what a communication that is about something is about

a commitment to tell the truth (especially in a court of law); to lie under oath is to become subject to prosecution for perjury

(religion) a solemn declaration that serves the same purpose as an oath (if an oath is objectionable to the person on religious or ethical grounds)

a verbal commitment by one person to another agreeing to do (or not to do) something in the future

a binding commitment to do or give or refrain from something