scholarly journals Characterization of the pseudo-symmetries of ideal Wintgen submanifolds of dimension 3

2010 ◽  
Vol 88 (102) ◽  
pp. 53-65
Author(s):  
Ryszard Deszcz ◽  
Miroslava Petrovic-Torgasev ◽  
Zerrin Şentürk ◽  
Leopold Verstraelen
Keyword(s):  
Extrinsic Curvature ◽  
Real Space ◽  
Point Of View ◽  
Intrinsic Curvature ◽  
Space Forms ◽  
3 Dimensional ◽  
Arbitrary Codimension ◽  
Sectional Curvatures ◽  
Wintgen Ideal Submanifolds ◽  
Normalized Scalar Curvature

Recently, Choi and Lu proved that the Wintgen inequality ? ? H2??? +k, (where ? is the normalized scalar curvature and H2, respectively ??, are the squared mean curvature and the normalized scalar normal curvature) holds on any 3-dimensional submanifold M3 with arbitrary codimension m in any real space form ~M3+m(k) of curvature k. For a given Riemannian manifold M3, this inequality can be interpreted as follows: for all possible isometric immersions of M3 in space forms ~M3+m(k), the value of the intrinsic curvature ? of M puts a lower bound to all possible values of the extrinsic curvature H2 ? ?? + k that M in any case can not avoid to ?undergo" as a submanifold of ?M. From this point of view, M is called a Wintgen ideal submanifold of ~M when this extrinsic curvature H2 ??? +k actually assumes its theoretically smallest possible value, as given by its intrinsic curvature ?, at all points of M. We show that the pseudo-symmetry or, equivalently, the property to be quasi-Einstein of such 3-dimensional Wintgen ideal submanifolds M3 of M~3+m(k) can be characterized in terms of the intrinsic minimal values of the Ricci curvatures and of the Riemannian sectional curvatures of M and of the extrinsic notions of the umbilicity, the minimality and the pseudo-umbilicity of M in ~M.

scholarly journals CLASSIFICATION OF LAGRANGIAN SURFACES OF CONSTANT CURVATURE IN THE COMPLEX EUCLIDEAN PLANE

2005 ◽  
Vol 48 (2) ◽  
pp. 337-364 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Constant Curvature ◽  
Euclidean Plane ◽  
Lagrangian Submanifolds ◽  
Real Space ◽  
Point Of View ◽  
Space Forms ◽  
Lagrangian Surfaces ◽  
Real Space Forms ◽  
Geometric Point

AbstractOne of the most fundamental problems in the study of Lagrangian submanifolds from a Riemannian geometric point of view is the classification of Lagrangian immersions of real-space forms into complex-space forms. In this article, we solve this problem for the most basic case; namely, we classify Lagrangian surfaces of constant curvature in the complex Euclidean plane $\mathbb{C}^2$. Our main result states that there exist 19 families of Lagrangian surfaces of constant curvature in $\mathbb{C}^2$. Twelve of the 19 families are obtained via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in $\mathbb{C}^2$ can be obtained locally from the 19 families.

scholarly journals Three-dimensional CR-submanifolds in the nearly Kaehler six-sphere satisfying B.Y. Chen's basic equality

2000 ◽  
Vol 31 (4) ◽  
pp. 289-296
Author(s):  
Tooru Sasahara
Keyword(s):  
Mean Curvature ◽  
Three Dimensional ◽  
Real Space ◽  
Sharp Inequality ◽  
Space Forms ◽  
Equality Case ◽  
3 Dimensional ◽  
Real Space Forms ◽  
Basic Equality ◽  
Cr Submanifolds

B. Y. Chen introduced in [3] an important Riemannian invariant for a Riemannian manifold and obtained a sharp inequality between his invariant and the squared mean curvature for arbitrary submanifolds in real space forms. In this paper we investigate 3-dimensional CR-submanifolds in the nearly Kaehler 6-sphere which realize the equality case of the inequality.

scholarly journals On the intrinsic Deszcz symmetries and the extrinsic Chen character of Wintgen ideal submanifolds

2010 ◽  
Vol 41 (2) ◽  
pp. 109-116 ◽  
Author(s):  
S. Decu ◽  
M. Petrovic-Torgasev ◽  
A. Sebekovic ◽  
L. Verstraelen
Keyword(s):  
Ricci Curvature ◽  
Curvature Tensor ◽  
Real Space ◽  
Space Forms ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Real Space Forms ◽  
Wintgen Ideal Submanifolds ◽  
Weyl Conformal Curvature Tensor

In this paper it is shown that all Wintgen ideal submanifolds in ambient real space forms are Chen submanifolds. It is also shown that the Wintgen ideal submanifolds of dimension $ >3 $ in real space forms do intrinsically enjoy some curvature symmetries in the sense of Deszcz of their Riemann--Christoffel curvature tensor, of their Ricci curvature tensor and of their Weyl conformal curvature tensor.

scholarly journals Ricci and Casorati principal directions of Wintgen ideal submanifolds

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 657-661 ◽  
Author(s):  
Simona Decu ◽  
Miroslava Petrovic-Torgasev ◽  
Aleksandar Sebekovic ◽  
Leopold Verstraelend
Keyword(s):  
Real Space ◽  
Space Forms ◽  
Real Space Forms ◽  
Principal Directions ◽  
Wintgen Ideal Submanifolds

We show that for Wintgen ideal submanifolds in real space forms the (intrinsic) Ricci principal directions and the (extrinsic) Casorati principal directions coincide.

