On Lagrange Epimorphisms and Lagrange Submersions

M. Popescu and P. Popescu

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract The aim of the paper is to define and study some aspects of Lagramge epimorphisms, which are generalizations for vector bundles of Lagrange submersions. We construct a Lagrangian on f^{∗}ξ′′ canonically associated with a Lagrange vector bundle ξ and an f_{0}-epimorphism of vector bundles f : ξ → ξ′′. The main result is that a Lagrangian L on ξ has a locally f-projectable metric on ξ′′ iff it is a local Lagrange submersion. All the definitions and results in the paper can be stated in particular for Lagrange submersions.

Mathematics Subject Classification: 58F05 Key words: Lagrangian, Legendre transform epimorphism, submersion

A Lagrangian on the vector bundle ξ = (E, π, M) is a real and differentiable function L : E → IR. A regular Lagrangian on ξ is a Lagrangian L which has Hess(L) non-degenerate.. All the Lagrangians considered in the sequel are regular. Notice that every Lagrangian on ξ define a (pseudo)metric on the fibres of the vertical bundle V ξ. In local coordinates, if (x^{i}) and (x^{i}, y^{a}) are coordinates on M and E respectively, then Hess(L) is a bilinear form on the vertical bundle V ξ, which has the matrix:

g_{bc}(x^{i}, y^{a}) = \frac{∂^{2}L}{∂y^{b}∂y^{c}} .

The Legendre transform associated with the Lagrangian L on ξ is:

L : E → E^{∗}, L(x^{i}, y^{a}) = (x^{i}, \frac{∂L}{∂y^{a}} (x^{i}, y^{a})) .

If L is regular, then L is a local diffeomorphism. Since most of our constructions are local, we can suppose, without loss of generality, that L is a global diffeomorphism.

In the sequel ξ = (E, π, M) and ξ′′ = (E′′, π′′, M′′) are two vector bundles, f_{0} : M → M′′ is a submersion and f : ξ → ξ′′ is an f_{0}-epimorphism (i.e. f is surjective on fibers). The restriction of the differential f_{∗} to the vertical bundle V ξ ⊂ τE defines an f-(epi)morphism of vector bundle

Balkan Journal of Geometry and Its Applications, Vol.5, No.2, 2000, pp. 97-101 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(1)  F : V ξ ≅ π^{∗}ξ → V ξ′′ ≅ (π′′)^{∗} ξ′′

If (ξ, L) is a Lagrange space, the vertical distribution and horizontal distribution of f are V f = ker F and Hf = (ker F)^{⊥} respectively, where the orthogonal is taken according the Lagrange metric, assuming that it exists. Notice that there is a canonical Whitney sum decomposition V ξ = V f ⊕ Hf.

We say that the Lagrangian L on ξ has an f- projectable metric if there exists a metric on V ξ′′such that the f -epimorphism F : V ξ → V ξ′′ is Riemannian, i.e. the fibers of Hf are isometric with the fibers of V ξ′′.

We say that f : ξ → ξ′ is a Lagrange epimorphism if there is a Lagrangian on ξ′′ such that the induced f-epimorphism F : V ξ| → V ξ′′ is Riemannian on fibres.

We use in the sequel local coordinates adapted to the submersion f_{0} and the epimorphism f: (x^{i}) = (x^{u}, xū) on M, (xū) on M′′, (x^{u}, xū, y^{a}, yā) on E and (xū, yā) on E′′, such that f_{0} and f has the local forms (x^{u}, xū) → (xū) and (x^{u}, xū, y^{a}, yā) → (xū, yā) respectively. We denote as {s_{u}, sū} and {s′′ ; ū} the local bases of sections in the vector bundles V ξ and V ξ′′ respectively. If g is a (pseudo)metric tensor on V ξ, then we denote as {g_{ij}} = {g_{uv}, g_{ūv} = g_{vū}, g_{ūv̄}} its components using the above base. Notice that {s_{u}} is a local base of sections in V f and {s̄ū = sū − g̃^{uv}g_{vū}s_{u}} is a local base of sections in Hf, where (g̃^{uv}) = (g_{uv})^{−1} as matrices. It is easy to see that we have g(s̄ū, s̄v̄) = g_{ūv̄} − g_{ūu}g̃^{uv}g_{vv̄}.

