No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Codazzi and totally umbilical hypersurfaces in $\mathrm {Sol}_1^4
Abstract
In this paper, we prove the non-existence of Codazzi and totally umbilical hypersurfaces, especially totally geodesic hypersurfaces, in the 4 -dimensional model space .
ResearchGate has not been able to resolve any citations for this publication.
We study geodesics and magnetic trajectories in the model space F 4 {{\rm{F}}}^{4} . The space F 4 {{\rm{F}}}^{4} is isometric to the 4-dim simply connected Riemannian 3-symmetric space due to Kowalski. We describe the solvable Lie group model of F 4 {{\rm{F}}}^{4} and investigate its curvature properties. We introduce the symplectic pair of two Kähler forms on F 4 {{\rm{F}}}^{4} . Those symplectic forms induce invariant Kähler structure and invariant strictly almost Kähler structure on F 4 {{\rm{F}}}^{4} . We explore some typical submanifolds of F 4 {{\rm{F}}}^{4} . Next, we explore the general properties of magnetic trajectories in an almost Kähler 4-manifold and characterize Kähler magnetic curves with respect to the symplectic pair of Kähler forms. Finally, we study homogeneous geodesics and homogeneous magnetic curves in F 4 {{\rm{F}}}^{4} .
We study hypersurfaces of the four‐dimensional Thurston geometry Sol04, which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor—including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form—and of totally umbilical hypersurfaces of Sol04. We also give a closed expression for the Riemann curvature tensor of Sol04, using two integrable complex structures.
In this paper, we give a classification of Codazzi hypersurfaces in a Lie group . We also give a characterization of a class of minimal hypersurfaces in with an example of a minimal surface in this class.
If N is a compact totally umbilical with constant mean curvature immersed in a Hermitian symmetric space M of compact type , then N is either totally geodesic or a real space form immersed in M as a purely real submanifold satisfying 2 dim N < dim M.
We classify left invariant metrics on the 4-dimensional, simply connected, unimodular Lie groups up to automorphism. When the corresponding Lie algebra is of type (R), this is equivalent to classifying the left invariant metrics up to isometry, but in general the classification up to automorphism is finer than that up to isometry. In the abelian case, all left invariant metrics are isometric. In the nilpotent case, the space of metrics can have dimension 1 or 3. In the solvable case, the dimension can be 2, 4, or 5. There are two non-solvable 4-dimensional unimodular groups, and the space of metrics has dimension 6 in both of these cases.
The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group Sol_1^4, and more generally, the crystallographic groups of Sol_1^4. The maximal compact subgroup of Isom(Sol_1^4) is D_4={\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2. We shall exhibit an infra-solvmanifold of Sol_1^4 whose holonomy is D_4. This implies that all possible holonomy groups do occur; the trivial group, {\mathbb{Z}}_2 (5 families), {\mathbb{Z}}_4, {\mathbb{Z}}_2{\times}{\mathbb{Z}}_2 (5 families), and {\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2 (2 families).
The main purpose of this paper is to study totally umbilical submanifolds in a Kahler manifold.
By using antipodal sets and fixed point sets, we introduce a new method to study compact symmetry spaces. In this paper, we also provide several applications of our method.
We show that a simply connected Riemannian homogeneous space M which admits a
totally geodesic hypersurface F is isometric to either (a) the Riemannian
product of a space of constant curvature and a homogeneous space, or (b) the
warped product of the Euclidean space and a homogeneous space, or (c) the
twisted product of the line and a homogeneous space (with the warping/twisting
function given explicitly). In the first case, F is also a Riemannian product;
in the last two cases, it is a leaf of a totally geodesic homogeneous
fibration. Case (c) can alternatively be characterised by the fact that M
admits a Riemannian submersion onto the universal cover of the group SL(2)
equipped with a particular left-invariant metric, and F is the preimage of the
two-dimensional solvable totally geodesic subgroup.
A submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.
The main purpose of this paper is to study totally umbilical submanifolds in a Kahler manifold.
In this paper we obtain all invariant, anti-invariant and CR submanifolds in (ℝ4, g, J) endowed with a globally conformal Kähler structure which are minimal and tangent or normal to the Lee vector field of the
g.c.K. structure.
In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Qm:=SO(m+2)/(SO(2)×SO(m)) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Qm by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745–755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Qm are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745–755]: The first type is constituted by manifolds isometric to CP1×RP1; their existence follows from the fact that Q2 is (via the Segre embedding) holomorphically isometric to CP1×CP1. The second type consists of 2-spheres of radius which are neither complex nor totally real in Qm.
We study homogeneous geodesics in 4-dimensional solvable Lie groups , , and .
Minimal invariant, minimal totally real, and minimal CR-submanifolds in the 4-dimensional homogeneous solvable Lie group equipped with standard globally conformal Kähler structure are studied.
The paper is concerned with totally geodesic submanifolds S of dimension r∈{2,⋯,n-1} of a general n-dimensional Riemannian manifold M ˜. As in this situation for every p∈S the subspace V:=T p S⊂T p M ˜ is curvature invariant, i.e., ∀x,y,z∈V:R ˜(x,y)z∈V, the author derives criteria for the curvature invariance of subspaces of Euclidean vector spaces with respect to “algebraic” curvature tensors. Thereby, it is possible to deduce specific results if M ˜ is a 3-dimensional unimodular Lie group with a left invariant metric or if M ˜ is a hypersurface of a real space form. In another theorem a 1:1 correspondence between totally geodesic submanifolds and integral manifolds of a certain horizontal distribution of the Grassmann bundle G r (TM ˜) over M ˜ is described.
We prove that a four-dimensional generalized symmetric space does not admit any non-degenerate hypersurfaces with parallel
second fundamental form, in particular non-degenerate totally geodesic hypersurfaces, unless it is locally symmetric. However,
spaces which are known as generalized symmetric spaces of type C do admit non-degenerate parallel hypersurfaces and we verify
that they are indeed symmetric. We also give a complete and explicit classification of all non-degenerate totally geodesic
hypersurfaces of spaces of this type.
KeywordsTotally geodesic–Parallel–Hypersurface–Generalized symmetric space
Classification, cobordism, and curvature of four-dimensional infra-solvmanifolds
- S Van Thuong
S. Van Thuong, Classification, cobordism, and curvature of four-dimensional infra-solvmanifolds, PhD Thesis (University
of Oklahoma, 2014). http://hdl.handle.net/11244/10360
Isometry groups of simply connected unimodular 4-dimensional Lie groups
- Y Ayad
- S Fahlaoui
Y. Ayad and S. Fahlaoui, Isometry groups of simply connected unimodular 4-dimensional Lie groups, 2024.
https://arxiv.org/pdf/2412.01588
Minimal submanifolds in $mathrm {Sol}^4_1$
- Z Erjavec
- J Inoguchi
- B.-Y Chen
- P Verheyen
B.-Y. Chen and P. Verheyen, Totally umbilical submanifolds of Kaehler manifolds. II., Bull. Soc. Math. Belg., Sér. B
35 (1983), 27-44.
Minimal submanifolds in Sol 4 1
- Z Erjavec
- J Inoguchi
Z. Erjavec and J. Inoguchi, Minimal submanifolds in Sol 4
1, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., RACSAM
117(4) (2023), 36.Paper no. 156