  Suppose $(X_n)$ is a sequence of positive-dimensional smooth projective
complete intersections over $\mathbb{F}_q$ with dimensions bounded from above
and with characteristic zero lifts $(\tilde{X}_n)$ to smooth projective
geometrically connected varieties. Suppose each complex variety
$\tilde{X}^{an}_n$ has (underlying real manifold equipped with) a Riemannian
metric $g_n$ of sectional curvature at least $-\kappa_n^2$, $\kappa_n\geq 0$,
and diameter at most $D_n$. In this note, we show that if
\[\lim_{n\rightarrow\infty}\min\{d:X_n(\mathbb{F}_{q^d})\neq\emptyset\}=+\infty,\]
then $\kappa_nD_n\rightarrow+\infty$. We deduce this theorem by proving a more
general theorem estimating the number of points over finite fields of the above
varieties in terms of the sectional curvature and diameter of Riemannian
structures on the analytification of characteristic zero lifts. We also prove a
version of this estimate for non-projective varieties equipped with complete
metrics of non-negative sectional curvature. Finally, we prove a characteristic
zero analogue of the above estimate by relating fixed-points of algebraic
endomorphisms of smooth projective complex complete intersections to Riemannian
structures on them.
