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Title: δ-invariants and their applications to centroaffine geometry
Authors: Chen, BY *
Dillen, Franki * ×
Verstraelen, Leopold * #
Issue Date: 2005
Publisher: Elsevier science bv
Series Title: Differential geometry and its applications vol:22 issue:3 pages:341-354
Abstract: We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in R^{n+1}. The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in R^{n+1}, n \ge 3, satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article. © 2005 Elsevier B.V. All rights reserved.
ISSN: 0926-2245
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Geometry Section
* (joint) first author
× corresponding author
# (joint) last author

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