NOTE DI MATEMATICA - UNIVERSITÀ DEL SALENTO - Pubblicazione Semestrale Volume 28 - suppl. numero 1 2009
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NOTE
DI
MATEMATICA
Pubblicazione Semestrale
Volume 28 – suppl. numero 1 – 2009
UNIVERSITÀ DEL SALENTOVolume 28, anno 2008, suppl. n° 1
ISSN 1123–2536 (printed version)
ISSN 1590–0932 (electronic version)
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La Rivista «Note di Matematica» esce in fascicoli semestrali.
Direttore Responsabile: Silvia Cazzato
Editor–in–Chief: V.B. Moscatelli, Dipartimento di Matematica, Università del Salento, Via per
Arnesano, 73 100 LECCE (Italy).
Secretaries: F. Catino, Dipartimento di Matematica, Università del Salento, Via per Amesano,
73100 LECCE, (Italy); R. A. Marinosci, Dipartimento di Matematica, Università del Salento,
Via per Arnesano, 73 100 LECCE, (Italy).
Editorial Board: M. Biliotti, Dipartimento di Matematica, Università del Salento, Via per
Arnesano, 73100 LECCE, (Italy); D. E. Blair, Michigan State University, Dept. of Mathematics,
East Lansing, MICHIGAN 48824 (U.S.A.); M. Carriero, Dipartimento di Matematica, Università
del Salento, Via per Arnesano, 73100 LECCE, (Italy); F. De Giovanni, Dipartimento di Matematica
e Applicazioni, Università di Napoli “Federico II”, Via Cintia, 80126 NAPOLI (Italy); S. Dierolf,
FB IV Mathematik, Universitàt Trier, Postfach 3825, D5500 TRIER (Germany); N. Fusco,
Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Via Cintia, 80126
NAPOLI (Italy); N. Johnson, Mathematics Dept. University of Iowa, Iowa City, IOWA 52242
(U.S.A.); O. Kowalski, MFF U.K. Sokolovska 83, 18600 PRAHA (Czech Republic); H. Laue,
Mathematisches Seminar der Universitêt Ludewig Meyn Str. 4, D–2300 KIEL 1 (Germany); D.
Perrone, Dipartimento di Matematica, Università del Salento, Via per Arnesano, 73100 LECCE,
(Italy); J.J. Quesada Molina, Departamento de Matemática Aplicada Universidad de Granada,
18071 GRANADA (Spain); A. Rhandi, Mathematisches Institut, Univ. Tübingen Auf der
Morgenstelle 10, D–72076 TÜBINGEN (Germany); K. Strambach, Mathematisches Institut
Universitêt Erlangen Numberg, Bismarckstrasse 1, D–8520 ERLANGEN (Germany).
Autorizzazione del Tribunale di Lecce n. 273 del 6 aprile 1981
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Preface
The international conference “ADVANCES IN DIFFERENTIAL GEOME-
TRY” held in Lecce (Italy) from 13 to 16 June 2007, was organized in honour
of Oldrich Kowalski, Emeritus Professor at the Charles University in Prague,
as an homage of his friends, collaborators and students. This issue of NOTE DI
MATEMATICA contains almost all lectures delivered during the conference.
They cover mainly the areas of Riemannian and pseudo-Riemannian geometry,
many of them connected with the interests of research of Oldrich Kowalski.
Professor Oldrich Kowalski began his professional career at the University of
Brno (Czech Republic) where he obtained his PhD in 1963. He has been teach-
ing as Associate Professor at that university until the academic year 1968–69;
from 1970 to 1991 he has been teaching, as Associate Professor, at the Charles
University in Prague where he was appointed Full Professor in 1992 and Pro-
fessor Emeritus in 2001. Professor Oldrich Kowalski has done research mainly
in the field of Differential Geometry and particularly Riemannian Geometry. In
the earlier period he gave relevant results on Generalized Symmetric Spaces.
