# On totally geodesic affine immersions into affine manifolds of recurrent projective curvature

Olszak, Zbigniew (1994) On totally geodesic affine immersions into affine manifolds of recurrent projective curvature. Accademia Peloritana dei Pericolanti, Classe di Scienze FF. MM. NN., LXXII. pp. 361-373.

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## Abstract

In the previous paper [8], the author has studied totally geodesic affine immersions $f:(M, \nabla \ ) \to \ ( \bar M \, \bar \nabla \ )$ in the case when $( \bar M \, \bar \nabla \ )$ is an affine manifold of recurrent curvature. It is shown there that $(M, \nabla \ )$ is flat or of recurrent curvature. And if $f$ is additionally umbilical with non-zero shape tensor and dim $M \ge \ 3$, then $(M, \nabla \ )$ is locally projectively flat. In the presented paper, the investigation of totally geodesic affine immersions is continued. We assume that the ambient manifold $( \bar M \ , \bar \nabla \ )$ is of recurrent projective curvature. It is proved that if dim $M \ge \ 3$, then $(M, \nabla \ )$ is locally projectively flat or of recurrent projective curvature. Moreover, if $f$ is additionally umbilical with non-zero shape tensor, then $(M, \nabla \ )$ is locally projectivvely flat.

Item Type: Article M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Scienze Fisiche, Matematiche e Naturali > 1994M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Scienze Medico-Biologiche > 1994M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Scienze Giuridiche, Economiche e Politiche > 1994M.U.S. - Miscellanea > Atti Accademia Peloritana > Classe di Lettere, Filosofia e belle Arti > 1994 Dr A F 19 Sep 2012 11:42 19 Sep 2012 11:43 http://cab.unime.it/mus/id/eprint/694