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Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
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Holonomy bounded in terms of area and the curvature
There are in fact more precise versions, expressing the parallel translation around a loop as the identity map plus a curvature integral over a homotopy. … References:
Section 3.1 of Werner Ballmann's lecture notes on vector bundles:
http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/conncurv1999.pdf
Deane Yang's notes "Holonomy equals curvature" on his web page …