The XI Edition of the International Conference on problems and trend of contemporary geometry

Current Geometry Lloyd’s Baia Hotel Vietri sul Mare (SA), Italy June 16–20, 2010

Leevvii-Ci -Ci vviitt a Inst it u utt e a nno u un ce s L nces

International Conference on problems and trends of contemporary geometry

Lloyd’s Baia Hotel, Vietri sul Mare (Salerno), Italy.

June 16-21, 2010.

Organized by Alexandre Vinogradov, Giuseppe Marmo, Gaetano Vilasi, Giovanni Sparano, Luca Vitagliano, Giovanni Moreno.

The power of synthesis of Geometry, which led in the past to the formulation of “grand unification theories”, has got an essential role nowadays, especially because of the growing fragmentation of knowledge due to scientific progress. In order to avoid too a big dispersion, geometers need a constant dialogue. Therefore, a stable experience of personal meetings, apart from telematic interchanges, cannot be renounced. Current Geometry was born to allow a periodic update about actual progresses in Geometry (and its applications) on the international scene.

Supported by G.N.S.A.G.A. (Italy); Dipartimento di Matematica ed Informatica (Salerno University, Italy); Dipartimento di Matematica e Applicazioni “R. Caccioppoli” (Naples University “Federico II”, Italy); Istituto Italiano per gli Studi Filosofici (Naples, Italy).

For more information visit http://www.levi-civita.org/Activities/Conferences/xicurrentgeometry

Program Wednesday, June 16 13:30—14:30 lunch 15:30—16:30 A. Vinogradov Chair: 16:30—17:00 coffe break G. Marmo 17:00—18:00 J. Kijowski 18:00—18:30 G. Moreno 20:00 dinner Thursday, June 17 9:30—10:30 G. Marmo 10:30—11:00 coffe break Chair: 11:00—12:00 D. Martin De Diego N. Poncin 12:00—13:00 G. Vilasi 13:30—14:30 lunch 15:30—16:30 F. Bogomolov 16:30—17:00 coffe break Chair: 17:00—17:30 K. S. Chahal J. Grabowski 17:30—18:00 O. G¨ ursoy 20:00 dinner Friday, June 18 9:30—10:30 G. Rudolph 10:30—11:00 coffe break Chair: 11:00—12:00 A. Kotov F. H´elein 12:00—13:00 A. Sossinsky 13:30—14:30 lunch FREE AFTERNOON 20:00 dinner Saturday, June 19 9:30—10:30 N. Poncin 10:30—11:00 coffe break Chair: 11:00—12:00 P. Damianou J. Kijowski 12:00—13:00 F. H´elein 13:30—14:30 lunch 15:30—16:00 L. Vitagliano 16:00—16:30 E. Vishniakova Chair: 16:30—17:00 coffe break F. Bogomolov 17:00—17:30 D. Malinin 17:30—18:00 A. K. Meraj 20:00 social dinner Sunday, June 20 9:30—10:30 D. Gurevich 10:30—11:00 coffe break Chair: 11:00—12:00 J. Grabowski A. Sossinsky 12:00—13:00 G. Gaeta 13:30—14:30 lunch

Titles and Abstracts of the Talks F. Bogomolov Geometry of complex surfaces and closed symmetric differentials The notion of a closed symmetric differential is a natural generalization of the notion of a closed differential. I will discuss topological implications of the existence of closed differentials on a complex surface.

K. S. Chahal Warped product CR–submanifolds of a T–manifold In the present talk, I will discuss about the existence and non-existence of warped product CR–submanifolds of a T–manifold and obtained a characterization result for the warped product of this type.

P. Damianou A Poisson approach to ADE singularities We give a brief general review of the ADE classification problem. The survey includes simple Kleinian singularities, symmetries of Platonic solids, finite subgroups of SU (2), the Mckay correspondence, integer matrices of norm 2 and Brieskorn’s theory of subregular orbits. We conclude with some joint work with H. Sabourin and P. Vanhaecke on transverse Poisson structures to subregular orbits in semisimple Lie algebras. We show that the structure may be computed by means of a simple Jacobian formula, involving the restriction of the Chevalley invariants on the slice. In addition, using results of Brieskorn and Slodowy, the Poisson structure is reduced to a three dimensional Poisson bracket, intimately related to the simple rational singularity that corresponds to the subregular orbit.

