### abstract

- A covariant canonical gauge theory of gravity free from torsion is studied. Using a metric conjugate momentum and a connection conjugate momentum, which takes the form of the Riemann tensor, a gauge theory of gravity is formulated, with form-invariant Hamiltonian. Through the introduction of the metric conjugate momenta, a correspondence between the Affine-Palatini formalism and the metric formalism is established. For, when the dynamical gravitational Hamiltonian $\tilde{H}_{Dyn}$ does not depend on the metric conjugate momenta, a metric compatibility is obtained from the equation of motions and the energy momentum is covariant conserved. When the gravitational Hamiltonian $\tilde{H}_{Dyn}$ depends on the metric conjugate momentum, an extension to the metric compatibility comes from the equation of motion and the energy momentum covariant conservation is violated. For a sample of the $\tilde{H}_{Dyn}$ which consists of a quadratic term of the connection conjugate momentum, the effective Lagrangian has the Einstein Hilbert term with a quadratic Riemann term in the second order formalism. A bouncing inflation is briefly discussed in the context of cosmological solutions of this action.