Differential Geometry Working Seminar

Amanda Petcu, Department of Pure Mathematics, University of Waterloo

"Some Calculations Regarding $G_2$ and the Isometric Flow I"

In the paper [1] the authors consider the following setup for a $7$-dimensional manifold $M$. Given a hypersymplectic structure $\underline{\omega}$ on $X^4$ they consider the manifold $M = X^4 \times T^3$ where $T^3 = S^1 \times S^1 \times S^1$. With this setup, they consider a closed $3$-form $\varphi$ on $M$ that gives a $G_2$ structure. This is due to the hypersymplectic structure on $X^4$. In the case where $X^4$ is compact the authors deform the hypersymplectic structure on $X^4$ to a hyperkahler triple. Then the $G_2$-structure $\varphi$ on $M = X^4 \times T^3 $ has vanishing torsion forms when $\underline{\omega}$ is a hyperkahler triple implying that $\varphi$ determines a $G_2$-metric.

In this talk, I will loosen the conditions on $X^4$ to pre-hypersymplectic and compute the forms $\varphi$ and $*_{7} \varphi = \psi$. We will also compute the four torsion forms for $M$ and determine what conditions are needed in order for the torsion forms to vanish. Finally given the full torsion tensor $T$ of $M$, we will compute $\operatorname{Div} T$ in terms of the four torsion forms and determine what conditions we might need in order for $\operatorname{Div} T$ to vanish.

[1] J.Fine and C. Yao, ``Hypersymplectic 4-manifolds, the $G_2$-Laplacian flow, and extension assuming bounded scalar curvature'', Duke University Press, vol. 167, no. 18, 2018, doi: 10.1215/00127094-2018-0040

MC 5417

Note special time for this seminar.