  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 extragalactic 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.
