# Quotient metrics in length spaces

If I collapse a (say, closed) set in a length space, I obtain a length space: is there some literature on this?

We consider length spaces as defined by Gromov and others. [However the case of a Riemannian distance already leads to interesting examples of what I am writing]. If $$X$$ is one such space, it has a distance $$d$$ which can be recovered by the length of curves. Suppose $$X$$ is path connected. Let $$c:[a,b]\to X$$ be a curve, define $$|c'(t)|=\lim_{\epsilon\to0}\sup_{|u-t|,|v-t|\le\epsilon}\frac{d(c(u),c(v))}{|u-v|}$$ and $$L(c)=\int_a^b|c'(t)|dt,$$ and set $$D(x,y)=\inf\{L(c) \text{ c is a curve having endpoints x and y}\}.$$ We are interested in cases where $$d=D$$ and where the distance $$D=d$$ is realized by lengths of geodesics (i.e. the hard work has been already done).

Now, let $$E\subseteq X$$ be closed and define a distance $$D_X$$ on $$X\setminus E\cup\{E\}$$ ($$X$$ with $$E$$ collapsed to a point): $$L_E(c)=\int_a^b|c'(t)|\chi_{X\setminus E}(c(t))dt,$$ and set $$D_E(x,y)=\inf\{L_E(c) \text{ c is a curve having endpoints x and y}\},$$ if $$x,y\in X\setminus E$$, and $$D_E(x,E)=D(x,E)=\inf_{y\in E}D(x,y)$$.

The space $$X\setminus E\cup\{E\}$$ is clearly a length space w.r.t. the distance $$D_E$$.

The reason I find these objects interesting is that, if $$E$$ is the smooth boundary of an open subset of $$X$$, a nice Riemannian manifold, then the points of $$E$$ play the role of the unit vectors in the tangent space of a point; this meaning that they can parametrize those geodesics leaving $$E$$ which, at least locally, minimize the distance from $$E$$. Even in the case of the Euclidean plane one obtains interesting pictures.

I would be very surprised if no one had developed this viewpoint in the past.

• Post Scriptum: one may view the definition of $L_E$ and $D_E$ as a limiting case in optics, with the light having infinite speed inside $E$. – Nicola Arcozzi Jun 11 '19 at 11:57
• you mean $\chi_{X\setminus E}$ instead of $\chi_E$? – Emanuele Paolini Jun 11 '19 at 13:36
• Yes! Thanks Emanuele... I edit. – Nicola Arcozzi Jun 11 '19 at 15:20
• On a side note: for a continuous, non-constant curve of finite length $c:[a,b]\to X$ one may have $|c'(t)|=0$ a.e., thus $\int_a^b|c'(t)dt=0$ (e.g. for the Cantor function). In fact, I think this is always true up to reparametrizations, so you would have $D(x,y)=0$ for all $x,y$! However, the equality "integral of |c'(t)| = length" is true for Lipschitz curves, which is ok for many purposes since continuous curves, finite length curves are Lipschitz up to reparametrization. – Pietro Majer Jun 12 '19 at 7:42
• (more precisely: if $c:[0,1]\to X$ is any curve and $C:[0,1]\to[0,1]$ is the Cantor function and ${\bf C}\subset[0,1]$ is the Cantor set, then $|(c\circ C)'|=0$ a.e., because $c\circ C$ is locally constant in ${\bf C^c}$) – Pietro Majer Jun 12 '19 at 8:14

## 1 Answer

If you take $$X=\mathbb R$$ and $$E=[1,2]\cup[3,4]$$ you see that $$D_E(0,5) = 3$$ while the geodesic distance between $$0$$ and $$5$$ in $$X\setminus E \cup\{E\}$$ is $$2$$ because you can join $$0$$ to $$E$$ with a curve of length $$1$$ and $$E$$ to $$5$$ with a curve of length $$1$$. I think you need to require $$E$$ to be geodesic-convex if you want the resulting space to be a length space.

Maybe what you really mean is $$d_E(x,y) = \min\{d(x,y), d(x,E) + d(y,E)\}.$$ This is the particular case of a quotient space. It is understood that the quotient space of a length space is a length space if the quotient has good properties (in your case $$E$$ being closed). See, for example, https://people.math.ethz.ch/~lang/LengthSpaces.pdf Proposition 3.2

• You are right Emanuele: with the definition I gave I am collapsing each connected component of $E$ to a distinct point (which is what I had in mind in this morning's mental picture), while you definition collapses the whole set to a point (which is my usual mental picture and which gives your definition). I have read the Proposition in Lang's notes. Both mental pictures correspond to his assumptions, with different equivalence relations. – Nicola Arcozzi Jun 11 '19 at 15:33