The nearby, middle-aged PSR B1055-52 has many properties in common with the Geminga pulsar. Motivated by the Gemingas enigmatic and prominent pulsar wind nebula (PWN), we searched for extended emission around PSR B1055-52 with Chandra ACIS. For an energy range 0.3-1 keV, we found a 4 sigma flux enhancement in a 4.9-20 arcsec annulus around the pulsar. There is a slight asymmetry in the emission close, 1.5-4 arcsec, to the pulsar. The excess emission has a luminosity of about 10^{29} erg s^{-1} in an energy range 0.3-8 keV for a distance of 350 pc. Overall, the faint extended emission around PSR B1055-52 is consistent with a PWN of an aligned rotator moving away from us along the line of sight with supersonic velocity, but a contribution from a dust scattering halo cannot be excluded. Comparing the properties of other nearby, middle-aged pulsars, we suggest that the geometry -- the orientations of rotation axis, magnetic field axis, and the sight-line -- is the deciding factor for a pulsar to show a prominent PWN. For PSR B1055-52, we also report on a flux decrease of at least 30% between the 2000 XMM-Newton and our 2012 Chandra observation. We tentatively attribute this flux decrease to a cross-calibration problem, but further investigations of the pulsar are required to exclude actual intrinsic flux changes.
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