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Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

Postal Subscription Code 80-964

2018 Impact Factor: 0.565

Front. Math. China    2015, Vol. 10 Issue (1) : 111-136    https://doi.org/10.1007/s11464-014-0383-5
RESEARCH ARTICLE
Wintgen ideal submanifolds with a low-dimensional integrable distribution
Tongzhu LI1,*(),Xiang MA2,Changping WANG3
1. Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
2. LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
3. College of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350108, China
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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 M?bius geometry. We classify Wintgen ideal submanfiolds of dimension m3 and arbitrary codimension when a canonically defined 2-dimensional distribution ?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 ?2 generates a k-dimensional integrable distribution ?k<?Pub Caret?>and k<m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.

Keywords Wintgen ideal submanifold      DDVV inequality      super-conformal surface      super-minimal surface     
Corresponding Author(s): Tongzhu LI   
Issue Date: 30 December 2014
 Cite this article:   
Tongzhu LI,Xiang MA,Changping WANG. Wintgen ideal submanifolds with a low-dimensional integrable distribution[J]. Front. Math. China, 2015, 10(1): 111-136.
 URL:  
https://academic.hep.com.cn/fmc/EN/10.1007/s11464-014-0383-5
https://academic.hep.com.cn/fmc/EN/Y2015/V10/I1/111
1 Bryant R. Some remarks on the geometry of austere manifolds. Bol Soc Bras Mat, 1991, 21: 122-157
https://doi.org/10.1007/BF01237361
2 Chen B Y. Some pinching and classification theorems for minimal submanifolds. Arch Math, 1993, 60: 568-578
https://doi.org/10.1007/BF01236084
3 Chen B Y. Mean curvature and shape operator of isometric immersions in real-space forms. Glasg Math J, 1996, 38: 87-97
https://doi.org/10.1017/S001708950003130X
4 Chen B Y. Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures. Ann Global Anal Geom, 2010, 38: 145-160
https://doi.org/10.1007/s10455-010-9205-5
5 Choi T, Lu Z. On the DDVV conjecture and the comass in calibrated geometry (I). Math Z, 2008, 260: 409-429
https://doi.org/10.1007/s00209-007-0281-6
6 Dajczer M, Florit L A, Tojeiro R. On a class of submanifolds carrying an extrinsic totally umbilical foliation. Israel J Math, 2001, 125: 203-220
https://doi.org/10.1007/BF02773380
7 Dajczer M, Tojeiro R. A class of austere submanifolds. Illinois J Math, 2001, 45: 735-755
8 Dajczer M, Tojeiro R. Submanifolds of codimension two attaining equality in an extrinsic inequality. Math Proc Cambridge Philos Soc, 2009, 146: 461-474
https://doi.org/10.1017/S0305004108001813
9 De Smet P J, Dillen F, Verstraelen L, Vrancken L. A pointwise inequality in submanifold theory. Arch Math, 1999, 35: 115-128
10 Dillen F, Fastenakels J, Van Der Veken J. Remarks on an inequality involving the normal scalar curvature. In: Proceedings of the International Congress on Pure and Applied Differential Geometry-PADGE, Brussels. Aachen: Shaker Verlag, 2007, 83-92
11 Ge J, Tang Z. A proof of the DDVV conjecture and its equality case. Pacific J Math, 2008, 237: 87-95
https://doi.org/10.2140/pjm.2008.237.87
12 Guadalupe I, Rodríguez L. Normal curvature of surfaces in space forms. Pacific J Math, 1983, 106: 95-103
https://doi.org/10.2140/pjm.1983.106.95
13 Li T, Ma X, Wang C P. Deformation of hypersurfaces preserving the Moebius metric and a reduction theorem. Adv Math, 2014, 256: 156-205
https://doi.org/10.1016/j.aim.2014.02.002
14 Li T, Ma X, Wang C P, Xie Z. Wintgen ideal submanifolds with a low-dimensional integrable distribution (II) (in preparation)
15 Liu H L, Wang C P, Zhao G S. M?bius isotropic submanifolds in Sn.Tohoku Math J, 2001, 53: 553-569
https://doi.org/10.2748/tmj/1113247800
16 Lu Z. On the DDVV conjecture and the comass in calibrated geometry (II). arXiv: Math.DG/0708.2921
17 Lu Z. Normal scalar curvature conjecture and its applications. J Funct Anal, 2011, 261: 1284-1308
https://doi.org/10.1016/j.jfa.2011.05.002
18 Petrovié-torga?ev M, Verstraelen L. On Deszcz symmetries of Wintgen ideal submanifolds. Arch Math, 2008, 44: 57-67
19 Wang C P. M?bius geometry of submanifolds in Sn. Manuscripta Math, 1998, 96: 517-534
https://doi.org/10.1007/s002290050080
20 Wintgen P. Sur l’inégalité de Chen-Willmore. C R Acad Sci Paris, 1979, 288: 993-995
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