  We review on the models of gravity with a constraint by the Lagrange
multiplier field. The constraint breaks general covariance or Lorentz symmetry
in the ultraviolet region. We report on the $F(R)$ gravity model with the
constraint and the proposal of the covariant (power-counting) renormalized
gravity model by using the constraint and scalar projectors. We will show that
the model admits flat space solution, its gauge-fixing formulation is fully
developed, and the only propagating mode is (higher derivative) graviton, while
scalar and vector modes do not propagate. The preliminary study of FRW
cosmology indicates to the possibility of inflationary universe solution is
also given.

High Energy Physics - Theory;Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology