Short answer
Bowing low to a person, a flag or a picture as a greeting is not allowed in the Hanafi school, and a bow close to ruku' (bowing in prayer) is treated like prostration. So the formal bow in karate, judo or taekwondo should be avoided. Martial arts themselves are permitted and even encouraged. A slight nod or a hand on the chest is fine.
Bowing to anyone other than Allah as a greeting is makruh tahrimi, and a bow that reaches near the position of ruku' is haram like prostration of greeting. It is not kufr unless done as worship. Learning martial arts for fitness and self-defence is permitted.
The four schools agree on this.
Explanation
A man asked the Prophet ﷺ whether he should bow to a friend when meeting him. He said no. Bowing and prostrating are acts of humility that belong to Allah. In the Hanafi books, bowing to a ruler or anyone else is disliked because it resembles the greetings of other religions, and a deep bow is counted like a prostration of greeting.
In many martial arts the bow (rei) is a sign of respect, not worship. That is why it is not kufr. But the act is still one Muslims are told not to do to people, so avoid:
- bowing to the instructor, to opponents or when entering the mat;
- bowing to a portrait of the founder or to a shrine (kamidana) in the dojo, which is more serious;
- kneeling bows that lower the head to the ground.
What you can do instead: stand straight, nod slightly, put your hand on your chest, or shake hands. Explain to the instructor before you start. Many clubs and competitions accept a nod for religious reasons.
The sports themselves are good. The Prophet ﷺ said the strong believer is better and more beloved to Allah than the weak believer, while there is good in both. Wrestling, archery and horse riding were practised in his time. Avoid strikes to the face in training where you can, keep your awrah covered, and keep the intention of health and self-defence. See sports that are sunnah and is boxing haram. On prostration, see prostrating to anyone but Allah.
The evidence
Jami' at-Tirmidhi 2728
Grading to be checked... أينحني له قال لا ...
"He asked, 'Should he bow to him?' He said, 'No.'"
Bowing to people as a greeting is not allowed.
Sahih Muslim 2664
Sahih... المؤمن القوي خير وأحب إلى الله من المؤمن الضعيف ...
"The strong believer is better and more beloved to Allah than the weak believer."
Building strength, including through martial arts, is praised.
Where the ruling is recorded
- al-Durr al-Mukhtar, Kitab al-Hazr wa al-Ibaha, section on greetings. Bowing to people is disliked, and a bow near ruku' takes the ruling of prostration of greeting.
- Radd al-Muhtar (Ibn Abidin), Kitab al-Hazr wa al-Ibaha, section on greetings. A prostration or bow of greeting is a major sin but not kufr unless it is worship.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.