TITLE: How can a cone have non-zero Riemann curvature yet can be made out of a piece of paper?
QUESTION [6 upvotes]: A cone with a $90^\circ$ vertex angle can be parameterized by
$$(x, y, z) = (z \cos \theta, z \sin \theta, z).$$
The metric on the cone can then be found to be
$$dx^2 + dy^2 + dz^2 = 2 dz^2 + z^2 d \theta.$$
The only non-zero Christoffel symbols are
$$\Gamma^z_{\space \space \theta \theta} = -\frac{1}{z} \hspace{1 cm}\Gamma^\theta_{\space \space \theta z} = \frac{z}{2}$$
and the only non-zero independent component of the Riemann curvature tensor is
$$R_{z \theta z \theta} = 2 z^{-2} + 1.$$
It's possible I made a mistake in the above computations, but I believe they are correct.
This shows that a cone is not locally flat.
However, a cone in real life can be formed by curving a piece of paper. For example, to compute the surface area of a cone, you can consider the surface area of a pac-man shape laying flat on a table. When the top and bottom of the mouth of the pac-man are brought together, the paper is in the shape of a cone.
The Christoffel symbols and Riemann tensor contain no information on the embedding of a manifold in ambient space. Therefore, it would seem to me that the Riemann tensor would have to be $0$, as it would be for flat piece of paper. How can this be?

REPLY [3 votes]: I was preparing an answer to this question on the same line of the answer of user1379857. This one is perfect, so I've nothing to add, but since I builded a little model that illustrate how the vertex is a special point and that a vector parallel transported  around this special point it comes back to the start position with a residual angle, I add here the images that can be useful.


The first image illustrate how a vector is moved on a plane witout an angle (the pac-man in the answer of user1379857 ), the second image  illustrate how this is equivalent to  path on a cone when the two sides of the angle are glued.
Note that this is true only if the closed path contain the vertex, so that the region is not smooth.