In a Distant Time Movie Harukanaru Toki no Naka de: Maihitoyo

One rainy day, Akane crosses path with a kind young man who tacitly offers to her his coat. On their second encounter, he confesses that he has no recollection of who he is, his name or his past, but feels contented by just being with her. Besotted, Akane sets out to find his name, and to unravel his enigmatic connection with a famous cursed dance rumoured to kill anyone who attempts to perform it.

(Source: Anime News Network)