Am I correct? - When a phrase defines directed action, we use akkusativ, and when it defines some state with no direction, we use dativ. (?) Like "Die Katze spielt auf dem Tisch" is dativ and ''Die Katze läuft auf den Tisch" is akkusativ. Then, "Die Maus läuft ums Glas" has a direction, so in this case it means the same as 'around the corner', I guess..
I may not have this 100% straight, but I think that I do. What you're saying is true for two-way prepositions. At the beginning of this unit, you can find which ones are which. There are a set of prepositions which always take dative, and there are a set of prepositions which always take accusative. Additionally, there are a set of so-called "two-way" prepositions which meet the criteria which you're specifying. These prepositions take dative if there is no movement (die Katze spielt auf dem Tisch (the cat plays on the table)) or if there is movement within a location without shifting to another location (Die Katze läuft auf dem Tisch (the cat walks on the table (perhaps from one end to the other))), and they take accusative if there is movement from one location to another (Die Katze läuft auf den Tisch (the cat walks onTO the table (perhaps from the arm of a chair))