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Curved flats in symmetric spaces. (English) Zbl 0870.53043

The authors introduce a new class of so-called curvature flat submanifolds, i.e., submanifolds of symmetric spaces defined by the property that the curvature tensor of the ambient space vanishes on them. These submanifolds are analogous to the developable surfaces in \(\mathbb{R}^3\), but they generalize other concepts such as isometric immersions, conformally flat subspaces, and isothermic surfaces. Starting with the Lie algebra definition, general properties of the concept are investigated, and the integration of the fundamental curved flat equations involving an additional “spectral” parameter is replaced by a hierarchy of finite-dimensional commuting ordinary differential equations in the Lax form.
Reviewer: J.Chrastina (Brno)

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C35 Differential geometry of symmetric spaces
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References:

[1] M. Berger,Les espaces symétriques non compacts, Ann. Ec. Norm. Sup.74 (1957), 85–177. · Zbl 0093.35602
[2] F. Burstall, U. Hertrich-Jeromin, F. Pedit. U. Pinkall,Curved flats and isothermic surfaces, SFB 288 Preprint No. 132 (1994)
[3] F.E. Burstall, D. Ferus, F. Pedit, U. Pinkall,Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop, algebras, Ann. of Math.138 (1993), 173–212. · Zbl 0796.53063 · doi:10.2307/2946637
[4] D. Ferus, F. Pedit,Isometric immersions of space forms and soliton theory, SFB 288 Preprint No. 154 (1995).
[5] D. Ferus, F. Pedit, U. Pinkall, I. Sterling,Minimal tori in S 4 J. Reine Angew. Math.429 (1992), 1–47. · Zbl 0746.53045
[6] J.D. Moore,Isometric immersions of space forms into space forms, Pacific J. Math.40 (1972), 157–166. · Zbl 0238.53033
[7] F. Pedit,A non-immersion theorem for space forms, Comment. Math. Helvetici63 (1988), 672–674. · Zbl 0655.53049 · doi:10.1007/BF02566784
[8] U. Hertrich-Jeromin,Conformally flat hypersurfaces and integrable systems, Thesis, TU-Berlin (1994). · Zbl 0820.53003
[9] U. Hertrich-Jeromin,On conformally flat hypersurfaces, curved flats and cyclic systems, in preparation. · Zbl 0870.53007
[10] H. Whitney,Elementary structure of real algebraic varieties, Ann. of Math.66 (1957), 545–556. · Zbl 0078.13403 · doi:10.2307/1969908
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