This work presents a novel framework for projection-based model order reduction (MOR) for cardiac mechanics simulations in a many query context by investigating the influence of varying geometries on heart motion. The underlying cardiac model couples a 3D solid mechanics model using anisotropic and actively contracting materials to a 0D cardiovascular blood circulation model [1]. Models like this contain uncertainty in their parameters and have to be solved multiple times, to either quantify the uncertainty or to inversely estimate said parameters resulting in great computational demand. Projection-based model order reduction meets this challenge by exploring the parameter space and computing a low-dimensional basis that approximates the high-fidelity solution space. To this date, MOR was successfully applied to material parameters of cardiovascular models achieving promising accelerations while maintaining sufficient accuracy [2]. In current MOR approaches the reduced basis is only valid for a single patient geometry. For a new patient, the expensive exploration phase must be repeated and a new basis must be computed from the ground up. The proposed approach uses methods from statistical geometry [3] to extract characteristic geometry-variation modes. This low-dimensional shape representation enables an exploration of feasible heart geometries by independently scaling the geometry modes in a sampling process and computing their solution. For each sample, deformation snapshots are collected. Exploiting these solutions snapshots, a reduced order basis is computed which encodes the complex relationship between shape and deformation. It can be used to solve the cardiac mechanics problem not only for a single but for a multitude of patient-specific geometries without having to repeat the exploration phase. The potential of this approach is demonstrated by showing good agreement of heart motion and temporally resolved cardiac performance measures such as ventricular pressures and volumes between reduced and full model.