Explanation:

- Tripura: Shares a border with Bangladesh. This northeastern state is surrounded by Bangladesh on three sides.

- Manipur: Does not share a border with Bangladesh. It is located to the east of Bangladesh and shares its borders with Myanmar and other Indian states.

- Meghalaya: Shares a long border with Bangladesh to the south of the Indian state.

- Mizoram: Also shares a border with Bangladesh to its west.

Therefore, Manipur is the only state among the options provided that does not share a border with Bangladesh.