An Indian woman in a flowing beaded gown glides through a pond. A mosquito net brushes over her lover. And to insistent drumbeats of the Indian heartland, a dancing phalanx of tunic-clad women twirls. These images appear in the trailer of a forthcoming movie, “Saawariya.” The melodramatic film, whose Hindi title means “beloved,” has an Indian director and cast. Its characters speak Hindi and burst into eight song-and-dance numbers. It is, in other words, vintage Bollywood—but for one thing. It is brought to you by Hollywood.