Hypersurfaces of Projective β –Changes

In 1984 C. Shibata investigated the theory of a change which is called a β -change of Finsler metric [10]. On the other hand, in 1985 a systematic study of geometry of hypersurfaces in Finsler spaces was given by M. Matsumoto [6]. In § 1, according to Shibata [10] we shall prepare the terminology and notations for the sake of argument. In § 2, we derive a condition for a β -change to be projective [10]. In § 3, we find that a totally geodesic hypersurface (n 1)-dimensional F remains to be a totally geodesic hypersurface (n 1)-dimensional F-bar under the projective β -change. In the last section, we get the main result that Finslerian hypersurfaces given by the projective β -change are projectively flat on condition the original Finsler space is projectively flat.