New Horizons in Differential Geometry and its Related Fields

257 Pages·2022·27.575 MB·English

by  Toshiaki Adachi (editor), Hideya Hashimoto (editor)

New Horizons in
Differential Geometry
and its Related Fields
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New Horizons in
Differential Geometry
and its Related Fields
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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Published by
World Scientific Publishing Co. Pte. Ltd.
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Cover image:
(front) Sunomata Castle in cherry blossoms
(back) A road in Hakuba
Photographed by Toshiaki Adachi
NEW HORIZONS IN DIFFERENTIAL GEOMETRY AND ITS RELATED FIELDS
Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 978-981-124-809-2 (hardcover)
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PREFACE
The year 2020 will be specially marked in our modern history after the
World War II. We experienced a pandemic disease of the coronavirus
COVID-19 which started in March. This pandemic is a huge wave com-
pared to the past 1918 flu pandemic, Spanish flu, and so on. More than
two million deaths were caused by this virus in that year. To make mat-
ters worse, as medical staff should take care of COVID-19, our ordinary
medical care system could not make the best contribution. In the fields of
mathematics, we saw the passing of some famous professors: J.H. Conway
(PrincetonUniversity),L.Nirenberg(NewYorkUniversity),C.K.Johnson
(NASA), amongst others. In our community, we sorely missed Professors
Georgi Ganchev (Bulgarian Acad. Sci.), Simeon Zamkovoy (Sofia Univer-
sity) and Vasile Oproiu (Iasi University) in 2020, and also Professors Aki-
hikoMorimoto(NagoyaUniversity)andSojiKaneyuki(SophiaUniversity)
in 2019.
During the pandemic, in order to prevent from being infected, people
needed to practice “social distancing” or even stay at home. This situ-
ation makes us change our lifestyle. In the academic area, students had
to take their online lectures, and many academic meetings were canceled
or were done through the internet. In 2020, we planned to have the 7th
International Colloquium on Differential Geometry and its Related Fields
(ICDG2020) at Iasi, Romania. Unfortunately, we had to cancel this meet-
ingamongtherest. Aswecouldnotdiscusswitheachotheratthemeeting,
thismadeouracademicexchangeprogrambetweenEastEuropeandJapan
somewhat stagnant. Although our circumstances for academic works were
not well, many scientists made huge efforts not to delay the speed of aca-
demic success.
This volume contains papers on the recent progress in differential ge-
ometry and also the results in discrete mathematics which are related to
geometry. We have ten original papers and two surveys. These cover
the Einstein metrics on symplectic groups, magnetic fields on contact
v
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vi PREFACE
manifolds, construction of contact homogeneous manifolds, submanifolds
by using fibre bundle structures and their geometric structures, K¨ahler
structures on cotangent bundles, geometric objects on graphs, and geo-
metric approach in coding theory. Some of these materials were prepared
for the canceled ICDG2020. The editors think and hope that this volume
would still influence and motivate researchers who study differential geom-
etry and its related fields, and also hope that it would be a good guide for
young scientists in this area.
Theeditorswouldliketothankthescientistswhocelebratedtheir60th
birthdays. The original editors of our ICDG series, Hideya Hashimoto,
Milen Hristov and Toshiaki Adachi became 60 in the 2020 academic year.
IneastAsia,wecelebrate60thbirthdaysbasedontheChinesecalendar. By
anoldChinesephilosophy,itisconsideredthattheuniverseisformedbyfive
elements: fire, water, wood, metal, and earth. On the other hand, it could
have been originated from astronomy; where the old calendar used twelve
horary signs. Combining these 5 elements and twelve signs we have sixty
