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

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.

Geometry of concircular curvature tensor of Kahler manifolds

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

Non-conformal harmonic maps into the 3-sphere

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.

Gravity and induced matter on nearly Kähler manifolds

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

In 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.