  In this paper, I construct a class of everywhere regular geometric sigma
models that possess bouncing solutions. Precisely, I show that every bouncing
metric can be made a solution of such a model. My previous attempt to do so by
employing one scalar field has failed due to the appearance of harmful
singularities near the bounce. In this work, I use four scalar fields to
construct a class of geometric sigma models which are free of singularities.
The models within the class are parametrized by their background geometries. I
prove that, whatever background is chosen, the dynamics of its small
perturbations is classically stable on the whole time axes. Contrary to what
one expects from the structure of the initial Lagrangian, the physics of
background fluctuations is found to carry 2 tensor, 2 vector and 2 scalar
degrees of freedom. The graviton mass, that naturally appears in these models,
is shown to be several orders of magnitude smaller than its experimental bound.
I provide three simple examples to demonstrate how this is done in practice. In
particular, I show that graviton mass can be made arbitrarily small.

General Relativity and Quantum Cosmology