A note on Canonical Ricci forms on 2-step nilmanifolds
Luigi Vezzoni
TL;DR
This paper proves that all left-invariant almost Hermitian structures on 2-step nilmanifolds are Ricci-flat with respect to the Chern connection, and characterizes when they are Ricci-flat for other canonical connections.
Contribution
It establishes Ricci-flatness of invariant structures on 2-step nilmanifolds and characterizes conditions for Ricci-flatness under other canonical connections.
Findings
All invariant structures are Ricci-flat w.r.t. Chern connection.
Ricci-flatness under other canonical connections occurs iff the structure is cosymplectic.
Abstract
In this note we prove that any left-invariant almost Hermitian structure on a 2-step nilmanifold is Ricci-flat with respect to the Chern connection and that it is Ricci-flat with respect to another canonical connection if and only if it is cosymplectic.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Advanced Differential Geometry Research