Differential Geometry Working Seminar

Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo

"Metric compatible connections in dimension 3"

Let $(M, g)$ be an oriented Riemannian manifold. If $D$ is a $g$-compatible connection on $TM$, then the difference $D - \nabla$, where $\nabla$ is the Levi-Civita connection of $g$, is uniquely determined by the torsion $T$ of $D$. The Ricci curvature $F_{ij}$ of $D$ is in general not symmetric. Its skew part can be expressed in terms of the torsion $T$ and its covariant derivative $\nabla T$. In dimension $3$, we can further exploit the fact that $\Lambda^2 T^* M \cong T^* M$ via the Hodge star to express the torsion as a $2$-tensor on $M$. Moreover, in dimension $3$, even if $T \neq 0$, the curvature $4$-tensor $F_{ijkl}$ of $D$ is still completely determined by the Ricci tensor $F_{ij}$. I will explain these various facts and briefly discuss why I am interested in such objects.

Zoom meeting:

• Meeting ID: 958 7361 8652
• Passcode: 577854