zbMATH — the first resource for mathematics

Some remarks on the geometry of austere manifolds. (English) Zbl 0760.53034
The condition of austerity on immersed submanifolds in Euclidean space is a pointwise condition on the second fundamental form which essentially requires that the non-zero eigenvalues of the second fundamental form in all normal directions occur in oppositely signed pairs. For example, complex submanifolds of \(\mathbb{C}^ n\) are austere. The author solves the pointwise problem of describing the set of all austere second fundamental forms in dimension at most four and the local problem of describing the austere three-folds in Euclidean space in all dimensions.
Reviewer: J.Hebda (St.Louis)

53C40 Global submanifolds
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53B25 Local submanifolds
Full Text: DOI
[1] M. Dajczer and D. Gromoll,Gauss parameterizations and Rigidity Aspects of Submanifolds, Journal of Differential Geometry22 (1985), 1-12. · Zbl 0588.53007
[2] R. Harvey and H. B. Lawson,Calibrated Geometries, Acta Mathematica148 (1982), 47-157. · Zbl 0584.53021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.