Chen Inequalities for Submanifolds of Real Space Forms with a Semi-Symmetric Non-Metric Connection

2012 ◽  
Vol 55 (3) ◽  
pp. 611-622 ◽  
Author(s):  
Cihan Özgür ◽  
Adela Mihai
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Real Space ◽  
Ambient Space ◽  
Space Forms ◽  
Real Space Forms ◽  
Metric Connection ◽  
Sectional Curvatures ◽  
The Mean

AbstractIn this paper we prove Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric non-metric connection, i.e., relations between the mean curvature associated with a semi-symmetric non-metric connection, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered.

Triharmonic Riemannian submersions from 3-dimensional space forms

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tomoya Miura ◽  
Shun Maeta
Keyword(s):  
Existence Theorem ◽  
Space Form ◽  
Dimensional Space ◽  
Riemannian Submersion ◽  
Partial Answer ◽  
Riemannian Submersions ◽  
Space Forms ◽  
3 Dimensional

Abstract We show that any triharmonic Riemannian submersion from a 3-dimensional space form into a surface is harmonic. This is an affirmative partial answer to the submersion version of the generalized Chen conjecture. Moreover, a non-existence theorem for f -biharmonic Riemannian submersions is also presented.

ON ROBERTS’ PROOF OF THE TURAEV-WALKER THEOREM

1996 ◽  
Vol 05 (04) ◽  
pp. 427-439 ◽  
Author(s):  
RICCARDO BENEDETTI ◽  
CARLO PETRONIO
Keyword(s):  
Euler Characteristic ◽  
A Priori ◽  
Algebraic Approach ◽  
Fusion Rule ◽  
Point Of View ◽  
Witten Invariant ◽  
Manifolds With Boundary ◽  
3 Dimensional ◽  
Skein Theory ◽  
Natural Way

In this paper we discuss the beautiful idea of Justin Roberts [7] (see also [8]) to re-obtain the Turaev-Viro invariants [11] via skein theory, and re-prove elementarily the Turaev-Walker theorem [9], [10], [13]. We do this by exploiting the presentation of 3-manifolds introduced in [1], [4]. Our presentation supports in a very natural way a formal implementation of Roberts’ idea. More specifically, what we show is how to explicitly extract from an o-graph (the object by which we represent a manifold, see below), one of the framed links in S3 which Roberts uses in the construction of his invariant, and a planar diagrammatic representation of such a link. This implies that the proofs of invariance and equality with the Turaev-Viro invariant can be carried out in a completely “algebraic” way, in terms of a planar diagrammatic calculus which does not require any interpretation of 3-dimensional figures. In particular, when proving the “term-by-term” equality of the expansion of the Roberts invariant with the state sum which gives the Turaev-Viro invariant, we simultaneously apply several times the “fusion rule” (which is formally defined, strictly speaking, only in diagrammatic terms), showing that the “braiding and twisting” which a priori may exist on tetrahedra is globally dispensable. In our point of view the success of this formal “algebraic” approach witnesses a certain efficiency of our presentation of 3-manifolds via o-graphs. In this work we will widely use recoupling theory which was very clearly exposed in [2], and therefore we will avoid recalling notations. Actually, for the purpose of stating and proving our results we will need to slightly extend the class of trivalent ribbon diagrams on which the bracket can be computed. We also address the reader to the references quoted in [2], in particular for the fundamental contributions of Lickorish to this area. In our approach it is more natural to consider invariants of compact 3-manifolds with non-empty boundary. The case of closed 3-manifolds is included by introducing a correction factor corresponding to boundary spheres, as explained in §2. Our main result is actually an extension to manifolds with boundary of the Turaev-Walker theorem: we show that the Turaev-Viro invariant of such a manifold coincides (up to a factor which depends on the Euler characteristic) with the Reshetikhin-Turaev-Witten invariant of the manifold mirrored in its boundary.

scholarly journals Manifold curvature and Ehrenfest forces with a moving basis

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Jessica Halliday ◽  
Emilio Artacho
Keyword(s):  
Extrinsic Curvature ◽  
Quantum States ◽  
Basis Set ◽  
Electronic Subsystem ◽  
Intrinsic Curvature ◽  
Geometrical Meaning

Known force terms arising in the Ehrenfest dynamics of quantum electrons and classical nuclei, due to a moving basis set for the former, can be understood in terms of the curvature of the manifold hosting the quantum states of the electronic subsystem. Namely, the velocity-dependent terms appearing in the Ehrenfest forces on the nuclei acquire a geometrical meaning in terms of the intrinsic curvature of the manifold, while Pulay terms relate to its extrinsic curvature.

scholarly journals f-Biharmonic Curves in the Three-dimensional Para-Sasakian Space Forms

2021 ◽  
pp. 369-379
Author(s):  
Murat Altunbaş
Keyword(s):  
Three Dimensional ◽  
Space Forms ◽  
3 Dimensional ◽  
Sasakian Space Forms ◽  
Sasakian Structures ◽  
Biharmonic Curves

In this paper, we give some characterizations for proper f-biharmonic curves in the para-Bianchi-Cartan-Vranceanu space forms with 3-dimensional para-Sasakian structures.

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