Are there any circumstances under which a body can move in a curve path without being accelerated?

Answer 1

No. Acceleration is the second derivative of displacement. By traveling in a curve, displacement is changing instantaneously. Thus the velocity and acceleration vectors will be changing as well.

Answer 2

If you are moving in a straight line through a pace-time that is curved relative to the observer you will 'appear' to move in a curved line. But you will also 'appear' to be being accelerated (even 'tho you, yourself will 'feel' no acceleration).

As a general case answer, no.

HTH

Doug