sich krümmen
to bend over
To bend or twist one's own body, often as a reaction to pain, laughter, or discomfort.
Er krümmte sich vor Schmerzen am Boden.
He bent over in pain on the floor.

to bend / to curve
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to bend over
To bend or twist one's own body, often as a reaction to pain, laughter, or discomfort.
Er krümmte sich vor Schmerzen am Boden.
He bent over in pain on the floor.

The verb 'krümmen' (to bend / to curve) is derived directly from the adjective 'krumm' (crooked / bent). It describes the action of making something crooked or becoming crooked.
Ein krummer Ast lag auf dem Weg.
A crooked branch was lying on the path.
This verb is often used reflexively ('sich krümmen') when the subject bends itself. If it is used transitively ('etwas krümmen'), the subject bends another object.
Die Katze krümmte sich genüsslich in der Sonne.
The cat stretched and curved its body contentedly in the sun.
The reflexive form 'sich krümmen' refers to living beings when it is about reactions such as pain or laughter. When it refers to objects, it describes deformation caused by external influences such as heat or pressure.
Das Kind krümmte sich vor Lachen über den Witz.
The child bent over with laughter at the joke.
When a person bends over, it often happens as a reaction to a strong feeling. Typical triggers are expressed with the preposition 'vor' + dative: 'sich vor Schmerzen krümmen' (to bend over in pain) or 'sich vor Lachen krümmen' (to double over with laughter).
Nach dem Sturz krümmte sich der Radfahrer vor Schmerzen.
After the fall, the cyclist bent over in pain.
The expression 'jemandem kein Haar krümmen' (to not harm a hair on someone's head) is a fixed unit. The person being protected is always in the dative. The expression is used almost exclusively in the negative form.
Keine Sorge, der Hund ist ganz lieb, der wird dir kein Haar krümmen.
Don't worry, the dog is very sweet; he won't harm a hair on your head.
to injure / hurt / violate
'verletzen' and 'jemandem kein Haar krümmen' (to not harm a hair on someone's head) can both have to do with whether someone is harmed. 'verletzen' directly says that someone suffers physical or emotional harm. The expression 'jemandem kein Haar krümmen', on the other hand, means that you do not do anything to someone at all. It is therefore mostly used in negative or reassuring statements. The expression can be confused with 'verletzen' because it also talks about harm. The important point is: 'krümmen' is not used here on its own like a normal verb, but belongs to the fixed expression 'jemandem kein Haar krümmen'.

Keine Sorge, der Hund wird dir kein Haar krümmen.
Don't worry, the dog won't harm a hair on your head.
Sie passt gut auf, damit sie sich beim Umzug nicht verletzt.
She is careful so that she doesn't injure herself during the move.