In a series of papers the general theory of these spaces was developed and it
was later summarized and completed in the unique existing and frequently cited
monograph. Together with several colleagues and collaborators, he investigated
differential operators on Riemannian manifolds and the converse of a Gelfands
theorem; a classification of naturally reductive and commutatives spaces in small
dimensions; the classification of D’Atri spaces in dimension three ; volumes of
tubes in Riemannian geometry (generalization of some results by H.Weyl from
the Euclidean case to the general Riemannian case). Classification of invariant
Einstein metrics on Aloff-Wallach spaces; solution of global extrinsic version of
the so-called Volume Conjecture by A. Gray and L. Vanhecke about the volume
of geodesic balls, using the methods of geometric measure theory ; introduction
of the notion of additive volume invariant and its application on the Volume
Conjecture by Gray and Vanhecke. Explicit classification (in the form of a finite
formula) of so-called non-elliptic semi-symmetric spaces and further generaliza-
tion of these results. Complete classification of curvature homogeneous spaces
in dimension 3, study of homogeneous geodesics on homogeneous Riemannian
manifolds (existence theorems), classification of Riemannian manifolds whose
all geodesics are homogeneous in dimension up to 7, classification of homoge-
neous affine connections in dimension 2. Geometry of the tangent bundle and
of the unit tangent sphere bundle on a Riemannian manifold.
There where almost one hundred partecipants at the conference; they came
from many countries in Europe, U.S.A. and Japan. The program featured twelve
invited lectures of forty five minutes, ten invited talks of twenty minutes and2
a section of poster communications. During the Conference, the accompanying
social events provided an inspiring atmosphere for scientific contacts and fruitful
discussions.
We thank all the participants who contributed, with their presence, to the
success of the meeting. Special thanks to those participants who offered us their
manuscripts for publication and also to the referees for their careful work.
We would also like to thank the members of the Organizing Committee:
R.A. Marinosci (coordinator) (Lecce, Italy), D. Perrone (Lecce, Italy),
S. Dragomir (Potenza, Italy), G. De Cecco (Lecce, Italy), E. Barletta (Potenza,
Italy), G. Calvaruso (Lecce, Italy).
Thanks are also due to the Sponsors listed below; the Conference would not
have been possible without their financial support.
Finally we would like to thank the Journal Note di Matematica for accept-
ing to publish the Proceedings of the Conference in a supplement of the same
Journal and for providing for the international diffusion of it.
Rosa Anna Marinosci
Domenico Perrone
Sorin Dragomir
Sponsors:
• Dipartimento di Matematica Ennio De Giorgi,
• University of Salento,
• PRIN 2005 (Project Riemannian metrics and differential manifolds, Local
Unity of Lecce),
• INDAM (GNSAGA),
• Banca Monte dei Paschi di Siena,
• Fondazione Caripuglia.3
PARTICIPANTS
Abbassi Mohamed (Morocco) mtk abbassi@Yahoo.fr
Alekseevsky Dmitry (UK) D.Aleksee@ed.ac.uk
Ancona Vincenzo (Italy) ancona@unifi.it
Arias-Marco Teresa (Spain) Teresa.Arias@uv.es
Bande Gianluca (Italy) gbande@unica.it
Barletta Elisabetta (Italy) barletta@unibas.it
Baum Helga (Germany) baum@mathematik.hu-berlin.de
Balmus Adina (Italy) balmus@unica.it
Bedulli Lucio (Italy) bedulli@math.unifi.it
Benyounes Michele (France) Michele.Benyounes@univ-brest.fr
Bordoni Manlio (Italy) bordoni@dmmm.uniroma1.it
Brunetti Letizia (Italy) brunetti@dm.uniba.it
Cabrera Francisco Martin (Italy) fmartin@ull.es
Caldarella Angelo (Italy) caldarella@dm.uniba.it
Calvaruso Giovanni (Italy) giovanni.calvaruso@unile.it
Cappelletti Montano Beniamino (Italy) cappelletti@dm.uniba.it
Chiossi Simon (Germany) sgc@math.hu-berlin.de
