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On totally umbilical submanifolds of conformally birecurrent manifolds. (English) Zbl 0667.53021

Let M be a pseudo-Riemannian manifold. M is called conformally birecurrent if its (0,4)-Weyl conformal curvature tensor C satisfies \[ C_{r_ 1r_ 2r_ 3r_ 4,vw}C_{s_ 1s_ 2s_ 3s_ 4}=C_{s_ 1s_ 2s_ 3s_ 4,vw}C_{r_ 1r_ 2r_ 3r_ 4,} \] where the comma denotes the covariant differentiation with respect to the metric of M. The authors study conformally birecurrent totally umbilical submanifolds of conformally birecurrent manifolds and give examples. The main result is a necessary and sufficient condition for an analytic, non conformally flat, totally umbilical submanifold of a conformally birecurrent manifold to be conformally birecurrent itself.
Reviewer: J.Deprez

MSC:

53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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