Differential Geometry Working Seminar

Timothy Ponepal, Wilfrid Laurier University

"The flow of the horizontal lift of a vector field"

Let $E$ be a vector bundle over a manifold $M$, and let $\nabla$ be a connection on $E$. Given a vector field $X$ on $M$, the connection determines its horizontal lift $X^h$, which is a vector field on the total space of $E$. We will show that the flow of $X^h$ is related to parallel transport with respect to $\nabla$. If time permits, we will show that in the special case when $E$ is a rank 3 oriented real vector bundle with fibre metric, the flow of $X^h$ preserves the cross product on the fibres.

MC 5403