DTI is an important tool to investigate the brain in vivo and non-invasively in spite of its shortcomings in regions of fiber-crossings. HARDI models such as QBI and Higher Order Tensors (HOT) were in- vented to overcome this shortcoming. HOTs, however, have not been explored extensively even though sophisticated estimation schemes were developed for DTI that guarantee positive diffusivity, such as the Rie- mannian framework. Positive diffusivity is an important constraint in diffusion MRI since it represents the physical phenomenon of molecular diffusion. It seems apt, to leverage the work done on DTI, to apply the positivity constraint to the HOT model. We, therefore, propose to extend the Riemannian framework from DTI to the space of 4th order diffusion tensors. We also review the existing methods for estimating 4th order diffusion tensors and compare all methods on synthetic, phantom and real datasets extensively to test for robustness and speed. Our contributions for extending the Riemannian framework from DTI to estimating 4th order diffusion tensors guarantees positive diffusivity, is robust, is fast, and can be used to discern multiple fiber directions.