Critical Homogeneous Metrics on the Heisenberg Manifold
Copyright policy )Critical Homogeneous Metrics on the Heisenberg Manifold
Let \(H\) be the real \(3\)-dimensional Heisenberg group and \(M = H/ \Gamma\) its compact quotient by a uniform discrete subgroup \(\Gamma\) with integral entries. In the present paper the author gets a necessary and sufficient condition for a left-invariant metric \(g\) on \(M\) to be a critical point for the total scalar curvature functional defined on a particular space of left-invariant metrics on \(M\). As a consequence, he gets that there exists no left-invariant Einstein metric on \(M\). Actually this follows also from the fact that on a nilpotent Lie group there does not exist any left-invariant Einstein metric by [\textit{J. Milnor}, Adv. Math. 21, 293--329 (1976; Zbl 0341.53030)].
- IE University Spain
- Pusan National University Korea (Republic of)
Special Riemannian manifolds (Einstein, Sasakian, etc.), Differential geometry of homogeneous manifolds, Critical metrics, Nilpotent and solvable Lie groups, Einstein metric, critical point, Heisenberg group, Global Riemannian geometry, including pinching
Special Riemannian manifolds (Einstein, Sasakian, etc.), Differential geometry of homogeneous manifolds, Critical metrics, Nilpotent and solvable Lie groups, Einstein metric, critical point, Heisenberg group, Global Riemannian geometry, including pinching
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