tract: Unsupervised Domain adaptation is an effective method in addressing the domain shift issue when transferring knowledge from an existing richly labeled domain to a new domain. Existing manifold-based methods either are based on traditional models or largely rely on Grassmannian manifold via minimizing differences of single covariance matrices of two domains. In addition, existing pseudo-labeling algorithms inadequately consider the quality of pseudo labels in aligning the conditional distribution between two domains. In this work, a deep spherical manifold Gaussian kernel (DSGK) framework is proposed to map the source and target subspaces into a spherical manifold and reduce the discrepancy between them by embedding both extracted features and a Gaussian kernel. To align the conditional distributions, we further develop an easy-to-hard pseudo label refinement process to improve the quality of the pseudo labels and then reduce categorical spherical manifold Gaussian kernel geodesic loss. Extensive experimental results show that DSGK outperforms state-of-the-art methods, especially on challenging cross-domain learning tasks.