Learn Russian: По окончании урока йоги мы чувствуем радость. - Upon finishing the yoga class, we feel joy.

Instead of using a full subordinate clause with a verb, like После того как мы окончили урок (After we finished the class), the sentence compresses the action into the verbal noun окончание (completion). This makes the sentence more compact, which is a common feature of written or slightly elevated Russian.

Why does окончании end in -ии? Doesn't the preposition по usually trigger the Dative case?

Usually, yes. However, when по is used to mean "upon" or "immediately after," it requires the Prepositional case. Because the nominative form окончание ends in -ие, its Prepositional case ending is -ии. You might sometimes hear native speakers mistakenly use the Dative here, but по окончании is the strictly correct formal grammar.

Why is the word for class (урока) in the Genitive case instead of the Accusative?

When an action is expressed by a verb (окончить урок - to finish the class), the target is a direct object in the Accusative. But when that action is compressed into a verbal noun (окончание), it loses the ability to take an Accusative object. Instead, the object shifts to the Genitive case, literally translating to "the finishing OF the class" (окончание урока).

We have урока and йоги right next to each other. How do these cases work together?