CLEVELAND — Gov. Mike Pence of Indiana, who has a weakness for impromptu stops at county fairs, took his future — and still only — wife ice-skating on their first date and is serious enough about his faith that he regularly meets with a small group of men who hold one another accountable to Christian principles.

He has now entered into a political alliance with a man, Donald J. Trump, whose idea of recreation is flying on his Boeing 757 to one of his oceanside golf courses, a man who met his current and third wife at a model-filled fashion week party in Manhattan and who referred last summer to holy communion as the “little wine” and “little cracker.”

By selecting Mr. Pence as his running mate, Mr. Trump has offered an olive branch to restive Christian conservatives in the Republican Party, while leaving little doubt that he will try to win the presidential election in the Midwest. He also signaled to party officials and donors that he was capable of making sober, even conventional decisions.

But Mr. Trump has also forged one of the most sharply dissimilar presidential tickets in modern political history, linking himself to somebody he scarcely knows and whose personality, politics and lifestyle are starkly different from his.