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I am doing the exercise 24(Local Convexity and Curvature) in p.174 of Do Carmo's Differential Geometry of Curves and Surfaces and have some questions about the statements of part (c) and (d). In part (d), there goes"...assume that there is a neighborhood $V \subset S$ of $p$ such that the principal curvatures on $V$ do not have different signs(this does not happen in the example of part (c)..."

In my opinion, $K \geq 0$ (which is the condition of the example to be proved in part (c)) has forced that the principal curvatures at each point on $U$ do not have different signs. Why does the author say that "(this does not happen in the example of part (c))"? This really confuses me.

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The English is confusing, I agree. I believe that he means in part d. that each of the principal curvatures is always of the same sign (or $0$), not that the two principal curvatures (at each point) do not have different signs. (Note that in the example, $k_1<0$ when $y=0$ and $x<0$, but, perhaps surprisingly, $k_2<0$ there as well.)

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    Hello, What do you mean about "each of the principal curvatures is always of the same sign"? At a first glance, I think of it as $k_{1}>0, k_{2}=0$ over the entire domain $V$ or $k_{1}>0, k_{2}>0$...but it might be the case that $k_{1}>0, k_{2}<0$, which can be seen from the case $z=f(x,y)=xy$ and this surface is indeed not locally convex at$(0,0)$. So some how the meaning you give to me needs some modification. – 2017-01-25
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    @Shen: The assumption that $K\ge 0$ still must hold, of course. – 2017-01-25