Abstract
IfM 2 is a nondegenerate surface in a 4-dimensional Riemannian manifold\(\tilde M\), then there is a natural affine metricg defined onM 2. It is shown that this affine metricg is conformal to the induced Riemannian metric onM 2 if and only ifM 2 is a minimal submanifold of\(\tilde M\) in the usual Riemannian sense. If the conformal factor is a constant, then the two metrics are said to be homothetic. It is shown that there does not exist a nondegenerate surface in Euclidean space ℝ4 or hyperbolic spaceH 4 whose affine metric is homothetic to the induced Riemannian metric. Furthermore, ifM 2 is a nondegenerate surface in the standard 4-sphereS 4 whose affine metric is homothetic to the induced Riemannian metric, thenM 2 is a Veronese surface.
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T. Cecil was supported by NSF Grant No. DMS-9101961.
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Cecil, T., Magid, M. & Vrancken, L. An affine characterization of the Veronese surface. Geom Dedicata 57, 55–71 (1995). https://doi.org/10.1007/BF01264060
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DOI: https://doi.org/10.1007/BF01264060