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# Giovanni Manno

INdAM-COFUND Outgoing fellow between 2012-07-01 and 2014-06-30

### Project information

#### Title

GMVMADS: Natural Hamiltonian systems admitting polynomial first integrals

#### Outgoing host organisation Return host organisation

Friedrich-Schiller-Universitaet Jena Politecnico di Torino

07737 Jena, Germany Corso Duca degli Abruzzi, 24

10129 Torino, Italy

http://www.uni-jena.de/ http://www.polito.it/

#### Abstract

The present project aims at solving some classical problems in Differential Geometry that are relevant also in Mathematical Physics.

In 1882 Sophus Lie formulated the problem of finding all 2-dim. metrics admitting a projective vector field, i.e. a vector

field whose local flow preserves geodesics without necessarily preserving the affine parameter. The set of the projective vector ields is a Lie algebra. Such problem has been tackled by the applicant, R. L. Bryant and V. Matveev and solved in the case when the projective action is locally transitive. However, a complete answer is still missing and it is an objective of the project.

The above problem lies into the more general problem of describing all “natural” Hamiltonian systems (i.e. the Hamiltonian is given by the sum of kinetic and potential energy) possessing some symmetries (we underline that, in the case when the

potential in equal to zero, the Hamiltonian system is the geodesic flow of some metric). An intriguing and hard problem would be to describe natural Hamiltonian systems admitting first integrals that are polynomial in momenta. Such problem is classical as its formulation dates back at least to Darboux. An objective of the project is to obtain a local description of the aforementioned systems at least in the case they admit a cubic integral.

### Fellow information

#### Research interests

- Metrics admitting geodesic symmetries
- Contact geometry of Monge-Ampère equations
- Symmetries, conservation laws and Baecklund transformations of PDEs
- Finite and infinite-dimensional integrable systems

#### Previous positions, awards

*September 2004-August 2008*: Research Fellow at the Department of

Mathematics “E. De Giorgi” , University of Salento*September 2008-June 2012*: Research Fellow at the Department of

Mathematics and Applications , University of Milano-Bicocca

#### Publications, preprints, other works

- S. Igonin, J. Van De Leur, G. Manno, V. Trushkov,
*Infinite-dimensional Prolongation Lie Algebras and*, J. Geom. Phys., 68 no. 8 (2013) 1-26.

Multicomponent Landau-Lifshitz Systems Associated with Higher Genus Curves - G. Manno, G. Metafune,
*On the extendability of conformal vector fields of 2-dimensional manifolds*, Differential Geom. Appl., 30 (2012), 365-369. - D.V. Alekseevsky, R. Alonso Blanco, G. Manno, F. Pugliese,
*Monge-Ampère equations on (para-)Khäler manifolds:*, Acta Appl. Math., 120, no. 2 (2012), 3â27.

from characteristic subspaces to special Lagrangian submanifolds - D.V. Alekseevsky, R. Alonso Blanco, G. Manno, F. Pugliese,
*Contact geometry of multidimensional Monge-Ampère*, Annales de l’Institut Fourier (Grenoble), 62,

equations: characteristics, intermediate integrals and solutions

no. 2 (2012), 497-524. - R. Alonso Blanco, G. Manno, F. Pugliese,
*Normal forms for lagrangian distributions on*, Differential Geom. Appl., 27 (2009), no. 2,

5-dimensional contact manifolds

212â229. - R. L. Bryant, G. Manno, V. S. Matveev,
*A solution of a problem of Sophus Lie: normal*, Math. Ann., 340, no. 2 (2008), 437-463.

forms of two-dimensional metrics admitting two projective vector

fields - G. Manno,
*On the geometry of Grassmannian equivalent connections*, Adv. Geom., 8 (2008) 329-342. - G. Manno, J. Pohjanpelto, R. Vitolo,
*Gauge invariance, charge conservation, and*, J. Geom. Phys., 58 no. 8 (2008) 996-1006.

variational principles - G. Manno, R. Vitolo,
*Geometric aspects of higher order variational*, Acta Appl. Math., 101 (2008) 215-229.

principles on submanifolds - R. Alonso Blanco, G. Manno, F. Pugliese,
*Contact relative differential invariants for*, Acta Appl. Math., 101 (2008) 5-19.

non-generic parabolic Monge-Ampère equations - G. Manno, F. Oliveri, R. Vitolo,
*On differential equations characterized by their*, J. Math. Anal. and Appl., 332, no. 2 (2007), 767-786.

