Non-singular weakly symmetric nilmanifolds

A Riemannian manifold is called weakly symmetric if any two points in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> can be interchanged by an isometry. We give a complete classification of simply connected non-singular weakly symmetric nilmanifolds. Besides previously known examples, there are new families with 3-dimensional center, and a one-parameter family of dimensions 14.