On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields |
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Authors: | Sorin Dragomir Domenico Perrone |
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Affiliation: | (1) Dipartimento di Matematica, Università degli Studi della Basilicata, Campus Macchia Romana, 85100 Potenza, Italy;(2) Dipartimento di Matematica Ennio De Giorgi, Università di Lecce, Via Provinciale Lecce-Arnesano, 73100 Lecce, Italy |
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Abstract: | We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S1 → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X↑ project on a nonlinear system of subelliptic PDEs on M. Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20 |
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Keywords: | Harmonic vector field Fefferman metric Tanaka-Webster connection Sasaki metric Reeb field Pseudoharmonic vector field |
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