Considering a standard nonlinear programming problem, one may view a subset of the equality constraints as an embedded Riemannian manifold. In this paper we investigate the differences between the Euclidean and the Riemannian approach for this problem. It is well known that the linear independence constraint qualification for both approaches are equivalent. However, when considering recently introduced constant rank constraint qualifications, the Riemannian approach provides a weaker condition as the rank of the gradients must remain constant only inside the manifold, while the Euclidean approach requires constant rank properties inside a full-dimensional neighborhood of the ambient space. Therefore by employing a Riemannian augmented Lagrangian method to a standard nonlinear programming problem we are able to obtain standard global convergence to a Karush/Kuhn-Tucker point under a new weaker constant rank condition that considers only lower dimensional neighborhoods. In this way we illustrate how the Riemannian perspective can provide new and stronger results to classical problems traditionally addressed through Euclidean theory. We also investigate the two alternative augmented Lagrangian algorithms in a comprehensive computational study, where we show some classes of problems where the Riemannian approach is much more robust in attaining better quality solutions.