Regularized continuum damage models such as those based on an order parameter (phase field) have been extensively used to characterize brittle damage of compressible elastomers. However, the prescription of the surface integral and the degradation function for stiffness lacks a physical basis. In this article, we propose a continuum damage model that draws upon the postulate that a damaged material could be mathematically described as a Riemannian manifold. Working within this framework with a well-defined Riemannian metric designed to capture features of isotropic damage, we prescribe a scheme to prevent damage evolution under pure compression. The result is a substantively reduced stiffness degradation due to damage before the peak response and a faster convergence rate with the length scale parameter in comparison with a second-order phase field formulation that involves a quadratic degradation function. We also validate this model using results of tensile experiments on double notched plates. © 2021 by ASME.