  We prove local existence for the second order Renormalization Group flow
initial value problem on closed Riemannian manifolds $(M,g)$ in general
dimensions, for initial metrics whose sectional curvatures $K_P$ satisfy the
condition $1+\alpha K_P > 0$, at all points $p \in M$ and planes $P \subset T_p
M$. This extends results previously proven for two and three dimensions.
