scholarly journals Clairaut Submersion

2021 ◽  
Author(s):  
Sanjay Kumar Singh ◽  
Punam Gupta
Keyword(s):  
Differential Geometry ◽  
Kähler Manifold ◽  
Total Space ◽  
Geometric Properties ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Kahler Manifold ◽  
Nearly Kähler ◽  
Nearly Kähler Manifold ◽  
Clairaut Submersion

In this chapter, we give the detailed study about the Clairaut submersion. The fundamental notations are given. Clairaut submersion is one of the most interesting topics in differential geometry. Depending on the condition on distribution of submersion, we have different classes of submersion such as anti-invariant, semi-invariant submersions etc. We describe the geometric properties of Clairaut anti-invariant submersions and Clairaut semi-invariant submersions whose total space is a Kähler, nearly Kähler manifold. We give condition for Clairaut anti-invariant submersion to be a totally geodesic map and also study Clairaut anti-invariant submersions with totally umbilical fibers. We also give the conditions for the semi-invariant submersions to be Clairaut map and also for Clairaut semi-invariant submersion to be a totally geodesic map. We also give some illustrative example of Clairaut anti-invariant and semi-invariant submersion.

On the Kähler-likeness on nearly Kähler manifolds

2020 ◽  
pp. 2150132
Author(s):  
Masaya Kawamura
Keyword(s):  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Hermitian Metrics ◽  
Kahler Manifold ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds ◽  
Nearly Kähler Manifold

We introduce Kähler-like, G-Kähler-like almost Hermitian metrics. We characterize the Kähler-likeness and the G-Kähler-likeness, and show that these properties are equivalent on nearly Kähler manifolds. Furthermore, we prove that a nearly Kähler manifold with the Kähler-likeness is Kähler.

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 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.

On 6-Dimensional Nearly Kähler Manifolds

2010 ◽  
Vol 53 (3) ◽  
pp. 564-570 ◽  
Author(s):  
Yoshiyuki Watanabe ◽  
Young Jin Suh
Keyword(s):  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Sufficient Condition ◽  
Kahler Manifolds ◽  
Kahler Manifold ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds ◽  
Nearly Kähler Manifold ◽  
Simply Connected

AbstractIn this paper we give a sufficient condition for a complete, simply connected, and strict nearly Kähler manifold of dimension 6 to be a homogeneous nearly Kähler manifold. This result was announced in a previous paper by the first author.

Differential forms on a Kähler manifold

1951 ◽  
Vol 47 (3) ◽  
pp. 504-517 ◽  
Author(s):  
W. V. D. Hodge
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Complex Manifold ◽  
Differential Forms ◽  
Kähler Manifold ◽  
Systematic Account ◽  
General Plan ◽  
Kahler Manifold ◽  
Frequent Use ◽  
Compact Kähler Manifold

While a number of special properties of differential forms on a Kähler manifold have been mentioned in the literature on complex manifolds, no systematic account has yet been given of the theory of differential forms on a compact Kähler manifold. The purpose of this paper is to show how a general theory of these forms can be developed. It follows the general plan of de Rham's paper (2) on differential forms on real manifolds, and frequent use will be made of results contained in that paper. For convenience we begin by giving a brief account of the theory of complex tensors on a complex manifold, and of the differential geometry associated with a Hermitian, and in particular a Kählerian, metric on such a manifold.

scholarly journals Geometry of concircular curvature tensor of Kahler manifolds

2020 ◽  
Vol 25 (2) ◽  
pp. 110
Author(s):  
Rawah A. Zaben1 ◽  
, Rana H. Jasim2
Keyword(s):  
Curvature Tensor ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Necessary Condition ◽  
Kahler Manifolds ◽  
Geometrical Meaning ◽  
Nearly Kähler ◽  
Concircular Curvature Tensor ◽  
Nearly Kähler Manifold ◽  
The Relationship

The study deals with the necessary condition where a nearly Kahler manifold of flat concircular tensor has been found. And the relationship between these invariants and additional properties of symmetry concircular tensor, as well as geometrical meaning of the reference in zero of these invariants .   http://dx.doi.org/10.25130/tjps.25.2020.037

scholarly journals Non-Umbilical Quaternionic Contact Hypersurfaces in Hyper-Kähler Manifolds

2017 ◽  
Vol 2019 (18) ◽  
pp. 5649-5673
Author(s):  
Stefan Ivanov ◽  
Ivan Minchev ◽  
Dimiter Vassilev
Keyword(s):  
Heisenberg Group ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Totally Umbilical ◽  
Kahler Manifold ◽  
Compact Case ◽  
Quaternionic Heisenberg Group

Abstract It is shown that any compact quaternionic contact (qc) hypersurfaces in a hyper-Kähler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. In the non-compact case, it is proved that a seven-dimensional everywhere non-umbilical qc-hypersurface embedded in a hyper-Kähler manifold is qc-conformal to a qc-Einstein structure which is locally qc-equivalent to the 3-Sasakian sphere, the quaternionic Heisenberg group or the hyperboloid.

scholarly journals H-Umbilical Lagrangian Submanifolds of the Nearly Kähler \( {\mathbb{S}^3\times\mathbb{S}^3} \)

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1427
Author(s):  
Miroslava Antić ◽  
Marilena Moruz ◽  
Joeri Van der Veken
Keyword(s):  
Lagrangian Submanifolds ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Nearly Kähler

H-umbilicity was introduced as an analogue of total umbilicity for Lagrangian submanifolds since, in some relevant cases, totally umbilical Lagrangian submanifolds are automatically totally geodesic. In this paper, we show that, in the homogeneous nearly Kähler S3×S3, also H-umbilical Lagrangian submanifolds are automatically totally geodesic.

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