I want to prove some Alexandrov space M is parabolic cone X x R.Since Alex has no Riemannian metric,so how to do?Is there any (triangle) formula about the relation of distance of two points in M and distance of the projection points in X?
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The metric of warp products $F\times_f B$ is fiber independent. It means that if $(x,a),(y,b)\in F\times_f B$ and $(x',a),(y',b)\in F'\times_f B$ then $$xy_F=x'y'_F \ \ \ \Rightarrow\ \ \ (x,a)(y,b) _{F\times_f B} = (x',a)(y',b) _{F\times_f B}$$ If I remember right, this statement is due to S. Alexander and R. Bishop. If you apply this to parabolic cone, you get a nice formula for distances, which depent only on $xy_F$ and two real values $a$ and $b$. 

