scholarly journals Almost Complex Surfaces in the Nearly Kähler SL(2,ℝ) × SL(2,ℝ)

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1160
Author(s):  
Elsa Ghandour ◽  
Luc Vrancken
Keyword(s):  
Complete Classification ◽  
Second Fundamental Form ◽  
Complex Surfaces ◽  
Kähler Structure ◽  
Totally Geodesic ◽  
Parallel Second Fundamental Form ◽  
Almost Complex ◽  
Nearly Kähler ◽  
Nearly Kähler Structure

The space S L ( 2 , R ) × S L ( 2 , R ) admits a natural homogeneous pseudo-Riemannian nearly Kähler structure. We investigate almost complex surfaces in this space. In particular, we obtain a complete classification of the totally geodesic almost complex surfaces and of the almost complex surfaces with parallel second fundamental form.

Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space

2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Space Form ◽  
Complete Classification ◽  
Second Fundamental Form ◽  
Normal Space ◽  
Space Forms ◽  
Parallel Surface ◽  
Parallel Surfaces ◽  
Point To Point ◽  
Lorentz Surface

AbstractA Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean 4-space $$ \mathbb{E}_2^4 $$ and in neutral pseudo 4-sphere S 24 (1) were classified in [14] and in [10], respectively. In this paper, we completely classify parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space H 24 (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H 24 (−1). Conversely, every parallel Lorentz surface in H 24 (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel Lorentz surfaces in 4D neutral indefinite space forms.

scholarly journals Non-conformal harmonic maps into the 3-sphere

2015 ◽  
Vol 12 (08) ◽  
pp. 1560012
Author(s):  
Bart Dioos
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
Kähler Manifold ◽  
Complex Surfaces ◽  
Kahler Manifold ◽  
Almost Complex ◽  
Nearly Kähler ◽  
Nearly Kähler Manifold

We present two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct from one non-conformal harmonic map a sequence of non-conformal harmonic maps. We also discuss the correspondence between non-conformal harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold S3 × S3.

scholarly journals TOTALLY REAL SURFACES IN $S^6$

2021 ◽  
Vol 23 (1) ◽  
pp. 11-14
Author(s):  
SHARIEF DESHMUKH
Keyword(s):  
Gaussian Curvature ◽  
Complex Structure ◽  
Second Fundamental Form ◽  
Real Surface ◽  
Almost Complex Structure ◽  
Constant Gaussian Curvature ◽  
Totally Geodesic ◽  
The Second Fundamental Form ◽  
Almost Complex ◽  
Totally Real

The normal bundle $\bar \nu$ of a totally real surface $M$ in $S^6$ splits as $\bar\nu= JTM\oplus \bar\mu$ where $TM$ is the tangent bundle of $M$ and  $\bar\mu$ is sub­bundle of $\bar\nu$ which is invariant under the almost complex structure $J$. We study the totally real surfaces M of constant Gaussian curvature K for which the second fundamental form $h(x, y) \in JTM$, and we show that $K = 1$ (that is, $M$ is totally geodesic).

scholarly journals Almost complex surfaces in the nearly Kähler $S^3\times S^3$

2015 ◽  
Vol 67 (1) ◽  
pp. 1-17 ◽  
Author(s):  
John Bolton ◽  
Franki Dillen ◽  
Bart Dioos ◽  
Luc Vrancken
Keyword(s):  
Complex Surfaces ◽  
Almost Complex ◽  
Nearly Kähler

scholarly journals Five-Dimensional Contact CR-Submanifolds in S 7 ( 1 )

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1278
Author(s):  
Mirjana Djorić ◽  
Marian Ioan Munteanu
Keyword(s):  
Contact Structure ◽  
Sasakian Manifold ◽  
Complete Classification ◽  
Second Fundamental Form ◽  
Remarkable Property ◽  
Dimensional Unit ◽  
Products Of Spheres ◽  
Kählerian Manifolds ◽  
Cr Submanifolds

Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure (φ,ξ,η), we study its five-dimensional contact CR-submanifolds, which are the analogue of CR-submanifolds in (almost) Kählerian manifolds. In the case when the structure vector field ξ is tangent to M, the tangent bundle of contact CR-submanifold M can be decomposed as T(M)=H(M)⊕E(M)⊕Rξ, where H(M) is invariant and E(M) is anti-invariant with respect to φ. On this occasion we obtain a complete classification of five-dimensional proper contact CR-submanifolds in S7(1) whose second fundamental form restricted to H(M) and E(M) vanishes identically and we prove that they can be decomposed as (multiply) warped products of spheres.

scholarly journals Gravity and induced matter on nearly Kähler manifolds

2015 ◽  
Vol 30 (03) ◽  
pp. 1550015 ◽  
Author(s):  
F. Naderi ◽  
A. Rezaei-Aghdam ◽  
F. Darabi
Keyword(s):  
Kähler Manifold ◽  
Fine Tuning ◽  
Potential Candidate ◽  
Conservation Of Energy ◽  
Kähler Structure ◽  
Kahler Manifold ◽  
Nearly Kähler ◽  
Nearly Kähler Structure ◽  
Nearly Kähler Manifold ◽  
Induced Matter

We show that the conservation of energy–momentum tensor of a gravitational model with Einstein–Hilbert like action on a nearly Kähler manifold with the scalar curvature of a curvature-like tensor, is consistent with the nearly Kähler properties. In this way, the nearly Kähler structure is automatically manifested in the action as a induced matter field. As an example of nearly Kähler manifold, we consider the group manifold of R×II ×S3×S3 on which we identify a nearly Kähler structure and solve the Einstein equations to interpret the model. It is shown that the nearly Kähler structure in this example is capable of alleviating the well known fine tuning problem of the cosmological constant. Moreover, this structure may be considered as a potential candidate for dark energy.

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