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Austere submanifolds of dimension four: examples and maximal types. (English) Zbl 1232.53008

Authors’ abstract: Austere submanifolds in Euclidean space were introduced by Harvey and Lawson in connection with their study of calibrated geometries. The algebraic possibilities for second fundamental forms of 4-dimensional austere submanifolds were classified by Bryant, into three types which we label A, B and C. In this paper, we show that type A submanifolds correspond exactly to real Kähler submanifolds, we construct new examples of such submanifolds in \(\mathbb R^{6}\) and \(\mathbb R^{10}\), and we obtain classification results on submanifolds with second fundamental forms of maximal type.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B35 Local differential geometry of Hermitian and Kählerian structures
53C38 Calibrations and calibrated geometries
58A15 Exterior differential systems (Cartan theory)
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References:

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