Conti Diego (Italy) diego.conti@unimib.it
Csikos Balazs (Hungary) csikos@cs.elte.hu
De Cecco Giuseppe (Italy) giuseppe.dececco@unile.it
De Leo Barbara (Italy) barbara.deleo@unile.it
De Nicola Antonio (Italy) denicola@dm.uniba.it
Diaz-Ramos (Ireland) jc.diazramos@ucc.ie
Di Leo Giulia (Italy) dileo@dm.uniba.it
Di Scala Antonio J. (Italy) antonio.discala@polito.it
Dragomir Sorin (Italy) dragomir@unibas.it
Djoric Mirjana (Serbia) mdjoric@matf.bg.ac.yu
Dusek Zdenek (Czech Republic) dusek@karlin.mff.cuni.cz
Ewert-Krzemieniewski Stanislaw (Poland) Ewert@ps.pl
Fernandez Marisa (Spain) mtpferol@lg.ehu.es
Fino Anna (Italy) fino@dm.unito.it
Galicki Krzysztof (USA) galicki@math.unm.edu
Gil-Medrano Olga (Spain) Olga.Gil@uv.es
Gori Anna (Italy) gori@math.unifi.it
Haydys Andriy (Germany) haydys@math.uni-bielefeld.de
Kowalski Oldrich ( Czech Republic) kowalski@karlin.mff.cuni.cz
Leitner Felipe (Germany) leitner@mathematik.uni-stuttgart.de4 Loi Andrea (Italy) loi@unica.it Loubeau Eric (France) loubeau@univ-brest.fr Lotta Antonio (Italy) lotta@dm.uniba.it Marchiafava Stefano (Italy) marchiaf@mat.uniroma1.it Manno Gianni (Italy) gianni.manno@unile.it Marinosci Rosa Anna (Italy) rosanna.marinosci@unile.it Markellos Michael (Greece) mark@upatras.gr Micelli Giuseppe (Italy) giuseppe.micelli@unile.it Michalis Dimitrios (Greece) mark@upatras.gr Mikes Josef (Czech Republic) Mikes@risc.upol.cz Montaldo Stefano (Italy) montaldo@unica.it Munteanu Marian Ioan (Romania) munteanu@uaic.ro Musso Emilio (Italy) musso@univaq.it Naveira Antonio M. (Spain) Antonio.Martinez@uv.es Nicolodi Lorenzo (Italy) lorenzo.nicolodi@unipr.it Nikcevic Stana (Yugoslavia) stanan@mi.sanu.ac.yu Nurowski Pawel (Poland) nurowski@fuw.edu.pl Opozda Barbara (Poland) Barbara.Opozda@im.uj.edu.pl Otway Thomas(USA) otway@yu.edu Palese Marcella (Italy) palese@dm.unito.it Pascali Maria Antonietta (Italy) mantonietta.cognome@gmail.com Perrone Domenico (Italy) domenico.perrone@unile.it Petit Robert (France) petit@math.univ-nantes.fr Piccinni Paolo (Italy) piccinni@mat.uniroma1.it Piu Paola (Italy) piu@unica.it Podesta’ Fabio (Italy) Podesta@math.unifi.it Rakic Zoran (Yugoslavia) zrakic@matf.bg.ac.yu Rinaldelli Mauro (Italy) rinaldelli@math.unifi.it Salamon Simon (Italy) salamon@sns.it Saltarelli Vincenzo (Italy) saltarelli@dm.uniba.it Sasso Vito (Italy) sasso@mat.uniroma2.it Savo Alessandro (Italy) savo@dmmm.uniroma1.it Sekizawa Masami (Japan) sekizawa@u-gakugei.ac.jp Slovak Jan (Czech Republic) slovak@muni.cz Spiro Andrea (Italy) andrea.spiro@unicam.it Szenthe Janos (Hungary) szenthe@ludens.elte.hu Urakawa Hajime (Japan) urakawa@math.is.tohoku.ac.jp Verdiani Luigi (Italy) verdiani@math.unifi.it Versori Luigi (Italy) luigi.vergori@unile.it Vezzoni Luigi (Italy) luigi.vezzoni@mail.dm.unito.it
5 Yampolsky Alexander (Ukraine) alexymp@gmail.com Winterroth Ekkehart (Italy) ekkehart@dm.unito.it Wolak Robert (Poland) wolak@im.uj.edu.pl Zelenko Igor (Italy) zelenko@sissa.it
Note di Matematica 28, suppl. n. 1, 2009, 6–35.
g-natural metrics: new horizons
in the geometry of tangent bundles
of Riemannian manifolds
M. T. K. Abbassi
Département des Mathématiques, Faculté des sciences Dhar El Mahraz,
Université Sidi Mohamed Ben Abdallah, Fès, Morocco.
mtk abbassi@yahoo.fr
Received: 20/11/07; accepted: 09/01/08.
Abstract. Traditionally, the Riemannian geometry of tangent and unit tangent bundles was
related to the Sasaki metric. The study of the relationship between the geometry of a manifold
(M, g) and that of its tangent bundle T M equipped with the Sasaki metric g s had shown some
kinds of rigidity. The concept of naturality allowed O.Kowalski and M.Sekizawa to introduce a
wide class of metrics on T M naturally constructed from some classical and non-classical lifts
of g. This class contains the Sasaki metric as well as the well known Cheeger-Gromoll metric
and the metrics of Oproiu-type.
