On nondegenerate umbilical affine hypersurfaces in recurrent affine manifolds
Abstract
Let M̃ be a differentiable manifold of dimension at least 5, endowed with a torsion-free affine connection of recurrent curvature. Let M be a nondegenerate umbilical affine hypersurface in M̃ whose shape operator does not vanish at every point. Under the additional assumption that the induced connection is related to the Levi-Civita connection of the affine metric by a specified formula, it is proved that the affine metric is conformally flat. Relations to totally umbilical pseudo-Riemannian hypersurfaces are also discussed.
References
B.Y. Chen, L. Verstraelen, Hypersurfaces of symmetric spaces, Bull. Inst. Math. Acad. Sinica 8 (1980), 201-236.
##plugins.generic.googleScholarLinks.settings.viewInGS##
L.P. Eisenhart, Non-Riemannian Geometry, Amer. Math. Soc. Colloq. Publ., Vol. VIII, New York, 1927 (reprinted 1986).
##plugins.generic.googleScholarLinks.settings.viewInGS##
P. Enghiș, Hypersurfaces dans un espace Riemannien récurrent, Studia Univ. Babeș-Bolyai, Ser. Math.-Mech. 19 (1974), 23-31.
##plugins.generic.googleScholarLinks.settings.viewInGS##
G.B. Folland, Weyl manifolds, J. Diff. Geometry 4 (1970), 145-153.
##plugins.generic.googleScholarLinks.settings.viewInGS##
C.M. Fulton, Umbilical hypersurfaces in affinely connected spaces, Proc. Amer. Math. Soc. 19 (1968), 1487-1490.
##plugins.generic.googleScholarLinks.settings.viewInGS##
S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. I, Interscience Publishers, New York, 1963.
##plugins.generic.googleScholarLinks.settings.viewInGS##
T. Miyazawa, G. Chuman, On certain subspaces of Riemannian recurrent spaces, Tensor N.S. 23 (1972), 253-260.
##plugins.generic.googleScholarLinks.settings.viewInGS##
P.T. Nagy, On the umbilical hypersurfaces, Acta Sci. Math. (Szeged) 36 (1974), 107-109.
##plugins.generic.googleScholarLinks.settings.viewInGS##
K. Nomizu, U. Pinkall, On the geometry of affine immersions, Math. Z. 195 (1987), 165-178.
##plugins.generic.googleScholarLinks.settings.viewInGS##
K. Nomizu, T. Sasaki, Affine Differential Geometry, Cambrige University Press, Cambridge, 1994.
##plugins.generic.googleScholarLinks.settings.viewInGS##
A.P. Norden, Spaces with Affine Connection (in Russian), Second edition, Izdat. Nauka, Moscow, 1976.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Z. Olszak, On totally umbilical surfaces in some Riemannian spaces, Colloq. Math. 37 (1977), 105-111.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Z. Olszak, Remarks on manifolds admitting totally umbilical hypersurfaces, Demonstratio Math. 11 (1978), 695-702.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Z. Olszak, On umbilical nondegenerate affine immersions, Reports of Institute of Mathematics, Wrocław University of Technology, No. 8, 1993 (unpublished).
##plugins.generic.googleScholarLinks.settings.viewInGS##
Z. Olszak, On certain affine immersions into locally symmetric spaces, In: Geometry and Topology of Submanifolds, Vol. VII, World Scientific Publ., Singapore, 1995, 204-207.
##plugins.generic.googleScholarLinks.settings.viewInGS##
W.A. Poor, Differential Geometric Structures, McGraw-Hill Book Company, New York, 1981.
##plugins.generic.googleScholarLinks.settings.viewInGS##
M. Prvanović, Certain theorems on hypersurfaces with indetermined lines of curvature of Riemannian recurrent spaces (in Serbo-Croatian), Math. Vesnik 1(16) (1964), 81-87.
##plugins.generic.googleScholarLinks.settings.viewInGS##
J. Schouten, Ricci Calculus, Springer-Verlag, 1954.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Y.C. Wong, Recurrent tensors on a linearly connected differentiable manifold, Trans. Amer. Math. Soc. 99 (1961), 325-334.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Y.C. Wong, Linear connexions with zero torsion and recurrent curvature, Trans. Amer. Math. Soc. 102 (1962), 471-506.
##plugins.generic.googleScholarLinks.settings.viewInGS##
