This work presents a novel framework for projection-based model order reduction (MOR) for cardiac mechanics simulations incorporating heart shape variability. The underlying cardiac model couples a 3D solid mechanics model using anisotropic and actively contracting materials to a 0D cardiovascular blood circulation model [1]. High-fidelity models like this can help to understand the origin and progression of cardiovascular diseases, which are the leading cause of mortality worldwide. But their widespread clinical adoption is impeded by the complex process of making models patient-specific and by their excessive computation time. The well-known MOR addresses the latter limitation by projecting the complex dynamic behavior of high-fidelity models onto a low-dimensional subspace leading to significantly fewer degrees of freedom. While traditional approaches in cardiac mechanics are mainly based on physical parameters such as material properties or boundary conditions, this work accounts for variability in heart shapes. By applying statistical shape analysis to a clinical dataset of 51 lower heart geometries, a population-average shape and principal modes of variation, also called shape modes (SM), are created. Independent scaling of the most relevant SM in a sampling process allows a low-dimensional representation of heart geometries for the computationally expensive offline phase. The recurring necessity of mapping surfaces onto each other is addressed using large deformation diffeomorphic metric mapping (LDDMM) [2], which perfectly supports this approach and distinguishes it from others. The reduced-order model generated in this way is not only valid for a single heart shape but generalizes to multiple unseen patients. Its online performance is demonstrated with five patient-specific heart geometries by comparing deformation fields, cardiac performance, and computational time with the full-order model and presenting characteristic patterns in the heart's shape and its deformation [3].