Please help me identify this curve. I strongly feel that it is a minimal geodesic, but I cannot show it.
Suppose that $M$ is a Hadamard manifold (i.e., complete, simply-connected smooth Riemannian manifold with nonpositive sectional curvature). Let $o,p,q \in M$. Is the curve defined for $t \in [0,1]$ by $$t \mapsto \exp_{o}(t \exp_{o}^{-1}(p) + (1-t)\exp_{o}^{-1}(q))$$ a minimal geodesic joining $p$ to $q$.
Thank you.