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Wintgen ideal submanifolds with a low-dimensional integrable distribution

Research paper by Tongzhu Li, Xiang Ma, Changping Wang

Indexed on: 09 May '14Published on: 09 May '14Published in: Frontiers of Mathematics in China



Abstract

Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are investigated in this paper using the framework of Möbius geometry. We classify Wintgen ideal submanfiolds of dimension m ⩽ 3 and arbitrary codimension when a canonically defined 2-dimensional distribution \(\mathbb{D}_2\) is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively. We conjecture that if \(\mathbb{D}_2\) generates a k-dimensional integrable distribution \(\mathbb{D}_k\) and k < m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.