On contact metric hypersurfaces in a real space form

For a (2n + 1)-dimensional N(k)-contact metric hypersurface in a real space form M(c), some main results are obtained as follows: (1) if k — c 0 then M is totally umbilical, and consequently, either M is a Sasakian manifold of constant curvature +1 or M is 3- dimensional and flat; (2) if k = c and M is Einstein then either M is totally geodesic or a developable hypersurface in M(k), in particular M is of constant curvature and consequently, either M is a Sasakian manifold of constant curvature +1 or M is 3- dimensional and flat; (3) if M is 3-dimensional non-Sasakian such that k = c then either M is flat or the shape operator of M is of a specific form (see Theorem 6); and (4) if M is 77-Einstein such that n 2 and k = c, then M is a developable hypersurface. An obstruction for M to be totally geodesic is also obtained