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Given a metric $$\mathrm{d}s^2 = g_{MN}\mathrm{d}x^M\mathrm{d}x^N$$ in $n$ dimensions you find the decomposition down to $d$ dimensions by rewriting the $n$-dimensional metric in terms of objects with no, one, and two $d$-dimensional indices (in the following indicated by Greek letters): $$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu + 2 A_{\... 1 The significance of the compactified circle as opposed to having a non-compact fifth dimension is that a compact dimension produces the discrete "Kaluza-Klein tower of states" in the effective four-dimensional theory - due to the scalar field then having a discrete Fourier series in the fifth coordinate, which, for small radii of the circle, produces one ... 0 The main consideration for the internal space, called a Calabi-Yau manifold, in a Kaluza-Klein (KK) theory is that it be Ricci flat. The reason for this is that if there is a nonzero Ricci tensor then a string that is wound on the internal space will grow in size. This has something to do with the Hamilton equations$$ \frac{dg_{ab}}{dt}~=~-2R_{ab}  The ...

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String theory postulates that of the elementary particles we currently know about, each relates directly to low-energy string vibrations, the presence of multiple holes causes the string patterns to fall into families. Each hole in the Calabi-Yau space is a group of low-energy string vibrational patterns. If the C-Y has three holes, then three families of ...

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It just means that the energy (e.g. in the GR language, the ADM energy) is minimized among all configurations with the same boundary conditions. It means that there are no gravitational or electromagnetic or other waves inside the space. In practice, it just means that the geometry is a Cartesian product $M^4\times Y$ where $Y$ is the manifold of compact ...

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