  Usual inflation is realized with a slow rolling scalar field minimally
coupled to gravity. In contrast, we consider dynamics of a scalar with a flat
effective potential, conformally coupled to gravity. Surprisingly, it contains
an attractor inflationary solution with the rapidly rolling inflaton field. We
discuss models with the conformal inflaton with a flat potential (including
hybrid inflation). There is no generation of cosmological fluctuations from the
conformally coupled inflaton. We consider realizations of modulated
(inhomogeneous reheating) or curvaton cosmological fluctuations in these
models. We also implement these unusual features for the popular
string-theoretic warped inflationary scenario, based on the interacting D3-anti
D3 branes. The original warped brane inflation suffers a large inflaton mass
due to conformal coupling to 4-dimensional gravity. Instead of considering this
as a problem and trying to cure it with extra engineering, we show that warped
inflation with the conformally coupled, rapidly rolling inflaton is yet
possible with N=37 efoldings, which requires low energy scales 1-100 TeV of
inflation. Coincidentally, the same warping numerology can be responsible for
the hierarchy. It is shown that the scalars associated with angular isometries
of the warped geometry of compact manifold (e.g. S^3 of KS geometry) have
solutions identical to conformally coupled modes and also cannot be responsible
for cosmological fluctuations. We discuss other possibilities.
