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Unsupervised domain adaptation is effective in leveraging the rich information from the source domain to the unsupervised target domain. Though deep learning and adversarial strategy make an important breakthrough in the adaptability of features, there are two issues to be further explored. First, the hard-assigned pseudo labels on the target domain are risky to the intrinsic data structure. Second, the batch-wise training manner in deep learning limits the description of the global structure. In this paper, a Riemannian manifold learning framework is proposed to achieve transferability and discriminability consistently. As to the first problem, this method establishes a probabilistic discriminant criterion on the target domain via soft labels. Further, this criterion is extended to a global approximation scheme for the second issue; such approximation is also memory-saving. The manifold metric alignment is exploited to be compatible with the embedding space. A theoretical error bound is derived to facilitate the alignment. Extensive experiments have been conducted to investigate the proposal and results of the comparison study manifest the superiority of consistent manifold learning framework.