Abstract
Locally strictly convex surfaces in four-dimensional affine space are studied from a perspective of the affine structure invented by Nuño-Ballesteros and Sánchez, which is especially suitable in convex geometry. The surfaces that are embedded in locally strictly convex hyperquadrics are classified under assumptions that the second fundamental form is parallel with respect to the induced connection and the normal connection is compatible with a metric on the transversal bundle. Both connections are induced by a canonical transversal plane bundle, which is defined by certain symmetry conditions. The obtained surfaces are always products of an ellipse and a conical planar curve.
Highlights
The main purpose of this paper is to characterize a class of surfaces embedded in locally strictly convex hyperquadrics contained in the four-dimensional affine space
We classify surfaces embedded in locally strictly convex hyperquadrics of R4 under the assumptions that the cubic form vanishes and a normal connection is compatible with a metric canonically defined on the transversal bundle
We study locally strictly convex surfaces embedded in hyperquadrics in R4
Summary
The main purpose of this paper is to characterize a class of surfaces embedded in locally strictly convex hyperquadrics contained in the four-dimensional affine space. A transversal bundle is called equiaffine if the metric volume element is parallel with respect to the affine connection induced on the surface by the bundle We classify surfaces embedded in locally strictly convex hyperquadrics of R4 under the assumptions that the cubic form vanishes (such submanifolds are called parallel because the second fundamental form is parallel with respect to the induced connections) and a normal connection is compatible with a metric canonically defined on the transversal bundle. We prove that if a surface satisfies the above properties and has no inflection points, it is locally affine equivalent to an open part of one of three types of surfaces: a product of two ellipses (affine equivalent to the Clifford torus), the product of an ellipse and a hyperbola or the product of an ellipse and a parabola We obtained another affine characterization of the Clifford torus as a parallel locally strictly convex surface contained in the ellipsoid (see Opozda’s work [5]). We note that in [6], a topological torus is characterized as a compact surface with the indefinite Burstin–Mayer metric
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