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On the connection between the nonholonomic metric on the Heisenberg group with the Grushin metric. (Russian, English) Zbl 1054.53062

Sib. Mat. Zh. 44, No. 6, 1377-1384 (2003); translation in Sib. Math. J. 44, No. 6, 1085-1090 (2003).
The author computes the geodesics of the Grushin metric \(dx^2 + 4x^{-2}dy^2\) on the upper-half plane \(x>0\) and shows that the projection of the Heisenberg group \(H\) onto the upper-half plane with the Grushin metric, which in cylindrical coordinates \((z,r,\varphi)\) on \(H\) takes the form \(z=y\), \(r=x\), is a submetry. Moreover, it maps geodesics of \(H\) with one end on the \(z\) axis into geodesics of the Grushin metric and preserves their lengths.

MSC:

53C22 Geodesics in global differential geometry
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