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Register a new userOn $3$-dimensional $left(varepsilon right)$-para Sasakian manifold
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Punam Gupta
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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.
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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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We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit a semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable simply connected symplectic 4-manifold with disjoint symplectic surfaces spanning the homology, all of them but one of genus 1 and the other of genus g>1, (b) to prove a bound on the second Betti number $b_2$ of an algebraic surface with $b_1=0$ and having disjoint complex curves spanning the homology when all of them but one are of genus 1 and the other of genus g>1.
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