G. Gaeta Twisted versus gauge symmetries We discuss how gauge transformations modify the symmetry properties of differential equations; it turns out standard symmetries in a given different gauge correspond to the so called “twisted symmetries” studied in recent years.

D. Gurevich Braided geometry and its applications By Braided Geometry I mean that related to a braiding, i.e. a special type solution of the Quantum Yang–Baxter equation. In the frameworks of this geometry certain notions of standard geometry (traces, Lie algebras, vector 2

fields) are replaced by their braided counterparts. In my talk I plan to deliver a shot review of basic notions and recent results (including applications to q– analogs of some dynamical models).

J. Grabowski Courant brackets, Dirac structures, and generalized geometries The Courant-Dorfman bracket, the concept of Dirac structure, and general Courant algebroids are introduced in the classical and super–geometrical setting. We also study Nijenhuis tensors N on Courant algebroids which are compatible with the pairing. For the Courant–Dorfman bracket this leads to Poisson– Nijenhuis and presymplectic–Nijenhuis structures in the sense of Magri and Morosi, and to generalized geometries with the Hitchin generalized complex geometry as a particular example.

O. G¨ ursoy The dual invariant of a line surface and space kinematics As known the geometry of a trajectory surfaces tracing by an oriented line (spear) is important in line geometry and spatial kinematics. Until early 1980’s, although two real integral invariants, the pitch of angle and the pitch of a trajectory surface were known, any dual invariant of the surface were not. Because of the deficiency, line geometry wasn’t sufficiently studied by using dual quantities. A global dual invariant of a closed trajectory surface is introduced and shown that there is a magic relation between the real invariants. It gives suitable relations, such as or and which have the new geometric interpretations of trajectory surface where is the measure of the spherical area on the unit sphere described by the generator of closed trajectory surface and and are the distribution parameters of the principal surfaces of the closed congruence. Therefore all the relations between the global invariants, and of c.t.s. are worth reconsidering in view of the new geometric explanations. Thus, some new results and new explanations are gained. Furthermore, as a limit position of the surface, some new theorems and comments related to space curves are obtained. References [1 ] Gursoy O., The Dual Angle of A Closed Ruled Surface, Mech. Mach. Theory, 25 (2), 131-140, 1990. [2 ] Gursoy O., On Integral Invariant of A Closed Ruled Surface, Journal of Geometry, vol.39, 80-91, 1990, S.W. [3 ] Gursoy O., Some Results on Closed Ruled Surfaces and Closed space Curves, Mech.Mach.Theory, 27, (1990), 323-330. [4 ] Gursoy O., Kucuk A., On the Invariants of Trajectory Surfaces, Mech.Mach. Theory, 34, (1999), 587-597.

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F. H´ elein A geometric framework for conservation laws and their charges Many dynamical equations in fields theories or some applications of Noether’s second theorem can be expressed in terms of a covariantly closed form with values in a vector bundle with connection. We present some speculations about a possible geometric framework to interpret and integrate these relations. This is based on the notion of solder (or soldering form) in differential geometry. This leads naturally to imagine a family of new geometries which has not yet a satisfactory definition in general. I’ll try anyway to explain through some examples and partial results that such a theory could make sense.

J. Kijowski Energy of gravitational field: a quasilocal, hamiltonian approach The notion of energy (mass) carried by the gravitational field will be thoroughly discussed. The non–localizability of gravitational energy will be related with the very essence of gravitational interaction. The “quasi–local” approach (proposed, among others, by R. Penrose in 1982) will be used. A general geometric setup for defining the quasi-local mass will be discussed and various implementations of this idea (due e.g. to Hawking, Brown–York, Yau and the present author) will be ralated to specific boundary–value problems for Einstei equations.

