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3D point cloud registration is a crucial technology for 3D scene reconstruction and has been successfully applied in various domains, such as smart healthcare and intelligent transportation. With theoretical analysis, we find that geometric structural relationships are essential for 3D point cloud registration. The 3D point cloud registration method achieves excellent performance only when fusing local and global features with geometric structure information. Based on these discoveries, we propose a 3D point cloud registration method based on geometric structure embedding into the attention mechanism (GraM), which can extract the local features of the non-critical point and global features of the corresponding point containing geometric structure information. According to the local and global features, the simple regression operation can obtain the transformation matrix of point cloud pairs, thereby eliminating the semantics that ignores the geometric structure relationship. GraM surpasses the state-of-the-art results by 0.548° and 0.915° regarding the relative rotation error on ModelNet40 and LowModelNet40, respectively.