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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)