Estimating \(\| d\phi ^ t\|\) for unit vector fields whose orbits are geodesics (Q752506)

To give a rough idea of this very interesting paper, we recall the following notations (in the category of real analytic manifolds). Let M be a connected, n-dimensional, complete Riemannian manifold and v a unit vector field with \(\nabla_ vv=0\). For \(x\in M\), set \(e_ x=\max \{| \nabla_ zv|^ 2-K_{zv}\},\) where z ranges over all unit vectors of \(T_ xM\) perpendicular to v and \(K_{zv}\) denotes the sectional curvature at x, with respect to the plane spanned by z and v. Also set \(E_{xt}=\max e_ y\) (x\(\in M\), \(t\geq 0)\), where \(y\in [\phi^{-\sqrt{2}t}(x),\phi^{\sqrt{2}t}(x)]\) and \(\phi^ t\) is the flow generated by v. Then, for \(x\in M\) with \(e_ x\geq 0\), it is proved that \[ | d\phi^ t(u)|^ 2+| d\phi^{-t}(u)|^ 2\leq 2 \cosh^ 2t\sqrt{E_{xt}}\quad (t\geq 0), \] for any unit vector \(u\in T_ xM.\) More generally, for any \(x\in M\) and any unit vector \(Z\in T_{(x,x)}D\) (D the diagonal of M), \(| d\phi^ t(Z)| \leq \cosh t\sqrt{\bar E_{xt}}\) for all t, or \[ | d\phi^ t(u)|^ 2+| d\phi^{-t}(u)|^ 2\leq 2 \cosh^ 2t\sqrt{2\bar E_{x\sqrt{2}t}}, \] for all t and every unit vector u. Here \(\bar E_{xt}\) is defined accordingly (although in a more complicated way). The previous study is based on the crucial idea to shift from the study of v to that of an appropriate vector field defined on the graph \({\mathfrak G}(v)\) of the 1-foliation defined by v [cf. also Ann. Global Anal. Geom. 1, No.3, 51-75 (1983; Zbl 0526.53039) by the same author]. The basic results obtained in this direction are heavily used in the proofs of the main theorems of the paper. Two theorems concerning lower bounds are also included.