Articles of metric geometry

Are all continuous bijective translations isometries?

Let $(M, d)$ be a metric space. I define a translation on $M$ to be a function $f$ from $M$ to $M$ such that $d(x,f(x))=d(y,f(y))$ for all $x$ and $y$ in $M$. In a previous question, I asked if every translation was an isometry in the post Are all metric translations isometries. The answer was […]

CAT(K) Finsler manifolds.

I was wondering if the following is true (and common knowledge): Let $(M,F)$ be a Finsler manifold. Let d be the induced distance by the norm in the usual sense. That is, $d(x,y)=\inf${lenghts of all piece-wise smooth curves…}. We consider then $(M,d)$ as a metric space. My questions, which are probably quite easy, are these: […]

why are CAT(0) spaces contractible?

In the book of Bridson and Haefliger it is said that ‘it follows easily’ from what they proved before. Does anyone know of a rigorous proof that CAT(0) spaces are contractible?

How can the notion of “two curves just touching” (vs. “two curves intersecting”) be expressed for a given metric space?

A popular introductory description of a “tangent (in geometry)” is presented as “the tangent line (or simply tangent) to a plane curve at a given point is the straight line that “just touches” the curve at that point.“ I like to find out whether and how this description might be expressed or translated in the […]

isometries of the sphere

There is a theorem by Pogorelov that if a $C^2$ surface $M$ in $\mathbb{R}^3$ is isometric to the unit 2-sphere, then $M$ is itself (a rigid motion of) the sphere. What is known about isometric deformations of the sphere, when the smoothness condition is relaxed? First, we need some notion of isometry for surfaces with […]

A complete metric space with approximate midpoints is intrinsic

Let $(X,d)$ be a metric space. For $x,y\in X$, define $A_{xy}$ the set of curves (the domain is supposed to be $[0,1]$) joining $x$ with $y$. For $\sigma\in A_{xy}$, define its length as $$L(\sigma)=\sup\sum_{i=1}^n d(\sigma(y_{i-1}),\sigma(y_i)),$$ where the supremum is taken over all partitions ${t_0,\ldots,t_n}$ of $[0,1]$. Define $$d_L(x,y)=\inf_{\sigma\in A_{xy}} L(\sigma).$$ The metric $d$ is said […]

Isometric immersions between manifolds with boundary are locally distance preserving?

Let $M$ be a compact, connected, oriented smooth Riemannian manifold with non-empty boudary. Let $f:M \to M$ be a smooth orientation preserving isometric immersion. Is it true that $f$ is locally distance preserving? (in the sense that around every $p \in M$ there exist a neighbourhood $U$ s.t $d(f(x),f(y))=d(x,y)$ for all $x,y \in U$) This […]

Can the isometry group of a metric space determine the metric?

Let $(X,d)$ be a metric space. There are always other metrics on $X$ which generates the same topology, and have the same isometry groups, for instance $\tilde d =\sqrt d$. (The same will be true for any injective continuous function $f:\mathbb{R}^{\ge 0}\to \mathbb{R}^{\ge 0}$, taking zero to itself and which satisfying some constraints to ensure […]

Linear bound on angles in an euclidean triangle.

I am trying to understand a proof in the book of Burago “A Course in metric geometry” (Lemma 10.8.13 page 383). I have difficulties with a certain inequality for the angle of euclidean triangles: Assume we are given a triangle $\triangle paq$ in euclidean space. By the cosine law the angle $\sphericalangle paq$ between the […]

Is there an explicit left invariant metric on the general linear group?

Consider $GL_n^+$, the group of (real) invertible matrices with positive determinant. Is it possible to find an explicit formula for a metric on $GL_n^+$ which is left-invariant, i.e $$d(A,B)=d(gA,gB) \, \,\forall A,B,g \in GL_n^+$$ and which generates the standard topology on $GL_n^+$. (Without the last requirement the discrete metric will do). Even finding a concrete […]