In the lecture [Notions of Scalar Curvature - IAS](https://youtu.be/uBrrKaBrPKU) around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully":



> 
> Suppose $(X,g\_X)$ and $(Y,g\_Y)$ are Riemannian manifolds, their sectional curvature satisfy $\sec(Y,g\_Y)\leq \kappa\leq \sec(X,g\_X)$ for some $\kappa\in\mathbb{R}$, and $X\_0$ is a subset of $X$. If $f\_0:X\_0\to Y$ is a map with Lipschitz constant $1$, then there exists a map $f:X\to Y$ with Lipschitz constant $1$ that extends $f\_0$, i.e. $f|\_{X\_0}=f\_0$.
> 
> 
> 


He mentions a few names before stating the result, but I cannot make out who they are.


He then discusses how this can be used to motivate a definition of "curvature" in the category of metric spaces with distance non-increasing maps, "except, of course, for normalization."


Does anyone know where I can read more about this? (Either in the setting of metric spaces or in the smooth setting of Riemannian manifolds.)



Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds
I can give a partial answer. The theorem you quote is a generalization of [Kirszbraun's theorem](https://en.wikipedia.org/wiki/Kirszbraun_theorem) (which covers the case where $X$ and $Y$ are Hilbert spaces), and a special case of a beautiful theorem of [Lang and Schroeder](https://link.springer.com/article/10.1007/s000390050018) (which applies to general metric spaces with synthetic curvature bounds defined via triangle comparison). These are the names Gromov mentions.


Personally, I do not know of a theory that takes this Lipschitz extension property as a *definition* of curvature, but that is likely my ignorance.


