### subset of sphere defined by inequalities

I need to draw several geodesics on a subset of a unit sphere, namely the set of all points with z-coordinate ~~-1<z<1.< ~~-0.5<z<0.5< p="">

1)How do I define the object?

A = sphere((0,0,0), 1)

How to cut off the domain that i need?
I'm a complete novice and failed to find the answer in the documentation.

2) What's the right way to draw a segment of a big circle between two points on a sphere given in spherical coordinates?