A. Kotov Superconnections in geometry This talk is to show the use of superconnections in real, complex, and quaternionic geometry. The exposition is illustrated by many important examples including A∞ –functors, direct images, and Atiyah–Drinfeld–Hitchin–Manin construction of instantons.

D. Malinin On finite arithmetic groups and their geometric applications The problems below originate from classification problems of positive definite quadratic lattices and their isometries. There is a number of applications to Finite Group Schemes, Arithmetic Algebraic Geometry and Galois cohomology. Let E/F be a Galois extension of finite degree of global fields, i.e. E, F are finite extensions of the field of rationals Q or a field of rational functions R(x) with a finite field R. Let us denote by OE and OF the maximal orders of E and F , and let Γ be the Galois group of E/F . Let E = F (G) be a field obtained via adjoining to F all matrix coefficients of all matrices g ∈ G for some finite subgroup G ⊂ GLn (E).

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We are interested in 3 basic conditions for the Γ-operation on G and the integrality of G. A. G is Γ-stable under the natural Galois operation. B. G ⊂ GLn (OE ). C. A primitive t-root of 1 ζt 6∈ E. We intend to discuss the following questions: 1. Do the conditions A., B. imply G ⊂ GLn (F Eab ), where Eab is the maximal abelian subextension of E/Q? 2. Do the conditions A., B., C. imply G ⊂ GLn (F )? 3. Classify the possible fields E = F (G). The case F = Q, the field of rationals, is specially interesting. The following theorem was proven using the classification of finite flat group schemes over Z, the ring of rational integers, annihilated by a prime p obtained by V. A. Abrashkin and J.- M. Fontaine: Theorem 1. Let K/Q be a normal extension with Galois group Γ, and let G ⊂ GLn (OK ) be a finite Γ-stable subgroup. Then G ⊂ GLn (OKab ) where Kab is the maximal abelian over Q subfield of K. Similar results for totally real extensions K/Q were considered earlier. In this case there are some interesting arithmetic applications to positive definite quadratic lattices and Galois cohomology. Theorem 2. Let F be a global field of a positive characteristic p, and let E be a splitting field of some irreducible polynomial f (y) ∈ F [y] whose roots are the conjugates of some element t ∈ E. Then E = F (G) for any positive integer n and an appropriate group G ⊂ GLn (E). Moreover, if t ∈ E is an element of OE , then G ⊂ GLn (OE ).

G. Marmo Geometry and quantum mechanics The standard formulation of Quantum Mechanics associates a Hilbert space with every quantum systems.The probabilistic interpretation requires, however, that quantum (pure) states should be identified with points of a corresponding complex projective space. This space turns out to be a Hilbert manifold and no more a Hilbert space, this implies, among other things, that the notion of operator does not apply any more and linear transformation should be replaced by diffeomorphisms which respect a Poisson Bracket and a probability–transition. When we consider also generic density states, convex combinations of pure states, the resulting state space carries a Poisson Bracket 5

and a compatible transition probability. In analogy with the notion of symplectic realization of a Poisson manifold we introduce the notion of Hermitian realization of Poisson spaces with probability transition. We show that these realizations are equivalent with the usual construction of Hilbert spaces associated with C-star algebras via the Gelfan–Naimark–Segal construction.

D. Martin De Diego Some new developments in nonholonomic dynamics: continuous and discrete In this talk, we will review some new aspects in nonholonomic mechanics. We will study the underlying geometry with special attention to the HamiltonJacobi equation for nonholonomic systems and the construction of nonholonomic integrators.

A. K. Meraj Totally Umbilical pseudo–slant submanifolds of nearly K¨ ahler manifold A. Carriazo defined the notion of bi–slant submanifold of an almost hermitian manifold, as a special case of these submanifolds, he introduced the notion of pseudo–slant submanifold. We have studied some differential geometric aspects of totally umbilical pseudo-slant submanifolds of a nearly K¨ahler manifold, that have led to the classification of totally umbilical pseudo–slant submanifolds of a nearly K¨ ahler manifold.

