The Codazzi-Mainardi equation is a set of equations in differential geometry that relates the second fundamental form of a submanifold to its curvature and the ambient space's curvature. These equations are crucial in understanding how submanifolds curve within higher-dimensional spaces, providing insights into the intrinsic and extrinsic geometry of the submanifold. They play a significant role in describing the compatibility conditions for the embedding of submanifolds.
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