Single-view 3D reconstruction methods like Triplane Gaussian Splatting (TGS) have enabled high-quality 3D model generation from just a single image input within seconds. However, this capability raises concerns about potential misuse, where malicious users could exploit TGS to create unauthorized 3D models from copyrighted images. To prevent such infringement, we propose a novel image protection approach that embeds invisible geometry perturbations, termed ``geometry cloaks'', into images before supplying them to TGS. These carefully crafted perturbations encode a customized message that is revealed when TGS attempts 3D reconstructions of the cloaked image. Unlike conventional adversarial attacks that simply degrade output quality, our method forces TGS to fail the 3D reconstruction in a specific way - by generating an identifiable customized pattern that acts as a watermark. This watermark allows copyright holders to assert ownership over any attempted 3D reconstructions made from their protected images. Extensive experiments have verified the effectiveness of our geometry cloak.