Lie point symmetries - G. Manno, F. Oliveri, R. Vitolo,
*Differential equations uniquely determined by*, Theoret. and Math. Phys., 151, no. 3 (2007), 843-850.

algebras of point symmetries - D. Catalano Ferraioli, G. Manno, F. Pugliese,
*Generalised symmetries of partial differential*, Bull. Austr. Math. Soc., 76 (2007), 243-262.

equations via complex transformations - D. Catalano Ferraioli, G. Manno, F. Pugliese,
*Contact symmetries of the elliptic Euler-Darboux*, Note Mat., no. 23, 2 (2004), 3â14.

equation - G. Manno, R. Vitolo,
*Relativistic mechanics, cosymplectic manifolds and symmetries*, Note Mat., no. 23, 2 (2004), 157-171. - G. Manno, F. Oliveri, R. Vitolo,
*Differential equations and Lie symmetries*, Wascom 2007, 14th Conference on Waves and Stability

in Continuous Media, World Sci. Publ., Hackensack, NJ, (2008),

459-468. - G. Manno, F. Oliveri, R. Vitolo,
*On the correspondence between differential*, Symmetry and Perturbation Theory 2007, Featured

equations and symmetry algebras

contribution, World Scientific Publishing, Hackensack, NJ, (2008),

164â171. - R. Alonso Blanco, G. Manno, F. Pugliese,
*Contact geometry of parabolic Monge-Ampère*, Symmetry and Perturbation Theory 2007, World

equations

Scientific Publishing, Hackensack, NJ, (2008). - G. Manno, F. Oliveri, R. Vitolo,
*Lie remarkable PDEs*, Asymptotic methods in nonlinear wave phenomena,

World Scientific Publishing, Hackensack, NJ, (2007), 119-131. - G. Manno, F. Oliveri, R. Vitolo,
*On an inverse problem in group analysis of PDEs*, Lie-remarkable equations, Wascom 2005 13th Conference on Waves and Stability in Continuous Media, World Scientific Publishing, Hackensack, NJ, (2006), 420- 431. - A. Kisselev, G. Manno,
*On the symmetry structure of the minimal surface*, Proc. of the IX int. conf. on Diff. Geom. and its

equation

Appl., Prague (Czech Rep.), Matfyzpress (2005), 499-506. - G. Manno,
*The geometry of the geodesic equation in the*, Conference proceedings of AIP, no. 729 (2004),

framework of jets of submanifolds

207-217. - G. Manno, R. Vitolo,
*Variational sequences on finite order jets of*, Proc. of the VIII int. conf. on Diff. Geom. and

submanifolds

its Appl., Opava (Czech Rep.), Math. Publ., 3 (2001), 435-446. - S. Igonin, G. Manno,
*Fundamental Lie algebras for multicomponent Landau-Lifshitz systems*, Preprint, 31 pp., available at http://www.staff.science.uu.nl/~igoni101/preprints/fa_mll.pdf - G. Manno, V.S. Matveev,
*Projectively homogeneous metrics near points where they are not*, Preprint, 33 pp.

projectively homogeneous - G. De Matteis, G. Manno,
*Lie algebra symmetry analysis of the Helfrich and Willmore surface*, Preprint, 24 pp.

shape equations - D.V. Alekseevsky, R. Alonso Blanco, G. Manno, F. Pugliese,
*Finding solutions of parabolic*, Preprint, 21 pp.

Monge-AmpÃšre equations by using the geometry of Cartan fields - A.J. Di Scala, G. Manno,
*On the extension of parallel sections of linear connections*, Preprint, 8 pp. - G. Manno,
*Classification of 2-dimensional metrics admitting*, In preparation

one projective vector field - G. Manno, B. Kruglikov,
*Normal forms for generic*, In preparation

C^{â}-smooth Legendrian distributions on 5-dim. contact

manifolds - S. Igonin, G. Manno,
*Fundamental algebras for scalar evolution equations*, In preparation - D.V. Alekseevsky, R. Alonso Blanco, G. Manno, F. Pugliese,
*Homogeneous contact manifolds and invariant*, In preparation

Monge-AmpÃšre equations of Goursat type - G. Manno, F. Oliveri, G. Saccomandi, R. Vitolo,
*Ordinary differential equations characterized by*, In preparation

their symmetries

#### Conferences, schools, other events

*Symmetries of Differential Equation*

Lecce (Italy), 2004*International Workshop on Global Analysis*

Ankara, 2004*Symmetries and Pertubation Theory*

Cala Gonone (Italy), 2004*Formal theory of PDEs and their applications*

Joensuu (Finland), 2006*Symmetries and Pertubation Theory*

Otranto (Italy), 2007*Geometry and Algebra of PDEs*

Tromsoe (Norway), 2007*Diff. Geometry and its Applications*

Olomouc (Czech Rep.), 2007*Geometry and Symmetries of Differential Equations*

Santa Marinella (Italy), 17-22 May 2010*Giornate di Geometria Algebrica e argomenti correlati X*

Gargnano del Garda (Italy), 25-29 May 2010*Cartan Connections, Geometry of Homogeneous Spaces, and Dynamics*

Vienna, 10-23 July 2011*The interaction of Geometry and Representation Theory. Exploring new frontiers*

Vienna, 3-14 Settembre 2012*Pseudogroups and Differential Equations*

Tromsoe (Norway), 13-16 March 2013*Differential Geometry and Its Applications conference*

Brno (Czech Rep.), 19-23 August, 2013*Workshop on Geometry of PDEs and Integrability*

Opava (Czech Rep.), 14-18 October, 2013