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Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y \in AB$. Is it true that then $d(x,y)\geq \frac{1}{4}$ ? I have tried a lot of things but non of them worked, mostly I tried to do it by the triangle inequaity. I even do not know if its true.

Greetings Lena

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  • $\begingroup$ Is it true or does one need further assumptions? $\endgroup$ – Lena Jun 30 '13 at 7:19
  • $\begingroup$ It seems to me that it might be a good idea to try and prove or disprove a local version, for $A$, $B$, $C$, $D$ in a small neighborhood; it might not be true in the large. $\endgroup$ – Robert Lewis Jun 30 '13 at 7:28
  • $\begingroup$ what do you mean by local version? $\endgroup$ – Lena Jun 30 '13 at 7:29
  • $\begingroup$ in a neighbourhood of what ? what would be the advantage if one considers the problem locally? $\endgroup$ – Lena Jun 30 '13 at 7:30
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    $\begingroup$ Note that the inequality does not hold in the plane. But in general, if triangle inequality does not work for you then nothing will --- any compact length-metric space can be approximated by a Riemannian manifold. $\endgroup$ – Anton Petrunin Jun 30 '13 at 12:22
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No (unless you assume something like a lower curvature bound).

Consider two very long segments (say, of length 100) in $\mathbb R^3$ that meet orthogonally at their midpoints. Let $M$ be a smoothened boundary of an $\varepsilon$-neighborhood of the union of these segments. It is a Riemannian manifold invariant under a 90 degrees rotation. Let $A$ be a point of $M$ near one of the segments' endpoints and $B$, $C$, $D$ be its consecutive images under this rotation. Due to rotation invariance, one has $d(A,B)=d(B,C)=d(C,D)=d(D,A)$. And all distances between $A,B,C,D$ are greater than 50.

On the other hand, geodesic segments $AC$ and $BD$ go through the central part of the construction, namely an $O(\varepsilon)$ neighborhood of the center. Take $x$ and $y$ in that region and observe that $d(x,y)=O(\varepsilon)$.

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  • $\begingroup$ But I do not take $x$ and $y$ in the region of the center. As I meantioned, I take $x \in CD$ and $y \in AB$, where by $CD$ and $AB$ I mean the geodesic segment with endpoints $C$ and $D$, and $A$ and $B$ respectively. $\endgroup$ – Lena Jun 30 '13 at 11:33
  • $\begingroup$ In this construction, the segments AB and CD pass really close to the "center". $\endgroup$ – Thomas Richard Jun 30 '13 at 11:49
  • $\begingroup$ @Lena: take $x$ in the intersection of $CD$ with the central region (this is possible because the intersection is nonempty) and $y$ in the intersection of $AB$ with the central region. Is this clearer? $\endgroup$ – Sergei Ivanov Jun 30 '13 at 13:44
  • $\begingroup$ yes, thanks ;). $\endgroup$ – Lena Jun 30 '13 at 13:46
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Surface   $z = y^2 - x^2$,   with proper scaling (say, one scaling per   $\epsilon > 0$),   will provide counter-examples, with   $\frac 14$   replaced by arbitrary   $\epsilon > 0$.   (Perhaps something like this was suggested by Anton Petrunin in his comment above).

By the way, all four mid-points of "edges" of   $ABCD$   will form a set of diameter   $\epsilon$   or less.

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