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Can you tell me the difference or physical application of Kaluza Klein approach and Gauss Codazzi approach?

In Kaluza Klein theory, 5 dimensional theory can be dimensional reduced to 4 dimensional theory with additional matter (Gauge fields, scalars, etc)

And In Gauss-Codazzi equation, for example, higher dimensional theory can be explained via submanifold information.

I want to know the similarity and difference between two approaches.

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    $\begingroup$ what do you call Gauss-Codazzi approach? If you refer to the relation of curvature of an ambient manifold and a submanifold, what does it have to do with Kaluza Klein? $\endgroup$ – MBN Jan 13 at 9:37

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