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Take the product $S^2 \times S^2$ of two two-spheres, but perturb the product metric as follows. Think of each $S^2$ as the unit two-sphere in Euclidean 3-space in the standard way so that for $p …

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$The free Lie algebra $\L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of https://en.wikipedia.org/wiki/Free …

I believe the standard example $dz - ydx = 0$ does the trick. This is the normal form for what many people call a ``quasi-contact distribution'-the even dimensional analogue of a contact form. Suppo …

Write $A = 1/a, B = 1/b, C = 1/c$ so that, in the problem solver's notation we have, for example, $<X_1, X_1 > = A^2$, and the metric is $$ds^2 _{a,b,c} = A^2 \sigma_1 ^2 + B^2 \sigma_2 ^2 + C^2 \s …

More of a comment than an answer. Chapter 5 of William Goldman's book `Complex Hyperbolic geometry' has a great detail of information on the bisectors in complex hyperbolic space (so real dimension …

I would argue, to the contrary, that there is NO explicit formula for the distance function, even for the standard 3 dimensional Heisenberg group. Look at eq (1.40) of your ref [1]. The variable $\th …

There is a wonderful trick - I think promulgated by Moser - for viewing Hamiltonian flows on the cotangent bundle of the sphere as the reduction of a Hamiltonian flow on the ambient phase space of ( …

Perhaps you mean to say that your metric behaves rather like those arising in Snell's law from geometric optics where to model the refraction of light as it passes from one medium to another you t …

Form the contact 1-form $\Theta = p \, dq -H \, dt$ on extended phase space $T^* Q \times {\mathbb R}$, the second factor being time and parameterized by $t$, the function $H = H(q,p,t)$ being the ti …

Feynman, in his lecture notes, argues convincingly that you will understand the guts of Quantum Mechanics [QM] as best you can by looking at the two-slit experiment whose Hilbert space is two-dimensi …

Re question 2. The answer is provided by the Ambrose Singer theorem. You can see how it connects with your question in my book A Tour of sub-Riemannian Geometry. When the connection form is analy …

The exponential map for any Riemannian metric on your compact manifold $M$, based at any point $p$ of $M$, maps the tangent space $T_p M$, an ${\mathbb R}^n$, onto $M$ and is a diffeo inside the cu …

I think a good starting place for your question regarding the moduli space for a flat 4-torus is the Fourier-Mukai' correspondence which came out of work of Nahm and which relates the moduli space o …

This is a kinda stupid answer but it is the simplest most useful test I know. Does your Hamiltonian vector field have a zero? If `yes': sorry, not a geodesic flow! (Could be the reduction of a geodes …

The reason the problem is hard is that we do not have a good handle on what abnormal (=singular) geodesics can look like. See the chapter of my book that describes abnormal geodesics. Progress is …