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Let $M^2$ be a closed surface, say the 2-sphere. Is there any example of metric on it such that there are uncountably many points are conic and the metric is smooth elsewhere?

We call $p\in M$ a conic point if there exists $\lambda_i\to \infty$ such that $(\lambda_i M, p)$ converge to a linear cone $C(S(\ell))$, the cone over a circle of length $\ell$, for $\ell\ne 2\pi$.

For convex surface, there at most countably many conic points, which is proven by AD Alexandrov.

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    $\begingroup$ It will be helpful if you include a general definition of a conic point. Usually people use the definition which implies that they are isolated. $\endgroup$ – Alexandre Eremenko Dec 23 '13 at 16:34
  • $\begingroup$ @AlexandreEremenko, I've edited my post. Thanks $\endgroup$ – J. GE Dec 23 '13 at 19:34
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The answer is NO.

Fix $\varepsilon>0$ and $\delta>0$.

Consider the set $X_{\varepsilon,\delta}$ of all the points in $M$ such that for any $r<\varepsilon$, the $r$-neighborhood of any $x\in X_{\varepsilon,\delta}$ is $r{\cdot}\delta$-close to $r$-ball in the cone over the circle with length $\ell$ such that $|\ell-2{\cdot}\pi|>\varepsilon$.

Note that $X_{\varepsilon,\delta}$ is uncountable for small $\varepsilon >0$ and any $\delta>0$. On the other hand, if $\delta$ is small then the set $X_{\varepsilon,\delta}$ is discrete, a contradiction.

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