Illinois Journal of Mathematics

Austere submanifolds of dimension four: Examples and maximal types

Marianty Ionel and Thomas Ivey

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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.

Article information

Source
Illinois J. Math., Volume 54, Number 2 (2010), 713-746.

Dates
First available in Project Euclid: 14 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1318598678

Digital Object Identifier
doi:10.1215/ijm/1318598678

Mathematical Reviews number (MathSciNet)
MR2846479

Zentralblatt MATH identifier
1232.53008

Subjects
Primary: 53B25: Local submanifolds [See also 53C40]
Secondary: 53B35: Hermitian and Kählerian structures [See also 32Cxx] 53C38: Calibrations and calibrated geometries 58A15: Exterior differential systems (Cartan theory)

Citation

Ionel, Marianty; Ivey, Thomas. Austere submanifolds of dimension four: Examples and maximal types. Illinois J. Math. 54 (2010), no. 2, 713--746. doi:10.1215/ijm/1318598678. https://projecteuclid.org/euclid.ijm/1318598678


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References

  • E. Abbena, S. Garbiero and S. Salamon, Almost Hermitian geometry in six-dimensional nilmanifolds, Sci. École Norm. Sup. (4) 30 (2001), 147–170.
  • R. L. Bryant, Some remarks on the geometry of austere manifolds, Bull. Braz. Math. Soc. (N.S.) 21 (1991), 133–157.
  • R. L. Bryant, S.-S. Chern, R. B. Gardner, H. L. Goldschmidt and P. A. Griffiths, Exterior differential systems, MSRI Publications, vol. 18, Springer-Verlag, New York, 1989.
  • M. Dajczer and L. Florit, A class of austere submanifolds, Illinois J. Math. 45 (2001), 735–755.
  • M. Dajczer and D. Gromoll, Gauss parametrizations and rigidity aspects of submanifolds, J. Differential Geom. 22 (1985), 1–12.
  • R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47–157.
  • T. A. Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, vol. 61, American Mathematical Society, Providence, 2003.
  • S. Karigiannis and M. Min-Oo, Calibrated subbundles in noncompact manifolds of special holonomy, Ann. Global Anal. Geom. 28 (2005), 371–394.
  • S. Kobayashi and K. Nomizu, Foundations of differential geometry, Wiley-Interscience, New York, 1969.
  • M. Stenzel, Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manuscripta Math. 80 (1993), 151–163.
  • A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), 243–259.