Annales AcademiŠ Scientiarum FennicŠ
Mathematica
Volumen 33, 2008, 35-63

THE HESSIAN OF THE DISTANCE FROM A SURFACE IN THE HEISENBERG GROUP

Nicola Arcozzi and Fausto Ferrari

Dipartimento di Matematica dell'UniversitÓ di Bologna
Piazza di Porta S. Donato, 5, 40126 Bologna, Italy; arcozzi 'at' dm.unibo.it

Dipartimento di Matematica dell'UniversitÓ di Bologna
Piazza di Porta S. Donato, 5, 40126 Bologna, Italy; ferrari 'at' dm.unibo.it

Abstract. Given a smooth surface S in the Heisenberg group, we compute the Hessian of the function measuring the Carnot-CharathÚodory distance from S in terms of the mean curvature of S and of an "imaginary curvature" which was introduced in [2] in order to find the geodesics which are metrically normal to S. Explicit formulae are given when S is a plane or the metric sphere.

2000 Mathematics Subject Classification: Primary 49Q15, 53C17, 53C22.

Key words: Heisenberg group, Carnot-CharathÚodory distance, horizontal Hessian, sub-Riemannian geometry.

Reference to this article: N. Arcozzi and F. Ferrari: The Hessian of the distance from a surface in the Heisenberg group. Ann. Acad. Sci. Fenn. Math. 33 (2008), 35-63.

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