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)

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

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′).

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 dconection 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