Cylindricity of tangent bundles of strongly parabolic metrics and strongly parabolic surfaces

It is proved that if the intrinsic zero-index of the Sasaki metric of a tangent bundleTMⁿ isk, thenk is even andMⁿ is the metric product of a Riemannian manifoldM^n–k/2 by a Euclidean spaceE^k/2, whileTMⁿ is the metric product ofTM^n–k/2 byE^k. An expression is obtained for the second fundamental forms of the imbeddingTF^lTMⁿ in terms of the second fundamental forms of the imbeddingF^lMⁿ and the curvature tensor ofMⁿ. It is proved thatTF^l is totally geodesic inTMⁿ if and only ifF^l is totally geodesic inMⁿ.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 12–32.