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Register a new userEinstein like $(varepsilon)$-para Sasakian manifolds
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Added by
Mukut Mani Tripathi Dr.
Publication date
2012
fields
Physics
and research's language is
English
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Einstein like $(varepsilon)$-para Sasakian manifolds are introduced. For an $(varepsilon) $-para Sasakian manifold to be Einstein like, a necessary and sufficient condition in terms of its curvature tensor is obtained. The scalar curvature of an Einstein like $(varepsilon) $-para Sasakian manifold is obtained and it is shown that the scalar curvature in this case must satisfy certain differential equation. A necessary and sufficient condition for an $(varepsilon) $-almost paracontact metric hypersurface of an indefinite locally Riemannian product manifold to be $(varepsilon) $-para Sasakian is obtained and it is proved that the $(varepsilon) $-para Sasakian hypersurface of an indefinite locally Riemannian product manifold of almost constant curvature is always Einstein like.
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The purpose of the present paper is to study the globally and locally $varphi $-${cal T}$-symmetric $left( varepsilon right) $-para Sasakian manifold in dimension $3$. The globally $varphi $-$ {cal T}$-symmetric $3$-dimensional $left( varepsilon right) $-para Sasakian manifold is either Einstein manifold or has a constant scalar curvature. The necessary and sufficient condition for Einstein manifold to be globally $varphi $-${cal T}$ -symmetric is given. A $3$-dimensional $% left( varepsilon right) $ -para Sasakian manifold is locally $varphi $-$ {cal T}$-symmetric if and only if the scalar curvature $r$ is constant. A $3 $-dimensional $left( varepsilon right) $-para Sasakian manifold with $% eta $-parallel Ricci tensor is locally $varphi $-${cal T}$-symmetric. In the last, an example of $3$-dimensional locally $varphi $-${cal T}$-symmetric $left( varepsilon right) $-para Sasakian manifold is given.
We study the local geometry of 4-manifolds equipped with a emph{para-Kahler-Einstein} (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated emph{twistor distribution}, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with nonvanishing Einstein constant this twistor distribution has exactly two integral leaves and is `maximally non-integrable on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-vanishing Einstein constant and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartans method of equivalence to produce a large number of explicit examples of pKE metrics with nonvanishing Einstein constant whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type $D,$ we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding emph{Cartan connections} satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.
We study a natural contact instanton (CI) equation on gauge fields over 7-dimensional Sasakian manifolds, which is closely related both to the transverse Hermitian Yang-Mills (tHYM) condition and the G_2-instanton equation. We obtain, by Fredholm theory, a finite-dimensional local model for the moduli space of irreducible solutions. We derive cohomological conditions for smoothness, and we express its dimension in terms of the index of a transverse elliptic operator. Finally we show that the moduli space of selfdual contact instantons (ASDI) is Kahler, in the Sasakian case. As an instance of concrete interest, we specialise to transversely holomorphic Sasakian bundles over contact Calabi-Yau 7-manifolds, and we show that, in this context, the notions of contact instanton, integrable G_2-instanton and HYM connection coincide.
We study 7D maximally supersymmetric Yang-Mills theory on 3-Sasakian manifolds. For manifolds whose hyper-Kahler cones are hypertoric we derive the perturbative part of the partition function. The answer involves a special function that counts integer lattice points in a rational convex polyhedral cone determined by hypertoric data. This also gives a more geometric structure to previous enumeration results of holomorphic functions in the literature. Based on physics intuition, we provide a factorisation result for such functions. The full proof of this factorisation using index calculations will be detailed in a forthcoming paper.
The aim of this paper is to study Sasakian immersions of (non-compact) complete regular Sasakian manifolds into the Heisenberg group and into $ mathbb{B}^Ntimes mathbb{R}$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the $eta$-Einstein case.
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