I just started to study exponential maps in differential geometry. I'm using the book Differential Geometry of Curves and Surfaces, by Manfredo P. Carmo. In section $4.6$ I'm having trouble at the exercise $5$. The problem is that I don't know how I'm supposed to use the exponential map to solve this. I need to someone solve only one of the items, then I can try for myself the other ones.
For which of the pairs of surfaces shown below there is a local isometry?
a. Revolution torus and a cone.
b. Cone and sphere.
c. Cone and cylinder.