Isothermic Submanifolds

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• 作者：Neil Donaldson (1) ndonalds@math.uci.edu
  Chuu-Lian Terng (1) cterng@math.uci.edu
• 关键词：Isothermic surfaces – Solitons – Loop groups
• 刊名：Journal of Geometric Analysis
• 出版年：2012
• 出版时间：July 2012
• 年：2012
• 卷：22
• 期：3
• 页码：827-844
• 全文大小：364.7 KB
• 参考文献：1. Brück, M., Du, X., Park, J., Terng, C.L.: The submanifold geometries associated to Grassmannian systems. Mem. Am. Math. Soc. 155, 735 (2002)
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  8. Terng, C.L.: Soliton equations and differential geometry. J. Differ. Geom. 45, 407–445 (1997)
  9. Terng, C.L., Uhlenbeck, K.: B?cklund transformations and loop group actions. Commun. Pure Appl. Math. 53, 1–75 (2000)
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  11. Tojeiro, R.: Isothermic submanifolds of Euclidean space. J. Reine Angew. Math. 598, 1–24 (2006)
• 作者单位：1. Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, USA
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X

文摘

We propose an answer to a question raised by F. Burstall: Is there any interesting theory of isothermic submanifolds of ? n of dimension greater than two? We call an n-immersion f(x) in ? m isothermic _k if the normal bundle of f is flat and x is a line of curvature coordinate system such that its induced metric is of the form ?_i=1n g_ii d x_i2\sum_{i=1}^{n} g_{ii}\,\mathrm{d} x_{i}^{2} with ?_i=1n e_i g_ii=0\sum_{i=1}^{n} \epsilon_{i} g_{ii}=0, where ε _i =1 for 1≤i≤n?k and ε _i =?1 for n?k<i≤n. A smooth map (f ₁,…,f _n ) from an open subset O{\mathcal{O}} of ? n to the space of m×n matrices is called an n-tuple of isothermic _k n-submanifolds in ? m if each f _i is an isothermic _k immersion, (f_i)_xj(f_{i})_{x_{j}} is parallel to (f₁)_xj(f_{1})_{x_{j}} for all 1≤i,j≤n, and there exists an orthonormal frame (e ₁,…,e _n ) and a GL(n)-valued map (a _ij ) such that df_i = ?_j=1n a_ij e_j d x_j\mathrm{d}f_{i}= \sum_{j=1}^{n} a_{ij} e_{j}\,\mathrm {d} x_{j} for 1≤i≤n. Isothermic₁ surfaces in ?3 are the classical isothermic surfaces in ?3. Isothermic _k submanifolds in ? m are invariant under conformal transformations. We show that the equation for n-tuples of isothermic _k n-submanifolds in ? m is the \fracO(m+n-k,k)O(m)×O(n-k,k)\frac{O(m+n-k,k)}{O(m)\times O(n-k,k)}-system, which is an integrable system. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries.