We show that images of TeV blazars in the GeV energy band should contain, along with point-like sources, degree-scale jet-like extensions. These GeV extensions are the result of electromagnetic cascades initiated by TeV gamma-rays interacting with ex
tragalactic background light and the deflection of the cascade electrons/positrons in extragalactic magnetic fields (EGMF). Using Monte-Carlo simulations, we study the spectral and timing properties of the degree-scale extensions in simulated GeV band images of TeV blazars. We show that the brightness profile of such degree-scale extensions can be used to infer the lightcurve of the primary TeV gamma-ray source over the past 1e7 yr, i.e. over a time scale comparable to the life-time of the parent active galactic nucleus. This implies that the degree-scale jet-like GeV emission could be detected not only near known active TeV blazars, but also from TeV blazar remnants, whose central engines were switched off up to ten million years ago. Since the brightness profile of the GeV jets depends on the strength and the structure of the EGMF, their observation provides additionally information about the EGMF.