G. Moreno Deformations of 3–dimensional real Lie algebra structures The problem of classifying 3–dimensional real Lie algebras, originally solved by L. Bianchi at the end of the eighteen century, was recently recast in a more elegant coordinate–free manner by several authors. However, none of these works allows a satisfactory description of deformations of 3–dimensional Lie algebras. In this talk I will introduce the algebraic variety Lie(V ) of all Lie algebra structures on a 3–dimensional vector space V , and describe it in a geometrically transparent way. In this pictures, linear deformations of a fixed Lie structure will appear as a special subvariety of Lie(V ), called the compatibility variety, and their equivalence will be defined in terms of a natural action of special subgroups of GL(V ). All compatibility varieties, and some corresponding to them moduli spaces of deformations, will be described, almost without computations, by using elementary aspects of the differential calculus over the symmetric algebra of V ∗ , in particular Poisson geometry. In the case of real Lie algebras, I will also show the effect of some deformations over the symplectic foliation. Reference: http://arxiv.org/abs/1005.5355v1

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N. Poncin Quantization of singular spaces Equivariant quantization is a new geometric theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces... In this talk, I intend to provide a survey on equivariant quantization and recent developments in this field, as well as to investigate existence of an equivariant quantization map for orbifolds. Our quantization, which commutes with reduction, combines an appropriate desingularization of a Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in a joint earlier work with F. Radoux and R. Wolak.

G. Rudolph On singular reduction and quantization of gauge models The models under consideration arise from lattice approximation of nonabelian gauge theories. First, we formulate them on the classical level as Hamiltonian systems, endowed with a gauge symmetry and with a natural momentum map. We discuss singular Marsden–Weinstein reduction, which yields the stratified reduced phase space, for simple examples [2]. Next, we shortly recall previous results [1] on the structure of the field and observable algebras of these models. In particular, there is a unique (generalized) Schr¨odinger representation. For implementing the stratified structure on quantum level [3] we use the generalized Bargmann–Segal transform for compact Lie groups as developed by B. C. Hall and the concept of a costratified Hilbert space as proposed by J. Huebschmann. The image of the Schr¨odinger representation space under the above transformation is the space of square integrable holomorphic functions on the complexification of the group manifold, endowed with a measure, which can be either obtained from heat kernel analysis or from half form K¨ahler quantization. We discuss a simple, exactly solvable example. Finally, we comment on the structure of the observable algebra for the same toy model [4]. It is a C ∗ –algebra generated by unbounded elements (obtained from geometric quantization of classical invariants) in the sense of Woronowicz. References [1 ] J. Kijowski, G. Rudolph, J. Math. Phys. Vol. 46, 032303 (2005) [2 ] E. Fischer, G. Rudolph and M. Schmidt, J. Geom. Phys. 57, 1193 (2007) [3 ] J. Huebschmann, G. Rudolph and M. Schmidt, Commun. Math. Phys. 286, 459 (2009) [4 ] G. Rudolph, M. Schmidt, J. Math. Phys. 50, 052102 (2009)

