I am not really familiar with the subject of geodesics, but I was wondering whether there is a notion of a continuous distance monotonic path, which is slightly more general than geodesics? Specifically, I was thinking that a continuous path $\gamma:[0,1]\to X$ from $x$ to $y$, in a path connected metric space $X$, if
- $\gamma(0)=x$ and $\gamma(1)=y$.
- Every $t_1\leq t_2$ satisfy $d\big(y,\gamma(t_2) \big) \leq d\big(y,\gamma(t_1) \big)$.
The only relevant thing I found was geodesics, but it seems there that they demand roughhly preserving the distance up to a constant. I am demanding less in some sense, and I assumed that someone had already dealt with this definition before in a name I don't know how to search for. I was also wondering whether perhaps, in a path connected locally compact, this notion coincides with the notion of geodesics?
I have never encountered such a definition, and what I can tell is that there are examples of such paths that are not geodesics.
For instance, consider $\Bbb R ^2$ with its usual euclidean metric, whose geodesics are straight lines. The path $\gamma(t) = \left(\cos(\frac{\pi}{2}t),\sin(\frac{\pi}{2}t)\right)$ from $(1,0)$ to $(0,1)$ is not a geodesic (it is a circular arc), but $$ d\left(\gamma(1),\gamma(t)\right)^2 = \cos(\frac{\pi}{2}t)^2+ (1-\sin(\frac{\pi}{2}t))^2 = 2(1 - \sin(\frac{\pi}{2}t)) $$ which you can check is decreasing on $[0,1]$, and so is $d\left(\gamma(1),\gamma(t)\right)$.