Title: ISONETRY : GEOMETRY OF CRITICAL INITIALIZATIONS AND TRAINING. Abstract: Recent work on critical initializations of deep neural networks has shown that by constraining the spectrum of input-output Jacobians allows for fast training of very deep networks without skip connections. The current understanding of this class of initializations is limited with respect to classical notions from optimization. In particular, the connections between Jacobian eigenvalues and curvature of the parameter space are unknown. Similarly, there is no firm understanding of the effects of maintaining orthogonality during training. With this work we complement the existing understanding of critical initializations and show that the curvature is proportional to the maximum singular value of the Jacobian. Furthermore we show that optimization under orthogonality constraints ameliorates the dependence on choice of initial parameters, but is not strictly necessary.