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.

Revised version: data analysis described in more detail, Figure 2 replaced; 6 pages, 2 color figures; accepted for publication in ApJ (v.643, 2006 June 1)