The annihilator of the vector subbundle V f ⊂ ξ is a vector subbundle V f^{∗} ⊂ ξ^{∗}, defined by the linear forms in ξ^{∗} which are null on vectors in V ξ. If (x^{u}, xū, p_{a}, pā) and (xū, pā) are local coordinates on E^{∗} and É^{′′∗} respectively, then the local coordinates on V f^{∗} are (x^{u}, xū, pā) and the inclusion V f^{∗} ⊂ ξ^{∗} has the local form (x^{u}, xū, pā) → (x^{u}, xū, pā, 0). Since L is a diffeomorphism, then W E = L^{−1}(V f^{∗}) ⊂ E is a submanifold. Notice the manifold W E has as coordinates (x^{u}, xū, Q^{u}(x^{u}, xū, yā), yā), such that

(2)  \frac{∂L}{∂y^{a}} (x^{u}, xū, Q^{a}(x^{u}, xū, yā), yā) = 0.

Let us observe that f_{|W E} : W E → f_{0}^{∗}E′′ is a diffeomorphism, where f_{0}^{∗}E′′ is the total space of the induced bundle:

M →^{f_{0}} M′′
↑ π′′
E′′

Indeed, the local form of f_{|W E} is

(3)  (x^{u}, xū, Q^{a}(x^{u}, xū, yā), yā) → (x^{u}, xū, yā).

Let us define

S = f_{|W E}^{−1} : f_{0}^{∗}E′′ → W E

and

L̄′′ = L ∘ S : π^{∗}E′′ → IR.

In local coordinates

L̄′′(x^{u}, xū, yā) = L(x^{u}, xū, Q^{u}(x^{u}, xū, yā), yā).

Notice that The map S is a section of the fibered manifold defined by the epimorphism E →^{f_{0}^{∗}f}

f_{0}^{∗}E′′ of vector bundles over M.

The couple (f_{0}^{∗}ξ′′, L̄′′) is a Lagrange vector bundle and we call L̄′′ the canonical Lagrangian on f_{0}^{∗}ξ′′ (induced by L and f).

Consider again the f-epimorphism (1), denote as I : W E → E the inclusion and consider the vector bundle I^{∗}Hf (i.e. the fibres of Hf along the submanifold W E).

Lemma 1 The restriction F_{|W E} : I^{∗}Hf → f_{0}^{∗}ξ′′ is an f_{|W E}-isomorphism which is an isometry on fibres according to the Lagrangians L_{|I^{∗}Hf} and L̄′′ respectively.

Proof. We use local coordinates. The local form of f_{|W E} is given by (3). The local correspondence of bases of sections by mean of f_{|W E} are s̄ā = sā− g̃^{ab}g_{bā}s_{a} → s′′ ; ā. We have

(4)  g(s̄ā, s̄b̄) = gāb̄ − g_{āa}g̃^{ab}g_{bb̄}

and

ḡ′′(s′′ ; ā, s′′ ; b̄) = \frac{∂^{2}L̄′′}{∂yā∂yb̄} ,

where

\frac{∂^{2}L̄′′}{∂yā∂yb̄} (x^{u}, xū, yā) = \frac{∂^{2}}{∂yā∂yb̄} L(x^{u}, xū, Q^{a}(x^{u}, xū, yā), yā).

But

\frac{∂}{∂yb̄} L(x^{u}, xū, Q^{a}(x^{u}, xū, yb̄), yb̄) = \frac{∂L}{∂yb̄} (x^{u}, xū, Q^{b}(x^{u}, xū, yb̄), yv̄)

thus

\frac{∂^{2}}{∂yā∂yb̄} L(x^{u}, xū, Q^{a}(x^{u}, xū, yā), yā) = \frac{∂^{2}L}{∂yā∂yb̄} (x^{u}, xū, Q^{v}(x^{u}, xū, yv̄), yv̄)+

+\frac{∂Q^{c}}{∂yā} (x^{u}, xū, yā)\frac{∂^{2}L}{∂yū∂y^{c}} (x^{u}, xū, Q^{a}(x^{u}, xū, yā), yā) = gāb̄ + \frac{∂Q^{c}}{∂yā} g_{b̄c}.

Differentiating partially the relation (2) with respect to yā we obtain that

\frac{∂Q^{a}}{∂yā} = −g̃^{ab}g_{bā},

where

(g̃^{ab}) = (g_{ab})^{−1}.

Thus

ḡ′′(s′′ ; ā, s′′ ; b̄) = gāb̄ − g_{āa}g̃^{ab}g_{bb̄}.

Comparing with relations (4), the conclusion follows. □

Notice that using a linear algebra computation we obtain the following equality of matrices:

(gāb̄ − g_{āa}g̃^{ab}g_{bb̄}) = (g^{āb̄})^{−1} .

Using the Lemma 1 we obtain:

Proposition 1 If f is a Lagrange epimorphism then ξ →^{f_{0}^{∗}f} f_{0}^{∗}ξ′′ is a Lagrange epimorphism, where the canonical Lagrangian L̄′′ is considered on f_{0}^{∗}ξ′′.

