# 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

## Related Questions

### Are universal geometric equivalences of DM stacks affine?

0  Asked on January 17, 2021 by harry-gindi

### Barycentric coordinates of weighted edges

0  Asked on January 17, 2021 by manfred-weis

### Question on limit in probability of the ratio of max to min of 2 sequences of non-ive, continuous iid random variables with support $[0, infty).$

1  Asked on January 16, 2021 by learning-math

### PhD dissertations that solve an established open problem

18  Asked on January 16, 2021

### Commutant of the conjugations by unitary matrices

3  Asked on January 16, 2021 by jochen-glueck

### Constructing intertwiners between representations of compact quantum groups

1  Asked on January 15, 2021

### A certain property for Heegaard splittings

1  Asked on January 15, 2021 by no_idea

### When have we lost a body of mathematics because errors were found?

10  Asked on January 14, 2021 by edmund-harriss

### Prove that there are no composite integers $n=am+1$ such that $m | phi(n)$

1  Asked on January 13, 2021 by david-jones

### Kernel of the map $mathbb{C}[G]^U to mathbb{C}[U^+]$

0  Asked on January 13, 2021 by jianrong-li

### Finite fast tests for periodicity of certain matrices

1  Asked on January 12, 2021

### Are these two kernels isomorphic groups?

0  Asked on January 12, 2021 by francesco-polizzi

### Definition of subcoalgebra over a commutative ring

2  Asked on January 11, 2021 by user839372

### Riesz Representation Theorem for $L^2(mathbb{R}) oplus L^2(mathbb{T})$?

0  Asked on January 10, 2021 by goulifet

### Morphism of distinguished triangles where one of the arrows is a quasi-isomorphism

0  Asked on January 10, 2021 by stabilo

### Cut points and critical points of the exponential map

0  Asked on January 9, 2021 by longyearbyen

### Eigendecomposition of $A=I+BDB^H$

0  Asked on January 9, 2021 by user164237