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Curvature as infinitesimal holonomy

The equation hol$ ( - \nabla, \partial D) = \exp \ (- \int_{D} F)$ has an infinitesimal counterpart. If $ X$ and $ Y$ are two tangent vectors and we let $ D_t$ be a parallelogram with sides $ tX$ and $ tY$ then the holonomy around $ D_t$ can be expanded in powers of $ t$ as

   hol$\displaystyle \ (\nabla, D_{t}) = 1 + t^2 \ F (X, Y) + 0 (t^{3}).
$



Michael Murray
1998-09-16