# Xerrades

Trobada de Tardor de Geometria i Física

Encuentro de Otoño de Geometría y Física

Fall Workshop on Geometry and Physics

(special session)

Vilanova i la Geltrú, 6-8 July 2000

### Abstracts of the talks

- José A. de Azcárraga (U. València)
*Superspaces, cohomology of superalgebras and the formulation of actions for supersymmetric extended objects* - José F. Cariñena (U. Zaragoza)
*Lie systems: differential equation systems admitting a superposition rule* - Manuel F. Rañada (U. Zaragoza)
*Symplectic formalism, symmetries, and super-integrable sytems* - Pedro Luis García (U. Salamanca)
*Lagrangian reduction and constrained variational calculus* - Óscar García Prada (U. Autónoma de Madrid)
*The geometry of coupled equations in gauge theory* - Olga Gil Medrano (U. València)
*Gradients and hessians of geometric functionals* - Joan Girbau (U. Autònoma de Barcelona)
*Linearization stability of the Einstein equation in the presence of matter* - Alberto Ibort (U. Carlos III, Madrid)
*Multisymplectic manifolds: general aspects and particular situations* - Manuel de León (CSIC, Madrid)
*Media with microstructure: constitutive theory and dynamics* - Antonio López Almorox (U. Salamanca)
*Geometry of the magnetic flow on complete riemannian manifolds and its quantization* - Franco Magri (U. Milano)
*Separable systems of Stackel and Gelfand-Zakharevich* - Giuseppe Marmo (U. Napoli)
*Quantum bihamiltonian systems* - Juan Carlos Marrero (U. La Laguna)
*Lie algebroids and Jacobi structures* - Mariano Santander (U. Valladolid)
*A self-dual approach to trigonometry in symmetric spaces* - Martintxo Saralegi Aranguren (U. Artois)
*Cohomology of riemannian flows* - Jaremir Tosiek (U. Lodz)
*The Weyl-Wigner-Moyal formulation of quantum mechanics in curved phase spaces*

#### José A. de Azcárraga

*Superspaces, cohomology of superalgebras and the formulation of actions for supersymmetric extended objects*

The relevance of Lie (super)algebra cohomology in its Chevalley-Eilenberg formulation is shown in the formulation of supersymmetric extended objects (p-branes).

#### José F. Cariñena

*Lie systems: differential equation systems admitting a superposition rule*

We will illustrate by means of simple examples the applications in physics of the result of the theorem by Lie and Scheffers's concerning the characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of a set of arbitrary, but independent, particular solutions, and some constants determining each solution.

The main idea consists on associating the equation on the given carrier space with an equation on a Lie group $G$ of transformations of a manifold $M$, with the given Lie algebra. Assuming the group to be written in terms of matrices and denoting by $g$ a generic element of the group, we would have

$$ \dot g\, g^{-1} = f^a\, \tau_a $$

with

$$ [\tau_a,\tau_b] = c_{ab}^m\, \tau_m , $$

$\{ \tau_k \mid k=1,\ldots,r \}$ being a basis of the Lie algebra. Thus, the original equation is then replaced by a corresponding equation on the group $G$.

We will emphasize by means of different examples the *universal character* of these equations, which may arise on the group $G$ either from *classical systems* or from *quantum dynamical systems*.

It will also be shown the usefulness of a generalization of the method proposed by Wei and Norman for solving these Lie-Scheffers systems.

The method will be illustrated with several examples of physical relevance.

[1] Cariñena J.F., Marmo G. and Nasarre J., "The non-linear superposition principle and the Wei--Norman method", Int. J. Mod. Phys. A 13, 3601--27 (1998).

[2] Cariñena J.F. and Ramos A., "Integrability of Riccati equation from a group theoretical viewpoint", Int. J. Mod. Phys. A 14, 1935--51 (1999).

[3] Cariñena J.F., Grabowski J. and Marmo G., {\sl Lie--Scheffers systems: a geometric approach}, Bibliopolis, Napoli, 2000.

[4] Cariñena J.F. and Nasarre J., "Lie-Scheffers systems in Optics", J. Optics B 2, 94--99 (2000)

[5] Cariñena J.F. and Ramos A., "Shape invariant potentials depending on $n$ parameters transformed by translation", J. Phys. A.: Math. Gen. 33, 3467--81 (2000)

[6] Cariñena J.F. and Ramos A., "Riccati equation, factorization method and shape invariance", Rev. Math. Phys. 12, (2000) (to appear).

[7] Cariñena J.F., Grabowski J. and Ramos A., "Reduction of time-dependent systems admitting a superposition principle", Acta Appl. Math., (2000) (to appear).

