Weak gravitation from a small extra 2D sphere

In order to explain weak gravitation in our 4-dimensional universe, a 6-dimensional model with a small extra 2D sphere is proposed. The traceless energy-momentum tensor is quite naturally appeared in our 6-dimensional model. The warp factor is given by $\phi (\theta ) = \epsilon + \sin{\theta }$, where $\epsilon$ plays a role of killing the singular point $\phi (\theta )=0$, and is assumed $0 < \epsilon \ll 1$. Any massive particle is rolling down into points along this geodesic line. The light ray can be shown to stay in our 4-dimensional universe. This suggest us that our 4-dimensional world can be located at $\theta =0$ and/or $\theta = \pi$, its background metric being $\epsilon ^2 \eta _{\mu \nu }$. As a result, we have the 4-dimensional Newton constant, which is given by $G_N \simeq G_6 \epsilon ^{10}$ and the fifth force coefficients appeared here are $\alpha _i\simeq \epsilon ^{2(i-4)}$, $i=1, 2, 3$. Here $G_6$ is the gravitational constant in 6-dimensional spacetime. If we take $\epsilon = 10^{-3.8}$ against $G_6\sim 1($GeV$)^{-2}$, we get $G_N\sim 10^{-38}($Gev$)^{-2}$, the present time gravitational constant.