We review some of the most interesting results, obtained recently, concerning the geometry
of the tangent and the unit tangent bundles equipped with an arbitrary Riemannian g-natural
metric.
Keywords: Riemannian manifold, tangent bundle, unit tangent sphere bundle, g-natural
metric, contact structure, curvatures
MSC 2000 classification: primary 53B20, 53C07, secondary 53A55, 53C25
Introduction and historical review
It is well-known, from different models of Management sciences and busi-
ness, that a typical life cycle of any product project passes through four main
stages: the introduction, the growth, the maturity (or saturation) and the de-
cline. A new product is first developed and then introduced to the market. Once
the introduction is successful, a growth period follows with wider awareness of
the product and increasing sales. The product enters maturity when sales stop
growing and demand stabilizes. Eventually, sales may decline by the repeated
facts of the competition, the economical hazards and the new tendencies until
the product is finally withdrawn from the market or redeveloped. The life cy-
cle of a research project or a research production in some field doesn’t escape
from this rule. Indeed, the introduction step of a research project or a research
activity in a scientific field is the step when the motivations of the subject or
the topic are stated and the first works on it are published. The growth stageg-natural metrics: Towards new horizons in the geometry ... 7
is then the stage when the scientific community acknowledges the interest of
the project and several groups of researchers are interested in the topic and a
real competition is engaged to solve its questions and problems. With the sta-
bilization of speed of competition and the identification of the bounds of the
research and problems related to the topic, a kind of saturation begins to occur
and the volume of works dedicated to the topic attains a regular level and be-
gins, in some sense, standard. Several research groups in the topic become then
disinterested and leave to other topics or subjects, and the production becomes
counter-optimal. Other tendencies or research ways tend then to replace the
initial topic or subject or to modify substantially its physiognomy, opening the
door to its decline.
The history of research in the topic of differential geometry of tangent bun-
dles over Riemannian manifolds looks very appropriate to illustrate explicitly
this case of figure: The introduction of the topic begun with S. Sasaki who
constructed, in its original paper [67] of 1958, a Riemannian metric gs on the
tangent bundle T M of a Riemannian manifold (M, g), which depends closely
on the base metric g. More precisely, the components of the metric gs depend
only on the components of the metric g and their first derivatives, i.e., using
the terminology related to jets, g s depends on the first jet of the metric g. Ge-
ometrically speaking, the Sasaki metric gs is completely characterized by the
following properties:
(1) The natural projection pM : (T M, gs ) → (M, g) is a Riemannian submer-
sion;
(2) The horizontal and vertical distributions are orthogonal;
(3) The induced metric on each fiber of T M is Euclidean.
The introduction of the Sasaki metric can be considered as the first stage of
the whole topic of differential geometry of tangent bundles, and we can even
say that the life cycle of the topic was considered by the specialists as the life
cycle of research on the Sasaki metric since all the works published on the topic
considered T M equipped with the Sasaki metric, although the introduction
during the sixteen’s of the 20-th century of other metrics on T M (cf. [80] and
[81]), using especially the various kinds of classical lifts of tensor fields from M
to T M . According to this concept of lift, the Sasaki metric is no other than
the diagonal lift of the base metric, but it is distinguished by the fact that it is
Riemannian, when the other constructed metrics are pseudo-Riemannian.
The decades 60-70 of the twentieth century had been the growth period of
the topic, with a massive interest of eminent geometers in geometrical proper-
ties of the tangent bundle, equipped with the Sasaki metric. According to the8 M. T. K. Abbassi
approaches adopted for research, we can distinguish between two schools, the
Japanese one led by Sasaki, Sato, Tanno who, influenced by physics, had chosen
to treat questions by means of coordinates, and the European school, repre-
sented by Dombrowski, Kowalski, Nagy, Walczak and others, and who chose to
work with coordinates-free formulas.
The middle of the 80’s of the previous century was actually the starting
period of the maturity stage in the life cycle of the topic, since it has been
shown in many papers that the Sasaki metric presents a kind of rigidity. In
[39], O. Kowalski proved that if the Sasaki metric gs is locally symmetric, then
the base metric g is flat and hence gs is also flat. In [44], E. Musso and F.