[8] Lie S. and Scheffers G., *Vorlesungen über continuierlichen Gruppen mit geometrischen und anderen Anwendungen*, Teubner, Leipzig, 1893.

[9] Wei J. and Norman E., "Lie algebraic solution of linear differential equations", J. Math. Phys. 4, 575--81 (1963).

#### Manuel F. Rañada

*Symplectic formalism, symmetries, and super-integrable sytems*

This talk can be considered as divided in two parts. In the first part, the theory of symmetries of a Hamiltonian system (dynamical symmetries, Noether symmetries, and Cartan symmetries) is analyzed using the symplectic formalism approach. Noether symmetries are regarded here as "projectable" Cartan symmetries, that is, Cartan symmetries induced by point transformations. It is known that the (generalized) Noether Theorem states that each (infinitesimal) Cartan symmetry determines a constant of the motion. The results are extended to the theory of higher-order symmetries.

A Hamiltonian system is called superintegrable if it is integrable in the Arnold-Liouville sense and, in addition, possesses more independent constants of motion than degrees of freedom. In particular, if a system with $n$ degrees of freedom possesses $2n-1$ independent first integrals, then it is called maximally superintegrable. In the second part, the properties of several superintegrable systems (as e.g., the Calogero-Moser system) are studied using as an approach the theory of dynamical symmetries and the theory of higher-order symmetries.

#### Pedro Luis García

*Lagrangian reduction and constrained variational calculus*

A general formalism for constrained variational problems defined by reducing free variational problems with reducible lagrangian is presented. The theory is illustrated with several typical examples (electromagnetism, Euler-Poincaré reductions, relativistic fluids, etc).

#### Óscar García Prada

*The geometry of coupled equations in gauge theory*

This talk will be devoted to certain differential equations which naturally emerge in the study of gauge theory on Kähler manifolds. These equations, which involve connections and Higgs fields, lead to moduli spaces whose geometry I will try to describe.

#### Olga Gil Medrano

*Gradients and hessians of geometric functionals*

The aim of this talk is to show how the use of riemannian metrics can be very helpful to understand, and to solve, many different variational problems.

At a first level, given finite dimensional manifolds $M$ and $N$, we can use any riemannian metric on $M$ to describe the natural structure of topological vector space of several spaces of tensor fields over $M$. Now, any riemannian metric on $N$ can be used to construct an atlas on $C^\infty(M,N)$ modeled over vector spaces of this type.

On the other hand, on $C^\infty(M,N)$ --and on any submanifold $\Cal S$-- we can define a weak riemannian metric, using again metrics on $M$ and $N$. This weak riemannian metric allows to give a concept of gradient of a functional $F: \Cal S \subset C^\infty(M,N) \to \R$; if such a gradient exists, then ${\operatorname{Grad}}F = 0$ is the Euler-Lagrange equation of the variational problem associated to $F$, that is, the condition characterizing critical points. We can also define a concept of weak gradient --and of critical point-- of a map $f: \Cal S \subset C^\infty(M,N) \to C^\infty(M)$. One of the advantages of this approach is that the equation that characterizes critical points of the restriction of the map to different submanifolds can be obtained by projecting its gradient. This riemannian point of view has been useful to see the relationship between several functionals and also to understand some properties of the Euler-Lagrange equation of certain functionals.

Concerning the second variation of a functional $F$ as above, the use of weak riemannian metric on $\Cal S$ is helpful to compute the hessian on critical points, because we only need to compute the differential of the gradient, and to deal with variational problems with constraints, by taking the adequate projections. But the use of a metric, or at least a connection, in the manifold of maps is completely necessary if we want to compute the hessian of a functional outside its critical points. To do so, a deeper knowledge of the geometry of this infinity dimensional manifold is needed. By the moment, the case better known is that of certain natural semimetrics in the manifold of nondegenerate $2$-covariant tensor fields, and some of its submanifolds --those consiting on all riemannian semimetrics with fixed signature, among others. To finish, we will show how to use the Levi-Civita connection of these metrics to study the second variation of a functional related with scalar curvature.

