You certainly don’t have to inform your manager. Most people don’t, so you wouldn’t be doing anything unethical or even out of the ordinary. In fact, most people inform their manager only after they’ve received and accepted another offer. In some cases, there could be advantages to telling your manager:
- He/she may be able do something to rectify the reason that you’re unhappy.
- He/she may be able to refer you to other companies. (This would be somewhat unusual though, as it’s sort of a conflict of interest for your manager to do such a thing.)
- ...