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Probably you are misinterpreting some statement in Krantz and Parks, since the claim that if $F$ is convex in the tangent variable for every $x\in U$ then straight lines are locally minimizing is clea …

A little algebra shows that, for $A=(1,0,0)$, $B=(0,1,0)$, and $C=(0,0,1)$ on the sphere, this surface is an irreducible algebraic surface of degree 5 that is singular at $A$, $B$, and $C$ but is othe …

Note: This is the third version of my answer, one that is, I hope, considerably clearer and cleaner than the previous two. This can be reduced to a standard problem in real arithmetic, one that, in …

The metric you have written down is the standard bi-invariant metric on the Lie group of nonzero quaternions. I.e., it is the metric such that the left-invariant 1-form $\omega = \mathbf{q}^{-1}\ d\m …

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 …

I was reminded of this question recently by a related problem, and I remembered a concept that I had not remembered when I originally saw the question, namely É. Cartan's notion of exterior square of …

The OP doesn't say what is meant by a 'geometric object', so it's hard to give a definitive answer. However, if one assumes that the geometric object is a smooth manifold $M^7$ and that the action is …

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 …

This is not even true on the circle. In that case, given any metric that comes from a Riemannian metric there is a constant $c>0$ such that there is a diffeomorphism of the circle with itself that ca …

In the smooth category, the problem has a completely different character for $n>2$ than when $n=2$. The first obstruction is whether there exists a positive definite quadratic form~$h$ on $S^n$ tha …

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 …

Here is a rigorous proof that non-horizontal curves are not rectifiable. First, recall that, given a metric space $(M,\delta)$ and a mapping $\gamma:[0,1]\to M$, the $\delta$-length of $\gamma$ is, b …

First, the metric that you write down has Gauss curvature $K\equiv-1$, so if you want to compute distances in this metric, you should write it in more standard coordinates and then use the known dista …

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 …

$\mathcal{T}$ is not a Lie group when $n>1$. Actually, the OP did not say whether he wanted $\mathcal{T}$ to be all possible sequences of compositions of these generating sets, but, if he did, then …