Using also Lemma 1 we obtain:

Proposition 2 Let f : ξ → ξ′′ be an epimorphism of vector bundles over the same base and L be a Lagrangian on ξ which is non-degenerate on V f and has an fprojectable metric.

Then f is a Lagrange submersion.

We say that a Lagrangian L̄′′ on f_{0}^{∗}ξ′′ projects on ξ′′ if there is a Lagrangian L′′ on ξ′′ such that L̄′′ = L′′ ∘h, where h : f_{0}^{∗}ξ′′ → ξ′′ is the canonical f_{0}-morphism which is the identity on fibers.

We are going to give sufficient conditions in order to induce a Lagrange submersion.

Proposition 3 Let f : ξ → ξ′′ be an f_{0}-epimorphism and L a Lagrangian on ξ which is non-degenerate on V f. Consider on f_{0}^{∗}ξ′′ the canonical Lagrangian.

If ξ →^{f_{0}^{∗}f} f_{0}^{∗}ξ′′ is a Lagrange epimorphism and the canonical Lagrangian on f_{0}^{∗}ξ′′ is projectable on ξ′′ then there is a Lagrangian L′′ on ξ′′ such that f is a Lagrange submersion.

Proof. Since h∘f_{0}^{∗}f = f, the conclusion follows using the composition of the Lagrange epimorphisms

ξ →^{f_{0}^{∗}f} f_{0}^{∗}ξ′′ →^{h} ξ′′,

which is a Lagrange epimorphism. □

Lemma 2 Let f_{0} : M → M′′ be a submersion, ξ′′ = (E′′, π′′, M′′) be a vector bundle and g : f_{0}^{∗}ξ′′ → ξ′′ be the canonical f_{0}-morphism of vector bundles.

Assuming that a Lagrangian L̄′′ on f_{0}^{∗}ξ′′ has a g-projectable metric on ξ′′, then L̄′′ projects locally on ξ′′.

Proof. Consider local coordinates (x^{u}, xū) on M, (xū) on M′′ and (xū, yā) on ξ′′, thus g has the local form (x^{u}, xū , yā) → (xū, yā). The condition that L̄′′ has a g-

projectable metric on ξ′′ reads that the local functions\frac{∂^{2}L̄′′}{∂yā∂yb̄} do not depend on x^{u}.

It follows that there are local and real functions L′′ : ξ′′_{U′′} → IR, where U′′ ⊂ M′′ and ξ′′_{U′′} is the restriction of the vector bundle ξ′′ to U′′, such that

\frac{∂^{2}L̄′′}{∂yā∂yb̄} (x^{u}, xū, yā) = \frac{∂^{2}L′′}{∂yā∂yb̄} (xū, yā).

□

Using Proposition 3 and Lemma 2 we obtain the main result:

Theorem 1 Let f : ξ → ξ′′ be an f_{0}-epimorphism and L a Lagrangian on ξ which is non-degenerate on V f.

Then L has a locally f-projectable metric on ξ′′ iff it is a local Lagrange submersion.

Let L : T M → IR be a regular Lagrangian on M and f : M → M′′ be a surjective submersion. Notice that all the definitions and the results stated above for vector bundles apply in this case, for example:

f is a Lagrange submersion if f_{∗} is a Lagrange submersion;

A Lagrangian L on M is f-projectable if it is f_{∗}-projectable.

The case when f is flat (i.e. the horizontal distribution Hf_{∗} is integrable) is need to Lagrange foliations (see [4]).

A theory of Lagrange submersion, in an analogous way of Riemannian submersion, will be done elsewhere. Aknowledgements. A version of this paper was presented at the Third Conference of Balkan Society of Geometers, Workshop on Electromagentic Flows and Dynamics, July 31 - August 3, 2000, University POLITEHNICA of Bucharest, Romania

Supported by MEN Grant No 21815/28.09.1998, CNCSU-31.

References

[1] Miron R., Anastasiei, M. The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Acad. Publ., 1994.

[2] Popescu Marcela, On the supplementary vector subbundles, Proc. of the Workshop on Global Analysis, Diff. Geom., Lie Alg., Thessaloniki (1995), Balkan Press, Bukarest, 1997, 65-70.

[3] Popescu Marcela, Semisprays induced on submanifolds, New developments in Diff. Geom., Kluwer Academic Publ., 1998, 317-328.

[4] Popescu Marcela and Popescu P., Projectable Non-linear Connections and Foliations, Proc. Summer School in Diff. Geometry, Coimbra, 1999, 159-166.

University of Craiova Department of Mathematics 11, Al.I. Cuza St., Craiova, 1100, Romania e-mail:paul@udjmatj2.sfos.ro