In multiple-input multiple-output (MIMO) device-to-device (D2D) networks, interference and rank-deficient channels are the critical bottlenecks for achieving high degrees of freedom (DoFs). In this paper, we propose a reconfigurable intelligent surface (RIS) assisted interference alignment strategy to simultaneous mitigate the co-channel interference and cope with rank-deficient channels, thereby improving the feasibility of interference alignment conditions and in turn increasing the achievable DoFs. The key enabler is a general low-rank optimization approach that maximizes the achievable DoFs by jointly designing the phase-shift and transceiver matrices. To address the unique challenges of the coupled optimization variables, we develop a block-structured Riemannian pursuit method by solving fixed-rank and unit modulus constrained least square subproblems along with rank increase. Finally, to reduce the computational complexity and achieve good DoF performance, we develop unified Riemannian conjugate gradient algorithms to alternately optimize the fixed-rank transceiver matrix and the unit modulus constrained phase shifter by exploiting the non-compact Stiefel manifold and the complex circle manifold, respectively. Numerical results demonstrate the effectiveness of deploying an RIS and the superiority of the proposed block-structured Riemannian pursuit method in terms of the achievable DoFs and the achievable sum rate.