  Nondegenerate geometry (T-geometry) with nonsymmetric world function is
considered. In application to the space-time geometry the asymmetry of world
function means that the past and the future are not equivalent geometrically.
T-geometry is described in terms of finite point subspaces and world function
between pairs of points of these subsets, i.e. in the language which is
immanent to geometry and free of external means of description (coordinates,
curves). Such a description appears to be simple and effective even in the case
of complicated T-geometry. Antisymmetric component of the world function
generates appearance of additional metric fields. This leads to appearance of
three sorts of Christoffel symbols and three sorts of geodesics. Three sorts of
the first order tubes (future, past and neutral) appear. If the fields
connected with the antisymmetric component are strong enough, the timelike
first order tube has a finite length in the timelike direction. It was shown
earler that the symmetric T-geometry explains non-relativistic quantum effects
without a reference to principles of quantum mechanics. One should expect that
nonsymmetric space-time T-geometry is also characteristic for microcosm, and it
will be useful in the elementary particle theory, because there is a series of
unexpected associations. First order tubes are associated with world tubes of
closed strings. Antisymmetry is associated with supersymmetry. Three sorts of
tubes are associated with three sorts of quarks. Limitation of the tube in time
direction is associated with confinement. At any rate, the space-time
T-geometry with additional parameters has more capacities, than usual
Riemannian geometry.
