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It is hard to imagine that Eddington's numbers could be anything but the imaginary part of the Clifford algebra $C$ of Minkowski space. Recall that the Clifford algebra for an n-dimensional real vec …

You will find a proof in Appendix D (Theorem D.1) of my book `A Tour of SubRiemannian Geometry'. It may not look like what you want at first glance, since that theorem is stated in a more general c …

You are missing a $\dot u$ in your equation! We want a dynamic vector field. The sign of your $\nabla_u u$ and $\Delta u$ are usually taken to be opposite, as with the sign of your $df^*$ and $\nabla …

Fix one $u$, its resulting geodesic probe, and vary $u$ slightly. Comparing the two geodesics at the same arclength values yields a close approximation to the solution to the Jacobi equation al …

I seriously doubt there is any general criteria. However there are more than one beautiful explicit examples of an external field which lead to an integrable problem. The simplest and probably best …

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 …

Joseph: this `answer' is perhaps more a comment than an answer, so apologies in advance. The deepest geometric work I know of is this wonderful piece by Mohammad Ghomi. I quote from his web page: ` …

The answers to question (1) for the 2 body problem are fine, and complete enough. Regarding (2). The 3 body problem (and N-body) with p =3 is significantly simpler than with $p \ne 3$. The added …

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 …

Arnol'd's ODEs. Hirsch and Smale. As a second best the `supersized version' of this with Devaney added as a co-author.