The geometry on a step 3 Grushin model

Abstract

In this article we study the geometry associated with the sub-elliptic operator

, where X ₁ = ∂ _x and X ₂ = x ²/2∂ _y are vector fields on R ². We show that any point can be connected with the origin by at least one geodesic and we provide an approximate formula for the number of the geodesics between the origin and the points situated outside of the y-axis. We show there are infinitely many geodesics between the origin and the points on the y-axis.

Keywords:

• Grushin operator
• Sub-Riemannian geometry
• Geodesics
• Hamilton-Jacobi theory
• Elliptic functions
• Euler's theta functions
• Jacobi's epsilon function
• AMS Classifications: Principal: 53C17
• Secondary: 34K10, 35H20

Acknowledgements

A part of this article is based on a lecture presented by the second author during the Spring School of EU Research and Training Network on “Geometric Analysis” which was held on March 1–6, 2004 at the Institute of Mathematics, Universität of Potsdam, Germany. The second author would like to thank the organizing committee, especially Professor B.-Wolfgang. Schulze for his warm hospitality during his visit to Germany. Partially supported by a William Fulbright Research grant and a competitive research grant at Georgetown University

Notes

Dedicated to Professor B.-Wolfgang Schulze on his sixtieth birthday

Additional information

Notes on contributors

Der-Chen Chang

Dedicated to Professor B.-Wolfgang Schulze on his sixtieth birthday