A function of direction in a Weyl hypersurface

Abstract

In an \((n+1)\)-dimensional space of Weyl \(W_{n+1}\) the author considers a hypersurface \(W_ n (g_{ij}, T_ k)\) with a fundamental tensor \(g_{ij}\) and a complementary vector \(T_ k\). In \(W_ n\) he chooses \(n\) congruences of an orthogonal ennuple of unit vectors \(\vec v_ r\) \((r=1,2,\dots, n)\) with contravariant components \(v_ r^ i\). Let \(k_{rr}\) be the normal curvature of \(W_ n\) in the direction of the vector of components \(v^ i_ r\), \(k_{rs}\) \((r\neq s)\) be the invariants of the geodesic torsion of the curve of the congruence with unit tangent vector of components \(v_ r^ i\) and \(\zeta_ r^ s\) be the geodesic curvatures of the orthogonal ennuple. Finally, let \(\delta_ r k_{rr}\) denote the derivative of \(k_{rr}\) and set \(P_ r:= v^ d_ r T_ d\). In the paper under review the author proves that the expression \[ -\delta_ r k_{rr}+ 2\zeta^ s_ r k_{rs}+ 2P_ r k_{rr} \qquad (r,s=1, 2,\dots,n;\;r\neq s) \] depends only on the direction of the unit vector of components \(v_ r^ i\). A similar problem, where the hypersurface belongs to an \((n+1)\)-dimensional Riemannian space, was investigated by \textit{A. Özdeğer} [J. Geom. 17, 1-6 (1981; Zbl 0483.53022)].