The concept of curvature is defined for any linear connection on any vector bundle $E \to M$, but the concept of torsion is only defined for connection on the tangent bundle $TM$ of a manifold $M^n$, or for a connection obtained as the pullback of a connection on a vector bundle $E \to M$ *isomorphic to $TM$* via an isomorphism $\theta \colon TM \to E$ equivalent to a solder form.

Why is that so ? If torsion can be interpreted as the twist of a moving frame along a curve, the same phenomena should occur for a connection on any vector bundle.

Is there a way to define a notion of torsion for any vector bundle ?