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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
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Approximating the parallel transport map on a curve with the covariant derivative
let $X,Y:M\to TM$ be vector fields on $M$. $\nabla_XY$ is the change in $Y$ along the flow curves of $X$. so for a point $p \in M$ let $\phi^X(t):\mathbb{R}\to M$ be a flow curve of $X$ passing throug …