#### Joan Girbau

*Linearization stability of the Einstein equation in the presence of matter*

In general relativity the equation governing gravitation is the Einstein's one: $G(g)=\chi T$, where $G$ is the Einstein tensor of $g$, $G(g)=\mbox{Ric}(g)-(1/2)Rg$, $T$ is the stress-energy tensor and $\chi$ is a constant. The Minkowski metric $\eta$ in ${\bf R}^4$ fulfils this equation in the vacuum (when $T=0$).In case $T$ is small Einstein wrote the metric $g$ in the form$g=\eta+h$ with $h$ small. Then he realized that the linear terms in $h$ in the equation $G(\eta+h)=\chi T$ were strongly related with the classical wave equation (gravitational waves). In the seventies a lot of papers appeared trying to answer the following question: under what mathematical conditions is it licit to linearize the Einstein equation? Most of them (Y. Choquet-Bruhat, S. Deser, A. Fischer, J.E. Marsden) concerned the vacuum. But little work was done in case of non-empty spaces.Recently L. Bruna and myself have studied the linearization stabilityof the Einstein equation in Robertson-Walker cosmological models and proved that such a model is stable in case of null curvature and unstable in case of positive curvature. The talk will be about these topics.

#### Alberto Ibort

*Multisymplectic manifolds: general aspects and particular situations*

Multisymplectic geometry is not a well defined body of knowledge. One of the reasons for this is that, contrary to the situation in the symplectic case, the generic and exceptional multisymplectic geometries are not well understood. We will present some observations that make these points clear and could eventually help to stablish the subject.

#### Manuel de León

*Media with microstructure: constitutive theory and dynamics*

An elastic medium with internal structure is modelled as a principal bundle moving and deforming into another principal bundle (the ambient space). The manifold of embeddings is just the configuration space and becomes another principal bundle whose structure group is just the gauge group. In this talk we show as the dynamics splits in two parts: the macromotion of the macromedium and the micromotion of the internal structure. Elastoplastic and viscoelastic phenomena will be also discussed.

#### Antonio López Almorox

*Geometry of the magnetic flow on complete riemannian manifolds and its quantization*

We analize some geometric properties of the so-called magnetic flow on a complete riemnannian manifold $(M,g)$. Namely, we will study the geometry of the trajectories of a charged particle under the effect of a magnetic field described by a closed 2-form $F$. In particular, we shall concentrate on Kaehler magnetic flows on manifolds of constant holomorphic sectional curvature due to their connection with the Landau-Hall problem.

Some dynamical properties of the magnetic flow can be given in terms of the magnetic Jacobi fields along the solutions. Using the magnetic-Dombrowski decomposition for $T(TM)$, some properties of the vector space of magnetic Jacobi fields along a magnetic curve can be given. In particular, we shall solve the magnetic Jacobi equation for uniform magnetic fields on concrete examples to illustrate the influence of the curvature and the strength of the magnetic field on the stability of the trajectories.

All these results can be used to look for invariant (under the magnetic flow) polarizations on the symplectic manifold $(TM,\Omega^{[F]})$ in order to give a geometric quantization of this problem.

#### Franco Magri

*Separable systems of Stackel and Gelfand-Zakharevich*

A geometrical scheme for the study of the separation of variables for the Hamilton-Jacobi equation associated with Hamiltonian systems on the Poisson manifolds is suggested. The scheme encompasses the classical theory of Stäckel, Eisenhart, and Levi-Civita for the quadratic Hamiltonians on the cotangent bundle of a Riemannian manifold. It also allows to deal with certain classes of Gelfand-Zakharevich systems on bihamiltonian manifolds. The talk is a report on a joint work with G. Falqui and M. Pechoni.

#### Giuseppe Marmo

*Quantum bihamiltonian systems*

In analogy with the classical case, we define bihamiltonian systems as derivations in operator algebras which are inner derivations with respect to alternative multiplicative associative structures on the space of observables. We introduce an associative version of Nijenhuis tensors. We also consider the Hilbert space version in terms of alternative Hermitian products. Some examples are used to illustrate the situation.

#### Juan Carlos Marrero

*Lie algebroids and Jacobi structures*

In this talk, we will show some relations between the theory ofLie algebroids and the theory of Jacobi structures. Roughly speaking, a Lie algebroid over a manifold M is a vector bundle $E$ on M such that its space of sections $\Gamma(E)$ has a structure of Lie algebra plus a mapping (the anchor map) from $E$ on $TM$ which provides a Lie algebra homomorphism from $\Gamma(E)$ into the Lie algebra of vector fields on $M$. On the other hand, a Jacobi structure on a manifold $M$is a local Lie algebra (in the sense of Kirillov) on the spaceof $C^{\infty}$ real-valued functions on $M$. Jacobi manifolds are natural generalizations of symplectic, Poisson and contactmmanifolds.

In the first part of the talk, we will exhibit the relation betweenLie algebroid structures on a vector bundle $E$ and linearJacobi structures on the dual bundle $E^*$. Some interesting examples will be given.

In the second part of the talk, we will show some classical resultsabout the relation between Poisson manifolds (resp. Lie-Poisson groups) and Lie bialgebroids (resp. Lie bialgebras). Extensions of these results to the Jacobi setting will be also presented.

