BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Viviana del Barco (Universidade Estadual de Campinas) DTSTART:20210623T150000Z DTEND:20210623T160000Z DTSTAMP:20260423T053134Z UID:DGSTO/18 DESCRIPTION:Title: ( Purely) coclosed G$_2$-structures on 2-step nilmanifolds\nby Viviana d el Barco (Universidade Estadual de Campinas) as part of Differential Geome try Seminar Torino\n\n\nAbstract\nIn Riemannian geometry\, simply connecte d nilpotent Lie groups endowed with left-invariant metrics\, and their com pact quotients\, have been the source of valuable examples in the field. T his motivated several authors to study\, in particular\, left-invariant G$ _2$-structures on 7-dimensional nilpotent Lie groups. These structures cou ld also be induced to the associated compact quotients\, also known as nil manifolds.\n\nLeft-invariant torsion free G$_2$-structures\, that is\, def ined by a simultaneously closed and coclosed positive $3$-form\, do not ex ist on nilpotent Lie groups. But relaxations of this condition have been t he subject of study on nilmanifolds lately. One of them are coclosed G$_2$ -structures\, for which the defining $3$-form verifies $\\mathrm{d} \\star _{g_\\varphi}\\varphi=0$\, and more specifically\, purely coclosed structu res\, which are defined as those which are coclosed and satisfy $\\varphi\ \wedge \\mathrm{d} \\varphi=0$. \n\nIn this talk\, there will be presented recent classification results regarding left-invariant coclosed and purel y coclosed G$_2$-structures on 2-step nilpotent Lie groups. \n\nOur result s are twofold. On the one hand we give the isomorphism classes of 2-step n ilpotent Lie algebras admitting purely coclosed G$_2$-structures. The anal ogous result for coclosed structures was obtained by Bagaglini\, Fernánde z and Fino [Forum Math. 2018]. \n\nOn the other hand\, we focus on the que stion of which metrics on these Lie algebras can be induced by a coclosed or purely coclosed structure. We show that any left-invariant metric is in duced by a coclosed structure\, whereas every Lie algebra admitting purely coclosed structures admits metrics which are not induced by any such a st ructure. In the way of proving these results we obtain a method to constru ct purely coclosed G$_2$-structures. As a consequence\, we obtain new exam ples of compact nilmanifolds carrying purely coclosed G$_2$-structures. \n \nThis is joint work with Andrei Moroianu and Alberto Raffero.\n LOCATION:https://researchseminars.org/talk/DGSTO/18/ END:VEVENT END:VCALENDAR