Table Of ContentProceedings of the Workshop
CONTEMPORAGREYO METRY
RELATEDT OPICS
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Proceedings of the Workshop
CONTEMPORAGREYO METRY
AND RELATED TOPICS
Be I g rad e , Y u g oslavia 15 - 21 May 2002
Editors
Neda Bokan
University of Belgrade, Yugoslavia
Mirjana DjoriC
University of Belgrade, Yugoslavia
Anatoly T. Fomen ko
Moscow State University, Russia
Zoran RakiC
University of Belgrade, Yugoslavia
Julius Wess
Universitat Munchen, Germany
vp
World Scientific
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CONTEMPORARY GEOMETRY AND RELATED TOPICS
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Preface
The Workshop Contemporary Geometry and Related Topics, which took
place in Belgrade from May 15th to May 21th, 2002, was organized by the
Faculty of Mathematics of the Belgrade University and the Mathematical
Institute of SANU (Serbian Academy of Science and Arts). The organizers
were motivated and inspired in their work by lengthy and excellent tradition
of collaboration between colleagues from the Belgrade University and the
Moscow State University, especially those from the Chair of Differential
Geometry and Applications, as well as the wonderful opportunity to be the
host to several guests with the similar field of their research. Although the
beginning was almost spontaneous, the final results were such that we felt
an urge to publish the Proceedings.
There were 60 participants from 12 countries: Belorussia, Bulgaria,
Czech Republic, China, France, Germany, Japan, Italy, FYR Macedonia,
Russia, Spain and Yugoslavia. Scientific activities were only in one section
with lectures of 45 min and 30 min, short communications of 15 min and
a poster section in order to give opportunity to the participants to follow
all activities without pressure and to obtain new information. Although at
first some topics seemed diverse, the participants have recognized as very
important a possible connection of their research to reach the interdisci-
plinary one. Therefore, in the similar spirit we have organized the content
of the Proceedings in the lexicographic order.
The success of a meeting depends on various facts. That is why we
would like to express our gratitude to all of those who have contributed to
the success of our Workshop and its Proceedings. First of all, we feel deep
appreciation for all the participants, whose style of communicating with
the organizers and quality of lectures have motivated us to implement our
plan with pleasure. As the financial support is always very important, we
would like to thank all the institutions who helped the organization of this
Workshop. The complete list of sponsors is enclosed. For their support and
encouragement we are grateful to Prof. Z. Kadelburg, Prof. A. Lipkovski
and Prof. D. RadunoviC, dean and vice-deans of the Faculty of Mathematics
in Belgrade, respectively. Referees played a significant role by reading the
manuscripts carefully and making very precise and useful comments, that
have contributed to the improvement of the final version of all the papers.
The cover page is based on the logo of the Woikshop. The logo is made
by software L2Primitives written by I. KnezeviC, R. SazdanoviC and dr
V
vi
S. VukmiroviC (available at www.mathsource.com). We are indebted to
our colleagues M. AntiC, 2. StaniC and dr S. VukmiroviC, members of the
Organizing Committee, who helped us very much, not only in the Workshop
but also in preparing the Proceedings. Finally, we are especially indebted
to World Scientific for providing us with the opportunity to publish the
Proceedings.
The Proceedings contains articles which were received by the Editorial
board till March 31st, 2003, and passed the usual referring process. Some
articles which had been scheduled to be communicated during the Work-
shop, but were not, due to various reasons, are also included.
The Editors
List of sponsors
Ministry of Science, Technology and Development, Republic of Serbia
Ministry of Education and Sport, Republic of Serbia
Faculty of Mathematics, Belgrade
Matematical Institute of SANU, Belgrade
Rectorate of University in Kosovska Mitrovica
Megatrend University, Belgrade
Yugoslav Chamber of Commerce and Industry, Belgrade
Prva preduzetnitka banka, Beograd
National Theater, Belgrade
Tourist Organization of Belgrade, Belgrade
CET, Belgrade
Minel, Beograd
Sinex, Beograd
Sekvenca, Beograd
Teledan, Beograd
Project MM1646, Geometry, Education and Visualization with Applica-
tions, of MNTR
Contents
V. V. Balashchenko
Invariant Structures Generated by Lie Group Automorphisms on
Homogeneous Spaces .................................................. 1
M. B. Banam
On Kenmotsu Hypersurfaces in a Six-Dimensional Hermitian
Submanifold of Cayley Algebra.. .................................... .33
N. B1a.M and S. VukmiroviC
Para-hypercomplex Structures on a Four-Dimensional Lie Group. ..... 41
A. V. Bolsinov and B. JovanoviC
Integrable Geodesic Flows on Riemannian Manifolds:
Construction and Obstructions ...................................... 57
M. DjoriC and M. Okumura
CR Submanifolds of Maximal CR Dimension in Complex Space
Forms and Second Fundamental Form.. ............................. 105
M. Carmen Domingo-Juan and V. Miquel
Pappus-Guldin’s Formulae versus Weyl’s Tube Formula: Old and
Recent Results., .................................................... 117
B. Dragovich
Non-Archimedean Geometry and Physics on Adelic Spaces ..........1 41
B. Dragovich and Z. RakiC
Lagrangian Aspects of Quantum Dynamics on a Noncommutative
Space.. ............................................................. 159
M. Ferncindez-Lbpez, E. Garcia-Rio and D. N. Kupeli
The Mobius Equation: A Local Analytical Characterization of
Twisted Product Structures ......................................... 173
M. Ferna’ndez, V. Mufioz and J. A. Santisteban
Symplectically Aspherical Manifolds with Nontrivial 7r2 and with
No Kahler Metrics.. ................................................ 187
A. T. Fomenko and P. V. Morozov
Some New Results in Topological Classification of Integrable
Systems in Rigid Body Dynamics ................................... 201
vii
...
