# Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds

MathOverflow Asked by Lawrence Mouillé on January 1, 2022

In the lecture Notions of Scalar Curvature – IAS 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 $$sec(Y,g_Y)leq kappaleq sec(X,g_X)$$ for some $$kappainmathbb{R}$$, and $$X_0$$ is a subset of $$X$$. If $$f_0:X_0to Y$$ is a map with Lipschitz constant $$1$$, then there exists a map $$f:Xto Y$$ 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.)

I can give a partial answer. The theorem you quote is a generalization of Kirszbraun's theorem (which covers the case where $$X$$ and $$Y$$ are Hilbert spaces), and a special case of a beautiful theorem of Lang and Schroeder (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.

Answered by user142382 on January 1, 2022