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A. Sossinsky Moduli spaces of planar mechanical linkages with two degree of freedom Planar mechanical linkages (or hinge mechanisms in another terminology) are a classical subject of mathematics and engineering going back to the end of the 19th century, with important work by James Watt, Maxwell, Cayley, Peaucellier, and Chebyshev. It was revived in the 1980ies by Thurston, who shifted the focus of interest to the study of configuration spaces (or moduli spaces) of linkages, and motivated many other researchers to work in this field, e.g. Connelly, King, Mnev, D.Zvonkine, Kapovich & Millson, Steiner, and the speaker. At present, the main area of application of the theory is robotics, in particular in the work of Farber (and many others). In the talk, I will concentrate on the study of the geometry of planar hinge mechanisms with two degrees of freedom. The mechanisms studied are the pentagon, which is a closed chain of five rectilinear links joined by hinges, one of the links being fixed on the plane, and the n-legged spider, which consists of a central hinge (the “body” of the spider) to which n “legs” (of two links joined by a mobile hinge, the extremity of each leg being a hinge fixed in the plane) are attached. A complete classification of moduli spaces of pentagons (independently obtained by many authors, including D. Zvonkine, Kapovich & Millson, Steiner & Curtis) will be described, in a version due to the speaker’s MS student, A. Kondakova. In this version, the classification of the 19 different moduli spaces is obtained via a kind of Vassiliev finite–type invariant defined on the parameter space of all pentagons; this approach leans heavily on the work of Kapovich & Millson, and involves Morse surgery of surfaces arising when ones crosses the discriminant in the parameter space. The moduli spaces of n–legged spiders are much more complicated than that of pentagons (actually, pentagons can be regarded as two–legged spiders). No complete classification result can be expected even for n = 3, however, a number of nontrivial results about n–legged spiders have been obtained by A.Kondakova, in particular, a finiteness theorem (for each n, there is only a finite number of different moduli spaces of generic n–legged spiders), and upper and lower bounds for the Euler characteristics, the number of connected components, the number and types of singularities of moduli spaces of n–legged spiders. It follows from these results that for any positive integer N there exists a spider whose moduli space is an orientable surface of genus M > N , but such values of M for large N are very scarce; in particular, there is no spider whose moduli space is a surface of genus 10. In conclusion, it will be explained why this talk may be interesting to experts in the differential calculus on R-algebras in the sense of A.M.Vinogradov. Namely, the speaker regards the kinematics and dynamics of nongeneric planar hinge mechanism as a testing ground for applying this calculus to the motion of these mechanisms in the vicinity of singular points. A first challenge would be to write a differential equation describing the motion of such a mechanism driven,

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say, by the force of gravity in the neighborhood of a singular point, e.g. a cone singularity (for nongeneric pentagons) or even a transversal self-intersection singularity in the simplest case of linkages with one degree of freedom.

G. Vilasi Einstein metrics with 2-dimensional Killing leaves and their physical interpretation Solutions of vacuum Einstein’s field equations, for the class of Pseudo–Riemannian 4–metrics admitting a non Abelian 2–dimensional Lie algebra of Killing fields, are explicitly described. When the distribution orthogonal to the orbits is completely integrable and the metric is not degenerate along the orbits, these solutions are parameterized either by solutions of a transcendental equation (the tortoise equation), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a 3–dimensional Lie algebra of Killing fields with 2–dimensional leaves. A formalism (j–complex analysis), allowing one to construct global Einstein metrics by matching together these solutions, is also described. It is shown that the remaining metrics, corresponding to the case in which the commutator of the two Killing fields is isotropic, represent nonlinear gravitational waves obeying to two nonlinear superposition laws. The energy and the polarization of this family of waves are explicitely evaluated; it is shown that they have spin–1 and their sources are also described.

E. Vishnyakova On complex homogeneous supermanifolds It is well known that the category of real Lie supergroups is equivalent to the category of the so called (real) Harish-Chandra pairs (Kostant). That means that a Lie supergroup depends only on the underlying Lie group and its Lie superalgebra with certain compatibility conditions. More precisely, the structure sheaf of a Lie group and the group morphisms can be explicitly described in terms of the corresponding Lie superalgebra. We will give a very simple proof of this result in the real and complex-analytic case. Further, a complex homogeneous V supermanifold may be non-split, i.e. the structure sheaf is not isomorphic to E, where E is a locally free sheaf. We will discuss necessary and sufficient conditions for a complex homogeneous supermanifold to be split. As an aplication of these two results we will give necessary conditions for a complex compact homogeneous supermanifold to have no non-constant holomorphic functions.

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L. Vitagliano Lagrangian-Hamiltonian higher derivative field theory We generalize the Lagrangian-Hamiltonian formalism of Skinner and Rusk to field theories whose action depends on derivatives of arbitarily high order. As a byproduct we solve the long standing problem of defining, in a coordinate free manner, a Hamiltonian formalism for Lagrangian field theories of arbitrarily high order. Namely, our formalism does only depend on the action functional and, therefore, unlike previously proposed ones, is free from any relevant ambiguity.

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