  I develop the idea that our world is a brane-like object embedded in
Euclidean bulk. In its ground state, the brane constituent matter is assumed to
be homogeneous and isotropic, and of negligible influence on the bulk geometry.
The analysis of this paper is model independent, in the sense that action
functional of bulk fields is not specified. Instead, the behavior of the brane
is derived from the universally valid conservation equation of the bulk stress
tensor. The present work studies the behavior of a $3$-sphere in the
$5$-dimensional Euclidean bulk. The sphere is made of bulk matter characterized
by the equation of state $p=\alpha\rho$. It is shown that stability of brane
vibrations requires $\alpha < 0$. Then, the stable brane perturbations obey
Klein-Gordon-like equation with an effective metric of Minkowski signature. The
argument is given that it is this effective metric that is detected in physical
measurements. The corresponding effective Universe is analyzed for all the
values of $\alpha<0$. In particular, the effective metric is shown to be a
solution of Einstein's equations coupled to an effective perfect fluid. The
effective energy density and pressure at the present epoch are calculated. So
are the age of the Universe, and the effective cosmological constant. All the
results are presented in two tables. As an illustration, one simple choice of
the brane constituent matter is studied in detail.
