Take the 2-minute tour ×  Let $M$ be an $n$-dimensional
Riemannian manifold with sectional curvature lower bound 1. Fix
a point say $O\in M$, let $S(r)$ denote the distance sphere
centered at $O$ with radius $r$. The classical Hessian
comparison theorem says that the principle curvatures of $S(r)$
is less than that of standard sphere ${S}^n(1)$. And Toponogov
triangle comparison implies that given any two point in $S(r)$
there distance in $M$ is less than or equal to the correspond
distance in round sphere with the same openning angle at $O$.
So is there any way to see how the intrinsic diameter (i.e. the
length metric induced from ambient metric) upper bound?  How
about the Ricci curvature case?  share|improve this question  1
Answer 1  up vote 2 down vote accepted  I guess you want to ask
is it true that $$\mathop{\rm
IntrinsicDiameter}[S(r)]\le\mathop{\rm IntrinsicDiameter}[\tilde
S(r)],$$ where $\tilde S(r)$ denotes the sphere of radius $r$ in
the standard sphere.    • This is true if $r\ge \tfrac\pi2$;