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Any surface of revolution in $3$-space with poles will have this property. The reason is that, in this case, any geodesic either goes through a pole (i.e., a point where the axis of revolution meets …

NB (3/1/13): I revised this answer to make it more complete (and, to be frank, more accurate). My original answer did not take into account the difference between the cut locus and the conjugate loc …

This may not reallly be an answer that you like, but I think that, maybe you misunderstood what Ben McKay was trying to describe. Here is a more explicit, extrinsic description that may help: Suppose …

There exist many algebraic surfaces of constant breadth with no continuous symmetries, even ones with no symmetries at all. To see this, consider the properties of the support parametrization: The su …

The OP specifically asked for a(n ordinary) metric, not a Riemannian metric. While Misha and Paul have given good answers, I think that it's worth pointing out that, if one just takes an arbitrary le …

Yes, $L_\delta$ is algebraic. You can find its equations by elimination theory as follows: Let $L$ be defined by the polynomial equation $F(x,y) = 0$. Now consider the polynomial equations $$ F(x,y …

I've added a few sentences to my answer to clarify something that some readers may be wondering about, which is why there isn't as simple an answer for surfaces in $3$-space as there is for curves in …

There's a different kind of answer to this that you might be interested in: Suppose that, when you fire off a probe along a unit speed geodesic starting at $p\in M$, you record the direction $\theta$ …

This is really more of a response to the OP's request for an 'insightful way to view the computation' than it is an answer to the specific problem; so keep this in mind. On the other hand, as I'll po …

Here's an answer to an analogous question, not Bill's original question, but also a question about how to specify (in a simple way) a non-positively curved metric on a compact Riemann surface $C$ of g …

Using the operator norm, as you have defined it, the fraction of the unit ball in real symmetric $n$-by-$n$ matrices that consists of positive definite matrices is $2^{-n(n+1)/2}$. Thus, this fractio …

I think it's important to keep two things separate here: First, if $(M,F)$ is a smooth Finsler manifold (which means that $F^2:TM\to [0,\infty)$ is smooth and strongly convex away from the zero sec …

Unfortunately, my original answer below was completely misguided (i.e., wrong), and the people who up-voted it should feel free to reverse their votes! I'm leaving the answer below so that people can …

Any smooth compact surface smoothly embedded in $\mathbb{R}^3$ that is not the $2$-sphere must have an infinite fundamental group and hence must have infinitely many distinct (in your sense) geodesics …

At least, in the classical case treated by Cayley, the $2{\times}2{\times}2$ hyperdeterminant, there is an interpretation in terms of volumes generalizing the classical determinant case. It goes like …