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Abstract
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We study locally strongly convex Tchebychev hypersurfaces,
namely the centroaffine totally umbilical hypersurfaces, in the
-dimensional
affine space
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We first make an ordinary-looking observation that such hypersurfaces are
characterized by having a Riemannian structure admitting a canonically defined
closed conformal vector field. Then, by taking advantage of properties about
Riemannian manifolds with closed conformal vector fields, we show that the ellipsoids
are the only centroaffine Tchebychev hyperovaloids. This solves the longstanding
problem of trying to generalize the classical theorem of Blaschke and Deicke on affine
hyperspheres in equiaffine differential geometry to that in centroaffine differential
geometry.
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Keywords
centroaffine hypersurface, Tchebychev hypersurface, shape
operator, difference tensor, hyperovaloid, ellipsoid.
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Mathematical Subject Classification 2010
Primary: 53A15
Secondary: 53C23, 53C24
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Milestones
Received: 22 October 2019
Revised: 4 July 2020
Accepted: 14 September 2021
Published: 13 December 2021
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