Can you tell me the name of this manga.The female lead is a high school student, who suddenly traveled back in old Japan, where she met a boy, she ended up in the boy's house/ garden who mistook her for someone else ( his fiance, to be exact). The boy brought her to his fiance's house, thinking she is his fiance. She met the fiance who is a Lady and they became friends and even switched places several times. She even covered for her when she had to meet the boy. The female lead realized that the boy was in love with the fiance, thinking it was her (the female lead) that he met before when they were young. But the fiance does not want to marry the boy. She already refused their engagement previously, before the female lead appeared.
Then the lead girl realized that she lost the necklace her grandmother gave her, and went back to the house of the boy, to look for it in the garden. She didn't know that the boy found it the day she arrived and kept it with him. Then she realized, she met the boy when she was younger.
The boy is also from an elite family. But he is adopted (or a son from a mistress). His older brother treated him well, and accepted him. The older brother was sick and died when he was young, so he became the "substitute heir" to replace him. The boy met the female lead when they were younger, she suddenly appeared in their garden, it was raining and he invited her inside their house. When the boy met his fiance for the first time, he thought it was the female lead that he fell in love with when they were young.