  The Finslerian extension of the Euclidean metric is proposed and studied
under rigorous conditions that the associated indicatrix is regular and convex.
The relativistic pseudo-Euclidean metric is extended, too. The extensions show
distinct violation of the $T-$parity, so that the future-past asymmetry of the
physical world can stem from the $T-$asymmetry of the Finslerian indicatrix.
Indeed the idea should compel much attention of physicists.
