To say it right away, I am not an expert in string theory, but I know well General Relativity. So I wonder how the curling up of extra-dimensions which is assumed in many "Kaluza-Klein" like theories (string theory, Randall-Sundrum etc.) is actually created. The extra dimensions in these theories are apparently warped, and therefore their geometry must be characterized by a non-zero curvature (Riemann tensor). According to General Relativity the latter one should be generated by a non-zero energy-momentum tensor which must have its origin in a significant matter Lagrangian $L\_{matter}$. $$T\_{\mu\nu}= -\frac{2}{\sqrt{-g}}\frac{\delta (L\_{matter})}{\delta(g\_{\mu\nu})}.$$ This finally would mean that the matter as origin of this warping must be significant in order to produce this warping of extra dimensions on an extremely small scale. Is this conclusion correct or not?