We build a variational theory of geodesics of the Tanaka-Webster connection $\nabla$ on a strictly pseudoconvex CR manifold $M$. Given a contact form $\theta$ on $M$ such that $(M , \theta)$ has nonpositive pseudohermitian sectional curvature ($k_\theta (\sigma ) \leq 0$) we show that $(M, \theta)$ has no horizontally conjugate points. Moreover, if $(M, \theta)$ is a Sasakian manifold such that $k_\theta (\sigma) \geq k_0 >0$ then we show that the distance between any two consecutive conjugate points on a lengthy geodesic of $\nabla$ is at most $\pi/(2 \sqrt{k_0})$. We obtain the first and second variation formulae for the Riemannian length of a curve in $M$ and show that in general geodesics of $\nabla$ admitting horizontally conjugate points do not realize the Riemannian distance.

Jacobi fields of the Tanaka-Webster connection on Sasakian manifolds

Abstract

We build a variational theory of geodesics of the Tanaka-Webster connection $\nabla$ on a strictly pseudoconvex CR manifold $M$. Given a contact form $\theta$ on $M$ such that $(M , \theta)$ has nonpositive pseudohermitian sectional curvature ($k_\theta (\sigma ) \leq 0$) we show that $(M, \theta)$ has no horizontally conjugate points. Moreover, if $(M, \theta)$ is a Sasakian manifold such that $k_\theta (\sigma) \geq k_0 >0$ then we show that the distance between any two consecutive conjugate points on a lengthy geodesic of $\nabla$ is at most $\pi/(2 \sqrt{k_0})$. We obtain the first and second variation formulae for the Riemannian length of a curve in $M$ and show that in general geodesics of $\nabla$ admitting horizontally conjugate points do not realize the Riemannian distance.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11563/16748