I have a question on Riemannian geometry or CAT(k) geometry, which might be simple for experts. Suppose $M$ is a complete smooth Riemannian manifold with sectional curvature bounded from above by $k>0$. Fix a ball $B\subset M$ with radius less than $\frac{\pi}{2\sqrt{k}}$. Fix three points $P,Q,R\in B$. Denote by $PQ$ the unique geodesic that connects $P$ and $Q$ in $B$ and $QR$ the unique geodesic that connects $P$ and $R$.

Let $Q_t\in PQ$ be the $t$-fractional point, that is, $d(p,Q_t)=td(p,Q)$ and $d(Q_t,Q)=(1-t)d(P,Q)$, and let $R_t\in PR$ be the $t$-fractional point as well. I wonder whether it is true that there exists a constant $c(k,t)>0$ such that $$d(Q_t,R_t)\le c(k,t)d(P,Q).$$ In case $k=0$, we know that we can choose $c(k,t)=t$.

Any comments or references would be greatly appreciated.

I have a question on Riemannian geometry or CAT(k) geometry, which might be simple for experts. Suppose $M$ is a complete smooth Riemannian manifold with sectional curvature bounded from above by $k>0$. Fix a ball $B\subset M$ with radius less than $\frac{\pi}{2\sqrt{k}}$. Fix three points $P,Q,R\in B$. Denote by $PQ$ the unique geodesic that connects $P$ and $Q$ in $B$ and $QR$ the unique geodesic that connects $P$ and $R$.

Let $Q_t\in PQ$ be the $t$-fractional point, that is, $d(p,Q_t)=td(p,Q)$ and $d(Q_t,Q)=(1-t)d(P,Q)$, and let $R_t\in PR$ be the $t$-fractional point as well. I wonder whether it is true that there exists a constant $c(k,t)>0$ such that $$d(Q_t,R_t)\le c(k,t)d(R,Q).$$ In case $k=0$, we know that we can choose $c(k,t)=t$.

Any comments or references would be greatly appreciated.