  The tubular geometry (T-geometry) is a generalization of the proper Euclidean
geometry, founded on the property of sigma-immanence. The proper Euclidean
geometry can be described completely in terms of the world function $\sigma
=\rho ^{2}/2$, where $\rho $ is the distance. This property is called the
sigma-immanence. Supposing that any physical geometry is sigma-immanent, one
obtains the T-geometry $\mathcal{G}$, replacing the Euclidean world function
$\sigma_{E}$ by means of $\sigma $ in the sigma-immanent presentation of the
Euclidean geometry. One obtains the T-geometry $\mathcal{G}$, described by the
world function $\sigma $. This method of the geometry construction is very
simple and effective. T-geometry has a new geometric property: nondegeneracy of
geometry. The class of homogeneous isotropic T-geometries is described by a
form of a function of one parameter. Using T-geometry as the space-time
geometry one can construct the deterministic space-time geometries with
primordially stochastic motion of free particles and geometrized particle mass.
Such a space-time geometry defined properly (with quantum constant as an
attribute of geometry) allows one to explain quantum effects as a result of the
statistical description of the stochastic particle motion (without a use of
quantum principles). Geometrization of the particle mass appears to be
connected with the restricted divisibility of the straight line segments. The
statement, that the problem of the elementary particle mass spectrum is rather
a problem of geometry, than that of dynamics, is a corollary of the particle
mass geometrization.
