Positive sectional curvature with no fixed lower bound on a noncompact manifold

The paraboloid is a complete, noncompact surface. [F3, F4, given] The map $f:{\mathbb{R}}^2\to \mathbb{R}$, $f(x,y)=x^2+y^2$, is smooth and continuous, so its graph $P=\{(x,y,f(x,y))\}$ is an embedded submanifold of ${\mathbb{R}}^2\times \mathbb{R}\cong {\mathbb{R}}^3$ by [F3] and is closed in ${\mathbb{R}}^3$ by [F3]. The Euclidean metric of ${\mathbb{R}}^3$ is complete [F4], and its Riemannian distance is the Euclidean distance; hence [F4] makes $P$ complete in the induced Riemannian metric. For noncompactness consider $p_t:=(t,0,t^2)\in P$ for $t=1,2,\dots$: the Euclidean norms $\mathrm{\mid }{p}_t{\mathrm{\mid }}^2=t^2+t^4$ are unbounded, so $(p_t)$ has no convergent subsequence and [F5] shows that $P$ is not compact.

The curvature at each point. [F1, F2, step 1.1] Parametrize $P$ by $X(x,y)=(x,y,f(x,y))$, so that $X_x=(1,0,2x)$, $X_y=(0,1,2y)$ and $g=\left(\begin{array}{cc}1+4x^2& 4xy\\ 4xy& 1+4y^2\end{array}\right),\det g=1+4(x^2+y^2)=1+4r^2,$ where $r^2=x^2+y^2$. The upward unit normal is $N=(1+4r^2{)}^{-1\mathrm{/}2}(-2x,-2y,1)$, and the second fundamental form has matrix $h=(1+4r^2{)}^{-1\mathrm{/}2}\mathrm{Hess}f=(1+4r^2{)}^{-1\mathrm{/}2}\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right),\det h=\frac{4}{1+4r^2}\mathrm{.}$ By [F2] the shape operator satisfies $S=g^{-1}h$ and $\det S=\frac{\det h}{\det g}=\frac{4\mathrm{/}(1+4r^2)}{1+4r^2}=\frac{4}{(1+4r^2{)}^2}\mathrm{.}$ By [F1] the sectional curvature of the tangent plane at $X(x,y)$, the only tangent two-plane of the surface, is $K=\det S=4\mathrm{/}(1+4r^2{)}^2$.

The curvature is positive but its infimum is zero. [F5, step 2.1] For every $(x,y)$ the numerator $4$ and the denominator $(1+4r^2{)}^2$ are positive, so $K=4\mathrm{/}(1+4r^2{)}^2>0$: the paraboloid has positive sectional curvature everywhere. On the other hand, given any $\delta >0$ choose $r>0$ with $(1+4r^2{)}^2>4\mathrm{/}\delta$; then the point $(r,0,r^2)\in P$ has $0<K<\delta$. Hence $0$ lies below every value of $K$ but no positive number is a lower bound, so the greatest lower bound of the set of values of $K$ is ${\inf }_{P}K=0$ by [F5]: a uniform lower bound $K\ge c>0$ fails on the complete noncompact manifold $P$, refuting the displayed claim. The paraboloid and the chosen radii are explicit, so the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ of [A1] is not drawn on beyond its declaration.