Vlll
F. Gavarini
The Crystal Duality Principle: From General Symmetries to
Geometrical Symmetries. ........................................... .223
2. Hu and H. Li
Willmore Submanifolds in a Riemannian Manifold, ..................2 51
Dj. Kadajevich
Some Aspects of Visualizing Geometric Knowledge: Possibilities,
Findings, Further Research. ......................................... 277
V. P. Karassiov
Dual Algebraic Pairs and Polynomial Lie Algebras in Quantum
Physics: Foundations and Geometric Aspects. ....................... 285
E. Mdkowsky and V. VelatkoviC
Visualisation and Animation in Differential Geometry ...............3 01
G. V. Nosovskiy
Computer Gluing of 2D Projective Images.. ........................ .319
A. A. Oshemkov
On the Topological Structure of the Set of Singularities of
Integrable Hamiltonian Systems ................................ .335
Th. Yu. Popelensky and Yu. P. Solovjev
Lie-Cartan Pairs and Characteristic Classes in Noncommutative
Geometry ...................................................... .351
S. Terzic'
On Rational Homotopy of Four-Manifolds ...................... ..... 375
S. Trapani
Dual Maps and Kobayashi Distance of Bounded Convex Domains
in CC" .......................................................... .... .389
K. TrenEevski
Osculating Spaces and Higher Order Curvature Tensors of
Submanifolds in R". ........................................... .407
Lj. S. VelimiroviC, S. M. Mantic' and M. S. Stankovic'
On Commutativity of the Lie Derivative and Covariant Derivative
at a Non-Symmetric Affine Connection Space. ...................... .425
L. Vrancken
Special Classes of Three Dimensional Affine Hyperspheres
Characterized by Properties of Their Cubic Form.. ................... 431
INVARIANT STRUCTURES GENERATED BY LIE GROUP
AUTOMORPHISMS ON HOMOGENEOUS SPACES *
VITALY V. BALASHCHENKO
Faculty of Mathematics and Mechanics
Belarusian State University
F. Scorina av. 4, Minsk 220080, BELARUS
E-mail: [email protected]
We collect the recent results on homogeneous regular @-spaces, specifically, ho-
mogeneous k-symmetric spaces. The structure of the commutative algebra of all
canonical affinor structures on any regular @-space is explicitly described. This
partitions all regular +-spaces into a countable set of disjoint classes with respect
to their algebra of canonical affinor structures. We prove the compatibility of
classical canonical structures with natural (pseudo)Riemannian metrics. In par-
ticular, this leads to applying canonical f-structures to the generalized Hermitian
geometry. As an example, we introduce a concept of nearly Kahler f-structures
naturally generalizing well-known nearly Kahler structures in Hermitian geometry.
A remarkable collection of invariant nearly Kahler f-structures is presented. It is
based on the canonical f-structures on homogeneous k-symmetric spaces. As a re-
sult, the analogy with homogeneous nearly Kahler manifolds is obtained. Finally,
some general and particular examples are considered in the above sense.
1. Introduction
An important place among homogeneous manifolds of Lie groups is occu-
pied by the homogeneous @-spaces, that is, the homogeneous spaces gen-
erated by automorphisms @ of Lie groups (see Refs. [53], [65], [19],
[ 401, [ 111). A special role is played by the regular @-spaces, which were
first introduced in Ref. [ 531. As is known [ 531, such @-spaces include all
(ak
k-symmetric spaces = id), are included, in their turn, into the well-
known class of homogeneous reductive spaces. At the same time, a distin-
* MSC2000: primary 53C15, 53C30; secondary 53C10, 53C35.
Keywords :h omogeneous space, regular @-space, k-symmetric space, invariant structure,
canonical affinor structure, nearly Kahler structure, f-structure.
t Partially supported by the Belarus State Program of Fundamental Research ”Mathe-
matical structures”.
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