×

zbMATH — the first resource for mathematics

On a concept of \(P\)-reducibility in Finsler spaces. (English) Zbl 0789.53014
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 403-409 (1992).
To consider \(P\)-reducible Finsler spaces, the author defines an \(A\)- indicatrix \(H^{n-2}\) of a Finsler space \(F^ n\) which is a hypersurface of the indicatrix \(I^{n-1}\) given by \(\log \sqrt{\overline{g}} = \text{const.}\), where \(\overline g\) is the restriction of \(g = \text{det}(g_{ij})\) to \(I^{n-1}\). \(H^{n-2}\) is normal to the vector field \(A = (A_ i)\) on \(I^{n-1}\). The second fundamental form, Gauss and Weingarten equations are found for \(H^{n- 2}\). Consequently the necessary and sufficient condition for \(H^{n-2}\) to be a congruence of totally geodesic hypersurfaces is written explicitly.
For the entire collection see [Zbl 0764.00002].
MSC:
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
PDF BibTeX Cite