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Currently I am reading about Heisenberg Group. And I understand that this group is one of the simplest examples of sub-Riemannian manifolds. I have read a lot about it structure, geodesics and e.t.c. Mostly from this Lecture Notes (attached below): https://cvgmt.sns.it/media/doc/paper/5339/sub-Riem_notes.pdf

But now I do not clearly understand, why we restrict the bundle and work on the Horizontal sub-bundle in case of the Heisenberg group. Why it is said that we cannot move directly along the Z-coordinate?

If you can share any materials or thoughts about this issue I will be very grateful!

Tat-iva
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  • Welcome to Math StackExchange. It's best practice to include explicit information about your question rather than referencing a text. This makes the question more searchable and increases the probability that someone will answer it! – J.V.Gaiter Apr 14 '24 at 18:52
  • Your title should be *sub*-Riemannian structure which is the main topic of Le Donne's notes which you can also follow on some very good Youtube videos. A good motivation for sub-Riemannian geometry is Dido's problem which is also contained in those notes. It is worth spending a lot of time with this. Having said that: the Heisenberg group admits also a quite natural Riemannian metric which leads to a scalar curvature of $-1/2,.$ An entirely different topic. – Kurt G. Apr 14 '24 at 18:52

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