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Affine metrics of locally strictly convex surfaces in affine 4-space

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Abstract

We introduce a new family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric planes coincide and contain the affine normal of the hyperquadric. In particular, any surface immersed in a locally strictly convex hyperquadric is affine semiumbilical with respect to the symmetric or antisymmetric equiaffine planes.

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Correspondence to Juan J. Nuño-Ballesteros.

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The first author has been partially supported by DGICYT Grant MTM2012-33073. The second author has been partially supported by FAPESP Grant BEPE 2011/21126-7.

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Nuño-Ballesteros, J.J., Sánchez, L. Affine metrics of locally strictly convex surfaces in affine 4-space. Geom Dedicata 183, 1–24 (2016). https://doi.org/10.1007/s10711-016-0140-x

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  • DOI: https://doi.org/10.1007/s10711-016-0140-x

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