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dg.differential geometry – Reference request: extendability of Lipschitz maps as an artificial judgement of curvature bounds retort

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dg.differential geometry – Reference request: extendability of Lipschitz maps as an artificial judgement of curvature bounds

In the lecture Notions of Scalar Curvature – IAS round 8:00, Gromov states the next final result, which he claims he does “slightly uncarefully”:

Suppose $(X,g_X)$ and $(Y,g_Y)$ are Riemannian manifolds, their sectional curvature respond $sec(Y,g_Y)leq kappaleq sec(X,g_X)$ for some $kappainmathbb{R}$, and $X_0$ is a subset of $X$. If $f_0:X_0to Y$ is a map with Lipschitz ceaseless $1$, then there exists a map $f:Xto Y$ with Lipschitz ceaseless $1$ that extends $f_0$, i.e. $f|_{X_0}=f_0$.

He mentions a number of names earlier than stating the result, however I can not make out who they’re.

He then discusses how this will breathe used to inspire a definition of “curvature” within the class of metric areas with distance non-increasing maps, “except, of passage, for normalization.”

Does anybody know the place I can learn extra about this? (Either within the setting of metric areas or within the gentle setting of Riemannian manifolds.)

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