Tricerri have demonstrated an extreme rigidity of gs in the following sense: if
(T M, gs ) is of constant scalar curvature, then (M, g) is flat. This made geometers
a bit reticent to the study of the geometry of (T M, gs ), but some research
groups (Borisenko, Yampolsky, Vanhecke, Boeckx, Blair, Kowalski, Sekizawa
and others) focused on the study of (unit) tangent sphere bundles endowed
with the Sasaki metric or with some homothetic one which confers to it the
structure of a contact manifold. Up to now, geometers remain interested on the
geometry of the unit tangent sphere bundle endowed with the Sasaki metric,
especially in the framework of harmonicity (G. Wiegmink, C.M. Wood, O.Gil
Medrano and others. . . ), but the geometers begun, during the 90’s of the 20-th
century, more and more convinced that this period was the beginning of the
stage of the decline of the life cycle of research on the geometry of the Sasaki
metric. With the evident historical relationship between the Sasaki metric and
the whole topic of differential geometry of tangent bundles, this could be also
the decline of the life cycle of the whole topic. Fortunately, there were a natural
thinking to the introduction of (Riemannian) metrics on the tangent bundle
other than the Sasaki metric, for which the rigidity of T M stops to be true.
A first step towards this end was initiated by Musso and Tricerri [44] in 1986,
who gave a process of construction of Riemannian manifolds on T M from basic
symmetric tensor fields of type (2, 0) on OM × m , where OM is the bundle of
orthonormal frames. As an example, they constructed a new Riemannian metric
on T M , i.e., the Cheeger-Gromoll metric gCG . M. Sekizawa [68] has shown that
the scalar curvature of (T M, gCG ) is never constant if the original metric on the
base manifold has constant sectional curvature (see also [34]). Furthermore, the
author and M. Sarih have proved that (T M, gCG ) is never a space of constant
sectional curvature (cf. [11]).
More generally, O. Kowalski and M. Sekizawa [40] used the developed con-
cept of naturality to give a full classification of metrics which are ‘naturally con-
structed’ from a metric g on the base M , supposing that M is oriented. Other
presentations of the basic results from [40] (involving also the non-oriented caseNote di Matematica Volume 28, suppl. number 1, 2009 Mohamed Tahar Kadaoui ABBASSI g-natural metrics: new horizons in the geometry of tangent bundles of Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Teresa ARIAS-MARCO Flat locally homogeneous attine connections with torsion on 2-dimensional manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Adina BALMUŞ, Stefano MONTALDO, Cezar ONICIUC Classification results and new examples of proper biharmo- nic submanifolds in spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Charles P. BOYER Sasakian geometry: the recent work of Krzysztof Galicki . . . . . . . . 63 Giovanni CALVARUSO Naturally Harmonic Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Sorin DRAGOMIR Subelliptic harmonic maps, morphisms, and vector fields . . . . . . . 131 Zdeněk DUŠEK Survey on homogeneous geodesics . . . . . . . . . . . . . . . . . . . . . . . . . 147 Anna FINO, Adriano TOMASSINI Solvmanifolds and Generalized Kähler Structures . . . . . . . . . . . . . 169 José Carmelo GONZÁLEZ-DÁVILA, Antonio MARTÍNEZ NAVEIRA Jacobi fields and osculating rank of the Jacobi operator in some special classes of homogeneous Riemannian spaces . . . . . . . 191 Jiang GUOYING 2-harmonic maps and their first and second variational for- mulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Toshiyuki ICHIYAMA, Jun-ichi INOGUCHI, Hajime URAKAWA Bi-harmonic maps and bi-Yang-Mills fields . . . . . . . . . . . . . . . . . . 233 Felipe LEITNER About Fefferman-Einstein metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Stefano MARCHIAFAVA Submanifolds of (para-)quaternionic Kähler manifolds . . . . . . . . . 295 Emilio MUSSO, Lorenzo NICOLODI The spinor representation of CMC 1 surfaces in hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Peter B. GILKEY, Stana NIKČEVIĆ The classification of simple Jacobi-Ricci commuting alge- braic curvature tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Georgi MIHAYLOV, Simon SALAMON Intrinsic torsion varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Alessandro SAVO Stability and the first Betti number . . . . . . . . . . . . . . . . . . . . . . . . . 377 Oldrich KOWALSKI, Masami SEKIZAWA On Riemannian geometry of orthonormal frame bundles . . . . . . . . 383 János SZENTHE Homogeneous pregeodesics and the orbits neighbouring a lightlike one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
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