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Elipsoid does not posess unbounded geodesics with no self-intersection. I do not know a conceptual explanation. My explanation is that (due to integrability of the geodesic flow of ellipsoid) we …

No, because otherwise we will have this property also for degenerate ellipses, which are intervals, which would imply that the euclidean distance between two (sufficiently close) points is $\lambda …

A pair (projective structure, volume form) allows one to canonically construct a torsion-free affine connection. This affine connection belongs to the projective structure and has the property …

The answer is positive; in fact any smooth manifold has two nonisometric metrics with conjugate geodesic flows. A construction is in C. Croke, B. Kleiner, Conjugacy and rigidity for manifolds with a p …

If $\mathcal{L}_v J= 0$ and $\mathcal{L}_v \Omega= 0$, then $\mathcal{L}_v g=0$ so $v$ is a Killing vector field. Indeed, the property $\mathcal{L}_v J= 0$ is equivalent to the condition that the …

This is both: answer on the new version of the question and comment on comment of Andrey Gogolev, who asked whether one can make the question more complicated assuming additionally that the set of …

The answer is no. The explanation of Anton is of course correct but there exist stronder statements in the literature: for example by Sinjukov (Dokl. Akad. Nauk SSSR (N.S.) 98, (1954) 21--23) any sy …

I do not know whether this is the answer you want since in my understanding it is well known: If we have a Riemannian metric $g$ on $M$, i.e. a (nondegenerate) bilinear form on each $T_xM$, then w …

The Riemannian distance has the following property: it is the so called length space. I recall (one of) the definitions of the length space. Having a distance function $d$, we can define the length o …

There exists examples of different point configurations in $\mathbb{R}^2$ having the same the set (but different matrices of!) distances. The simplest example contains 4 points and could be found i …