^{α′}g_{β′}^{β}g_{γ′}^{γ}C_{βγ}^{α} , C_{v′w′}^{u′} = g_{u}^{u′}g_{v′}^{v}g_{w′}^{w}C_{vw}^{u}.

If we compare the above formulas with those of a d-connection on the v.b. i^{∗}ξ [2, ec. (7.5), pag. 72], we obtain:

Theorem 3.1 Let ξ′ be a v.sb. of the v.b. ξ, N a non-linear connection on ξ and S a Finsler splitting of the inclusion i : ξ′ → ξ.

1) Every linear d-connection on ξ defines canonicaly a restricted d-conection on ξ related to ξ′.

2) Every restricted d-connection on ξ related to ξ′ defines canonicaly a d-conection on ξ′ and a linear Finsler ξ′-connection on ξ′′ = ξ/ξ′.

3) A d-connection on ξ′ and a linear Finsler ξ′-connection on ξ′′ = ξ/ξ′, defines canonicaly, using the Finsler splitting S, a restricted d-conection on ξ related to ξ′.

The proof of the theorem will be given elsewhere. For the definition of a linear Finsler ξ′-connection on a v.b. ξ′′, over the same base, see [4].

Acknowledgements. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

References

[1] S. Kobayashi, K. Nomizu Foundation of differential geometry, I, Wiley Interscience, NY, 1963.

[2] R. Miron, M. Anastasiei Vector bundles, Lagrange spaces. Application to Relativity, (in Romanian), Editura Academiei, Bucureşti, 1987; in English, Balkan Society of Geometers, Monographs and Textbooks, 1, Geometry Balkan Press, Bucharest, Romania, 1997.

[3] Marcela Popescu, P. Popescu The structure equations of induced non-linear connections, The National Seminar on Finsler spaces, Brasov, 1996 (to appear in Analele Univ. Timisoara).

[4] Marcela Popescu On Finsler bundles and Finsler connections, The 25^{th} National Conference of Geometry and Topology, Iassy, 1995 (to appear in Analele Univ. Iassy).

Department of Mathematics, University of Craiova 11, Al.I.Cuza St., Craiova, 1100, Romania e-mail paul@udjmath2.sfos.ro