Component-centric reduced order models (ROMs) have recently been developed in the context of linear structural dynamics. They lead to an accurate prediction of the response of a part of structure (referred to as the β component) while not requiring a similar accuracy in the rest of the structure (referred to as the α component). The advantage of these ROMs over standard modal models is a significantly reduced number of generalized coordinates for structures with groups of close natural frequencies. This reduction is a very desirable feature for nonlinear geometric ROMs, and thus, the focus of the present investigation is on the formulation and validation of component-centric ROMs in the nonlinear geometric setting. The reduction in the number of generalized coordinates is achieved by rotating close frequency modes to achieve unobservable modes in the β component. In the linear case, these modes then completely disappear from the formulation owing to their orthogonality with the rest of the basis. In the nonlinear case, however, the generalized coordinates of these modes are still present in the nonlinear stiffness terms of the observable modes. A closure-type algorithm is then proposed to finally eliminate the unobserved generalized coordinates. This approach, its accuracy and computational savings, is demonstrated first on a simple beam model and then more completely on the 9-bay panel model considered in the linear investigation.