general relativity - Extra dimensions

String theory, Kaluza Klein theory etc. need extra-dimensions.
For string theory these are compactified.
My feeling is that these extra dimensions are not of our 4-D space, but are only of the space in which our universe is EMBEDDED, and what we feel as electro-magentism and the other interactions are related to the extrinsic curvature of our 4-D space in these extra dimensions.
I.e. gravity is the only one which derives from intrinsic curvature, all other forces are "tidal forces" connected with extrinsic curvature.

I.e. a spiral curve in 3-D space is intrinsic flat, but it should be somehow different for a 1-D creature to go on a straight line or on a spiral.
From the point of view of a 3-D creature looking to the 1-D creature, the latter has to experience a centrifugal force.