  The superb spatial resolution of Chandra has allowed us to detect a 20''-long
tail behind the Geminga pulsar, with a hard spectrum (photon index 1.0+/-0.2)
and a luminosity (1.3+/-0.2) 10^{29} ergs/s in the 0.5 - 8 keV band, for an
assumed distance of 200 pc. The tail could be either a pulsar jet, confined by
a toroidal magnetic field of about 100 microGauss, or it can be associated with
the shocked relativistic wind behind the supersonically moving pulsar confined
by the ram pressure of the oncoming interstellar medium. We also detected an
arc-like structure 5'' - 7'' ahead of the pulsar, extended perpendicular to the
tail, with a factor of 3 lower luminosity. We see a 3-sigma enhancement in the
Chandra image apparently connecting the arc with the southern outer tail that
has been possibly detected with XMM-Newton. The observed structures imply that
the Geminga's pulsar wind is intrinsically anisotropic.