#### Mariano Santander

*A self-dual approach to trigonometry in symmetric spaces*

The talk will be mainly devoted to introducing a new self-dual approach to trigonometry in rank-one symmetric spaces. The context is a viewpoint, also to be described, which relates the full classification of symmetric homogeneous spaces to the division algebras.

The method encapsulates into a single group equation the whole trigonometry of the rank-one spaces of constant (holomorphic) curvature and metric of any (even degenerate) signature over either the reals, complex, quaternions and octonions. Thus it could be described as *`curvature/signature (in)dependent trigonometry'*, and its distinctive traits are `universality' and`self-duality': every equation is meaningful for nine (three signs for curvature, three types of signature) spaces at once, and displays explicitly invariance under a duality transformation.

In a sense this brings to its logical end the Bolyai idea of an `absolute trigonometry', and allows to cover with ease the whole `spectrum' from the `high-level' single basic group equation to the down-to-earth *bestiarium* of specific trigonometric equations, both in the real and in the complex hermitian cases as well.

Physically, most of these rank-one spaces are very relevant; for instance this approach gives the trigonometry for the DeSitter and antiDeSitter space-times (where the presence of horizons comes out as an analogue of the hyperbolic angle of paralelism).

The complex case is specially interesting, because the trigonometry of quaternionic and octonionic type spaces reduce to the complex one. As far as I know, this approach to `hermitian trigonometry' is completely new, yet it is physically relevant, because the elliptic hermitian space is the Quantum Space of States, whose trigonometry can thus be studied. A substantial part of the talk will be devoted to this `hermitian trigonometry'.

At the end of the talk, some comments on the as yet unknown trigonometry of grassmannians will be also given.

This work is being done in collaboration with F.J. Herranz and R. Ortega.

#### Martintxo Saralegi Aranguren

*Cohomology of riemannian flows*

Given a smooth action of the circle ${\mathbb S}^1$ on a manifold $M$ the deRham cohomologies of $M$ and of the orbit space $B$ are related by a long exact sequence

$$\cdots \to H^i(B,F) \to H^{i+2}(B) \to H^{i+2}(M) \to H^{i+1}(B,F) \to \cdots,$$

called the *Gysin sequence*. Here $F$ denotes the submanifold of fixed points. Notice that, although $B$ is a singular manifold, we still can talk about the deRham cohomology of $B$ by using the controlled differential forms of Verona

A first generalization of this formula is obtained by considering a smooth action $\Phi \colon {\mathbb R} \times M \to M$ preserving a riemannian metric $\mu$ on $M$, that is, an *isometric action*. Since the orbit space can be very wild (even totally disconnected!!) the right cohomology to study the transverse structure is the *basic cohomology* $H^*(M/\mathcal{F})$ of the flow determined by the action. Of course, when the action is periodic we are in the previous case and moreover $H^*(M/\mathcal{F}) = H^*(B)$. In this context we have constructed the following Gysin sequence

$$\cdots \to H^i((M,F)/\mathcal{F}) \to H^{i+2}(M/\mathcal{F}) \to H^{i+2}(M) \to H^{i+1}((M,F)/\mathcal{F}) \to \cdots.$$

A second generalization of this formula is obtained by considering a smooth action $\Phi \colon {\mathbb R} \times M \to M$ preserving not a riemannian metric $\mu$ on $M$, but just the restriction of $\mu$ to the normal bundle of $\mathcal{F}$, that is, a *riemannian action*. In this context we have constructed the following Gysin sequence

$$\cdots \to H^i_\kappa ((M,F)/\mathcal{F}) \to H^{i+2}(M/\mathcal{F}) \to H^{i+2}(M) \to H^{i+1}_\kappa ((M,F)/\mathcal{F}) \to \cdots,$$

where $\kappa$ is the mean curvature of the flow and $H_\kappa^*((M,F)/\mathcal{F})$ is the twisted basic cohomology

This work has been done in collaboration with J.I. Royo Prieto (Univ. of the Basque Country).

#### Jaremir Tosiek

*The Weyl-Wigner-Moyal formulation of quantum mechanics in curved phase spaces*

The short review of the formulation of quantum mechanics on the classical phase space is given. The definition and the main properties of the Moyal product "*" in the R^2n space are presented. Problems with using star product "*" in the curvilinear coordinates are illustrated. The bundle of Weyl algebras is constructed. The relations between geometry of the Weyl bundle and quantum product "*" are shown. The opportunity of using computer in computations is presented. The possible physical generalisations of the Moyal product are given.