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

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