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Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures

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Abstract

Wintgen proved (C. R. Acad. Sci. Paris, 288:993–995, 1979) that the Gauss curvature K and the normal curvature K D of a surface in Euclidean 4-space \({\mathbb {E}^4}\) satisfy K + |K D| ≤ H 2, where H 2 is the squared mean curvature. A surface in \({\mathbb {E}^4}\) is called Wintgen ideal if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in \({\mathbb {E}^4}\) form an important family of surfaces, namely, surfaces with circular ellipse of curvature. In this article, we completely classify Wintgen ideal surfaces in \({\mathbb E^4}\) satisfying |K| = |K D| identically.

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  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824–1027, USA

    Bang-Yen Chen

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  1. Bang-Yen Chen
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Chen, BY. Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures. Ann Glob Anal Geom 38, 145–160 (2010). https://doi.org/10.1007/s10455-010-9205-5

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  • DOI: https://doi.org/10.1007/s10455-010-9205-5

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