2012-09-19T11:42:50Z2012-09-19T11:43:29Zhttp://cab.unime.it/mus/id/eprint/694This item is in the repository with the URL: http://cab.unime.it/mus/id/eprint/6942012-09-19T11:42:50ZOn totally geodesic affine immersions into affine manifolds of recurrent projective curvatureIn 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.Zbigniew Olszak