In Dorpat he taught Karl Peterson and supervised his doctoral thesis that established the Gauss Bonnet theorem and derived Gauss Codazzi equations.
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These equations can also be Pierre Bonnet proved that two quadratic forms satisfying the Gauss-Codazzi equations always uniquely determine an embedded surface locally.
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For this reason the Gauss-Codazzi equations are often called the fundamental equations for embedded surfaces, precisely identifying where the intrinsic and extrinsic curvatures come from.
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