Page 346 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 346
DIFFERENTIAL GEOMETRY: OLD AND NEW (MS-15)
In particular we obtain similarity curvature, similarity Frenet formula and fundamental theorem
of plane curves in similarity geometry.
On the other hand, in industrial design (CAGD), log-aesthetic curves are studied extensively.
In this talk, we give a similarity geometric reformulation of log-aesthetic curves and discuss
relations to curve flows derived form Burgers flows.
Contact CR submanifolds in odd-dimensional spheres: new examples
Marian Ioan Munteanu, marian.ioan.munteanu@gmail.com
University Alexandru Ioan Cuza of Iasi, Romania
The notion of CR-submanifold in Kähler manifolds was introduced by A. Bejancu in 70’s,
with the aim of unifying two existing notions, namely complex and totally real submanifolds in
Kähler manifolds. Since then, the topic was rapidly developed, mainly in two directions:
• Study CR-submanifolds in other almost Hermitian manifolds.
• Find the odd dimensional analogue of CR-submanifolds. Thus, the notion of semi-
invariant submanifold in Sasakian manifolds was introduced. Later on, the name was changed
to contact CR-submanifolds.
A huge interest in the last 20 years was focused on the study of CR-submanifolds of the
nearly Kähler six dimensional unit sphere. Interesting and important properties of such sub-
manifolds were discovered, for example, by M. Antic, M. Djoric, F. Dillen, L. Verstraelen, L.
Vrancken. As the odd dimensional counterpart, contact CR-submanifolds in odd dimensional
spheres were, recently, intensively studied. In this talk we focus on those proper contact CR-
submanifolds, which are as closed as possible to totally geodesic ones in the seven dimensional
spheres endowed with its canonical structure of a Sasakian space form. We give a complete
classification for such a submanifold having dimension 4 and describe the techniques of the
study. We present also some very recent developments concerning dimension 5 and 6 and pro-
pose further problems in this direction.
This presentation is based on some papers in collaboration with M. Djoric and L. Vrancken.
Keywords: (contact) CR-submanifold, Sasakian manifolds, minimal submanifolds, mixed and
nearly totally geodesic CR-submanifolds
References
[1] M. Djoric, M.I. Munteanu, L. Vrancken: Four-dimensional contact CR-submanifolds in
S7(1), Math. Nachr. 290 (2017) 16, 2585–2596.
[2] M. Djoric, M.I. Munteanu, On certain contact CR-submanifolds in S7, Contemporary
Mathematics, Geometry of submanifolds, Eds. (J. van der Veken et al.) 756 (2020) 111-
120.
[3] M. Djoric, M.I. Munteanu, Five-dimensional contact CR-submanifolds in S7(1), Math-
ematics, Special Issue Riemannian Geometry of Submanifolds, Guest Editor: Luc
Vrancken, 8 (2020) 8, art. 1278.
344
In particular we obtain similarity curvature, similarity Frenet formula and fundamental theorem
of plane curves in similarity geometry.
On the other hand, in industrial design (CAGD), log-aesthetic curves are studied extensively.
In this talk, we give a similarity geometric reformulation of log-aesthetic curves and discuss
relations to curve flows derived form Burgers flows.
Contact CR submanifolds in odd-dimensional spheres: new examples
Marian Ioan Munteanu, marian.ioan.munteanu@gmail.com
University Alexandru Ioan Cuza of Iasi, Romania
The notion of CR-submanifold in Kähler manifolds was introduced by A. Bejancu in 70’s,
with the aim of unifying two existing notions, namely complex and totally real submanifolds in
Kähler manifolds. Since then, the topic was rapidly developed, mainly in two directions:
• Study CR-submanifolds in other almost Hermitian manifolds.
• Find the odd dimensional analogue of CR-submanifolds. Thus, the notion of semi-
invariant submanifold in Sasakian manifolds was introduced. Later on, the name was changed
to contact CR-submanifolds.
A huge interest in the last 20 years was focused on the study of CR-submanifolds of the
nearly Kähler six dimensional unit sphere. Interesting and important properties of such sub-
manifolds were discovered, for example, by M. Antic, M. Djoric, F. Dillen, L. Verstraelen, L.
Vrancken. As the odd dimensional counterpart, contact CR-submanifolds in odd dimensional
spheres were, recently, intensively studied. In this talk we focus on those proper contact CR-
submanifolds, which are as closed as possible to totally geodesic ones in the seven dimensional
spheres endowed with its canonical structure of a Sasakian space form. We give a complete
classification for such a submanifold having dimension 4 and describe the techniques of the
study. We present also some very recent developments concerning dimension 5 and 6 and pro-
pose further problems in this direction.
This presentation is based on some papers in collaboration with M. Djoric and L. Vrancken.
Keywords: (contact) CR-submanifold, Sasakian manifolds, minimal submanifolds, mixed and
nearly totally geodesic CR-submanifolds
References
[1] M. Djoric, M.I. Munteanu, L. Vrancken: Four-dimensional contact CR-submanifolds in
S7(1), Math. Nachr. 290 (2017) 16, 2585–2596.
[2] M. Djoric, M.I. Munteanu, On certain contact CR-submanifolds in S7, Contemporary
Mathematics, Geometry of submanifolds, Eds. (J. van der Veken et al.) 756 (2020) 111-
120.
[3] M. Djoric, M.I. Munteanu, Five-dimensional contact CR-submanifolds in S7(1), Math-
ematics, Special Issue Riemannian Geometry of Submanifolds, Guest Editor: Luc
Vrancken, 8 (2020) 8, art. 1278.
344
