I guess it is not. The example I have in mind is: X2 is the spherical suspension of a circle S1(t) of length 0<t<2π. Then X has constant curvature =1 except at two suspension points, say N and S.

But I cannot convince myself, since it seems this manifold can be approximated by a sequence of smooth Riemannian metrics gi with positive curvature by smoothing arbitrary small neighbourhood of N and S, then by Pogorelov, if we assume sufficient smoothness on gi, we get desired embedding by taking limit.

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