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Can anyone give me a hint or a reference that would help to understand the following result :

The only compact $\mathcal{C}^{\infty}$-submanifold of $\mathbb{R}^3$ of constant curvature $\frac{1}{R}$ is the sphere of radius $R$

Thank you for your answers

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See p. 61 here: http://math.berkeley.edu/~reshetik/140/ShifrinDiffGeo.pdf