A nonstandard theory of Finsler subspaces. (English) Zbl 0641.53021

Topics in differential geometry, Pap. Colloq., Hajduszoboszló/Hung. 1984, Vol. 2, Colloq. Math. Soc. János Bolyai 46, 815-851 (1988).
[For the entire collection see Zbl 0628.00010.]
A generalized Finsler space (GFS) is nothing but a real differentiable manifold whose vertical bundle is endowed with a pseudo-Riemannian metric. The purpose of the paper is to obtain a theory of submanifolds in a GFS. From the results we only mention that we obtain all structure equations of Gauss, Codazzi and Ricci for such submanifolds. The first author has first obtained such results for hypersurfaces of GFS [Finsler spaces, Proc. 3rd Natl. Semin., Braşov/Rom. 1984, 117-131 (1984; Zbl 0557.53012)].
We finally note that the whole study is based on the following hypothesis: the induced nonlinear connection is a vector subbundle of the nonlinear connection on the ambient GFS. Later on, the above hypothesis was dropped and the second author obtained a coordinate-free theory of submanifolds of a GFS [Math. Rep., Toyama Univ. 10, 133-168 (1987; Zbl 0629.53023)].
Reviewer: R.Miron


53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53B25 Local submanifolds