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.
Best Answer
The question is about curvature and if the surface can isometrically transformed to the plane. The cone has zero curvature so it is only locally isometric with another surface of zero curvature. The cone and also be unfolded ? or opened up to lie on the plane. Eg, you can make a cone from a piece of paper, but not a torus or a sphere. I think the exercise is in reference to Minding's theorem.