I am currently looking into the compactification of spacetimes as it is often done in (super-)stringtheory.
So, say I start with a ten-dimensional Lorentz manifold $(N, g)$, where $N$ denotes the underlying smooth manifold and $g$ denotes the Lorentzian metric on $N$.
Compactification would then lead to a ten-dimensional Lorentz manifold $(\widetilde{N}, g_{\widetilde{N}})$ that decomposes into product of a four-dimensional Lorentz manifold $(M^4, g_M)$ and a six-dimensional Riemannian manifold $(C^6, g_C)$.
My questions pertains to the Lorentzian metric $g_{\widetilde{N}}$ of $\widetilde{N}$.
Is $(\widetilde{N}, g_{\widetilde{N}})$ assumed to be a direct product manifold or are people also interested in warped products as compactified spacetimes?
By a warped product I mean the following construction:
Let $(M, g_M)$ and $(N, g_N)$ be two semi-Riemannian manifolds and let $f\in C^\infty(M, \mathbb{R}_+)$. Then the warped product of $(M, g_M)$ and $(N, g_N)$ is the following semi-Riemannian manifold:
$M\times_f N :=(M\times N, g_M + f^2g_N)$.
The direct product is obviously a special case of the warped product where $f\equiv 1$.
I'd love to know whether warped products are being considered by physicists working with compactifications at all or not and of how much interest they are compared to direct product compactifictions.
