Unsupervised reinforcement learning (URL) aims to learn general skills for unseen downstream tasks. Mutual Information Skill Learning (MISL) addresses URL by maximizing the mutual information between states and skills but lacks sufficient theoretical analysis, e.g., how well its learned skills can initialize a downstream task's policy. Our new theoretical analysis shows that the diversity and separatability of learned skills are fundamentally critical to downstream task adaptation but MISL does not necessarily guarantee them. To improve MISL, we propose a novel disentanglement metric LSEPIN and build an information-geometric connection between LSEPIN and downstream task adaptation cost. For better geometric properties, we investigate a new strategy that replaces the KL divergence in information geometry with Wasserstein distance. We extend the geometric analysis to it, which leads to a novel skill-learning objective WSEP. It is theoretically justified to be helpful to task adaptation and it is capable of discovering more initial policies for downstream tasks than MISL. We further propose a Wasserstein distance-based algorithm PWSEP can theoretically discover all potentially optimal initial policies.