types. More precisely, as each kind is divided into two, we have 120 types.
But 60 is considered as an important period. The editors are not familiar
witholdhistory,sothisisonlytheirimagination,assexagesimalwasusedin
Babylonianmathematics,thereappeartobesomeglobalinfluence. Evenin
the olden days, there was a worldwide interchange of cultures. The editors
hope that the ICDG program will spread the exchange of mathematical
ideas within the society.
The Editors
3 June 2021
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CONTENTS
Preface v
Magnetic curves in quasi-Sasakian manifolds of product type 1
Marian Ioan MUNTEANU and Ana Irina NISTOR
Motion of charged particles in a compact homogeneous
Sasakian manifold 23
Osamu IKAWA
A note on Legendre trajectories on Sasakian space forms 41
Qingsong SHI and Toshiaki ADACHI
Non naturally reductive Einstein metrics on the symplectic
group via quaternionic flag manifolds 51
Andreas ARVANITOYEORGOS and Yusuke SAKANE
A Lie theoretic interpretation of realizations of some contact
metric manifolds 71
Takahiro HASHINAGA, Akira KUBO,
Yuichiro TAKETOMI and Hiroshi TAMARU
About code equivalence — a geometric approach 91
Iliya BOUYUKLIEV and Stefka BOUYUKLIEVA
vii
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viii CONTENTS
An algorithm for computing the covering radius of a linear
code based on Vilenkin-Chrestenson transform 105
Paskal PIPERKOV, Iliya BOUYUKLIEV and
Stefka BOUYUKLIEVA
Geometric properties of non-flat totally geodesic surfaces in
symmetric spaces of type A 125
Misa OHASHI and Kazuhiro SUZUKI
On the relationships between Hopf fibrations and Cartan
hypersurfaces in spheres 139
Hideya HASHIMOTO
Bochner curvature of cotangent bundles with natural
diagonal K¨ahler structures 151
Simona-Luiza DRUT¸A˘-ROMANIUC
Isotropicity of surfaces with zero mean curvature vector in
4-dimensional spaces 177
Naoya ANDO
Geometry of Lie hypersurfaces in a complex hyperbolic space 193
Sadahiro MAEDA and Hiromasa TANABE
K¨ahler graphs whose principal graphs are of Cartesian
product type 209
Toshiaki ADACHI
In memory of Professor Akihiko MORIMOTO 233
Naoko MORIMOTO and Toshiaki ADACHI
In memory of Professor Georgi GANCHEV 243
Velichka MILOUSHEVA
December22,2021 8:45 ws-procs9x6 WSPCProceedings-9inx6in 01 page1
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MAGNETIC CURVES IN QUASI-SASAKIAN MANIFOLDS
OF PRODUCT TYPE
MarianIoanMUNTEANU
“Alexandru Ioan Cuza” University of Iasi,
Department of Mathematics,
Bd. Carol I, n. 11, Iasi, 700506, Romania
[email protected]
AnaIrinaNISTOR
“Gheorghe Asachi” Technical University of Iasi,
Department of Mathematics and Informatics,
Bd. Carol I, n. 11A, Iasi, 700506, Romania
[email protected]
In Memory of Professor Vasile I. OPROIU (1941–2020)
In this paper we give an affirmative answer to sustain the conjecture about
the order of a magnetic curve in a quasi-Sasakian manifold. More precisely,
we show that the magnetic curves in a quasi-Sasakian manifold obtained as
the product of a Sasakian and a K¨ahler manifold have maximum order 5.
Moreover, we obtain the explicit parametrizations, the periodicity conditions
andexamplesinthestudyofmagneticcurvesinS3×S2.
Keywords:Contactmagneticfields;quasi-Sasakianmanifolds;magneticcurves.
1. Introduction
A strong motivation to work on the quasi-Sasakian manifolds was given
by the Blair’s approach in his PhD Thesis [6], back in 1966, as follows:
“Sasakian manifolds have often been considered the odd-dimensional ana-
loguesofK¨ahlermanifolds. However,ifM2n isaK¨ahlermanifold,M2n×R
can be considered an odd-dimensional analogue, but M2n×R carries a nat-
ural cosymplectic (quasi-Sasakian of rank 1) structure. Thus, in a certain
sense, quasi-Sasakian manifolds are better analogues of K¨ahler manifolds.”
Moreover,inthesamework[6],itisshowedthatthetypesofquasi-Sasakian
manifoldsM2n+1 rangefromthecaseofcosymplecticmanifolds(rank=1)
to the case of contact manifolds (rank = 2n+1). In particular, Sasakian
manifolds are quasi-Sasakian manifolds of rank 2n+1.
1