Effects of spatial curvature and anisotropy on the asymptotic regimes in
  Einstein-Gauss-Bonnet gravity
  In this paper we address two important issues which could affect reaching the
exponential and Kasner asymptotes in Einstein-Gauss-Bonnet cosmologies --
spatial curvature and anisotropy in both three- and extra-dimensional
subspaces. In the first part of the paper we consider cosmological evolution of
spaces being the product of two isotropic and spatially curved subspaces. It is
demonstrated that the dynamics in $D=2$ (the number of extra dimensions) and $D
\geqslant 3$ is different. It was already known that for the $\Lambda$-term
case there is a regime with "stabilization" of extra dimensions, where the
expansion rate of the three-dimensional subspace as well as the scale factor
(the "size") associated with extra dimensions reach constant value. This regime
is achieved if the curvature of the extra dimensions is negative. We
demonstrate that it take place only if the number of extra dimensions is $D
\geqslant 3$. In the second part of the paper we study the influence of initial
anisotropy. Our study reveals that the transition from Gauss-Bonnet Kasner
regime to anisotropic exponential expansion (with expanding three and
contracting extra dimensions) is stable with respect to breaking the symmetry
within both three- and extra-dimensional subspaces. However, the details of the
dynamics in $D=2$ and $D \geqslant 3$ are different. Combining the two
described affects allows us to construct a scenario in $D \geqslant 3$, where
isotropisation of outer and inner subspaces is reached dynamically from rather
general anisotropic initial conditions.
