## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Thursday, April 21, 2022 — 1:00 PM EDT

**Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo**

**"Calibrated subbundles of $\mathbb R^7$****"**

One can view $\mathbb R^7$ as the total space of the bundle $E = \Lambda^2_- (\mathbb R^4)$ of anti-self-dual 2-forms on $\mathbb R^4$. In this way we can describe the standard flat $G_2$-structure in terms of 4-dimensional geometry. Given an oriented surface $M^2$ in $\mathbb R^4$, the restriction of $E$ to $M$ decomposes as a direct sum of a line bundle and a rank 2 bundle. We determine conditions on the immersion of $M^2$ in $\mathbb R^4$ that are equivalent to the total spaces of the subbundles being (respectively) associative and coassociative submanifolds. This is (very old) work of myself, Ionel, and Min-Oo from 2005. I will also discuss several ways in which it has already been generalized and ways in which it can potentially still be generalized further.

MC 5403

Event tags

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

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