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## Curved flats, exterior differential systems, and conservation laws, Complex, contact and symmetric manifolds (2005)

Venue: | 235–254, Progr. Math., 234, Birkhauser |

Citations: | 8 - 3 self |

### Citations

265 |
Exterior differential systems
- Bryant, Chern, et al.
- 1982
(Show Context)
Citation Context ... h −1 vh for some constant h ∈ (U0)A. 3. Basics of exterior differential systems We give a brief account of Cartan-Kähler theory based on the lectures given by R. Bryant at MSRI in 1999 and 2003 (cf. =-=[3]-=- for details and references). Let M be a smooth manifold, and Ω ∗ (M) the graded algebra of differential forms on M. An ideal I of Ω ∗ (M) is called a differential ideal if I satisfies the following c... |

95 |
Coifman, Scattering and inverse scattering for first order systems
- Beals, R
- 1984
(Show Context)
Citation Context ... of the U/U0-system has a unique local solution for any given local real analytic initial data on the x1-axis. But it was also proved in [12] using the inverse scattering theory of Beals and Coifman (=-=[1]-=-) that given any smooth rapidly decaying function v0 : R → U1 ∩ A ⊥ , there exists a unique smooth solution v : R r → U1 ∩ A ⊥ so that v(x1,...,xr) = v0(x1,0,... ,0). Although the theory exterior diff... |

68 | Bäcklund transformations and loop group actions
- Terng, Uhlenbeck
(Show Context)
Citation Context ...rse scattering theory to solve the Cauchy problem globally with smooth rapidly decaying initial data ([12]), used loop group factorization to construct infinitely many families of explicit solutions (=-=[14]-=-), and noted that many these systems occur as the GaussCodazzi equations for submanifolds in space forms ([12, 15]). The main goals of this paper are: (i) review some of these known results, (ii) use ... |

48 | Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems, in Integrable systems, Geometry and Topology
- Burstall
- 2006
(Show Context)
Citation Context ...2) These systems occur naturally in submanifold geometry. For example, the Gauss-Codazzi equations for isometric immersions of space forms in space forms ([9, 12, 2]), for isothermic surfaces in R n (=-=[7, 2]-=-), and for flat Lagrangian submanifolds in C n or in CP n ([15]). The U/U0-system also arises naturally from soliton theory (cf. [12]). In fact, given 1 ≤ i ≤ r, b ∈ A, and a positive integer j, the (... |

28 | Soliton equations and differential geometry
- Terng
- 1997
(Show Context)
Citation Context ...doctoral fellowship of MSRI. 12 CHUU-LIAN TERNG ∗ AND ERXIAO WANG † involutive exterior differential system Iσ such that integral submanifolds of Iσ in U project down to curved flats in U/U0. Terng (=-=[12]-=-) used r first flows in the U/U0-hierarchy of commuting soliton equations to construct the U/U0-system. She showed that the U/U0-system and the curved flat system are gauge equivalent, used the invers... |

28 | K.,Poisson actions and scattering theory for integrable systems, Surveys in Differential Geometry: Integrable Systems, A supplement to
- Terng, Uhlenbeck
- 1999
(Show Context)
Citation Context ...sition 2.1 we see that ∑ i (aiλ + [ai,v])dxi is a flat Gvalued connection on Rr for all λ ∈ C. Hence for each fixed t, v(· · · ,t) is a solution of the U/U0-system. □ The following is well-known (cf. =-=[11, 13]-=-):CURVED FLATS AND CONSERVATION LAWS 15 Theorem 5.5. (1) Qb,j(x,t) is a polynomial in u,∂xv, · · · ,∂ j−1 x v, (2) Qb,j satisfies the following recursive formula (3) Qb,0 = b, Qb,1 = [b,v]. Proof. A ... |

26 | Curved flats in symmetric spaces
- Ferus, Pedit
- 1996
(Show Context)
Citation Context ...s of σ on U. Then [U0, U0] ⊂ U0, [U0, U1] ⊂ U1, [U1, U1] ⊂ U0. The quotient space U/U0 is a symmetric space, and the eigen-decomposition U = U0 + U1 is called a Cartan decomposition. Ferus and Pedit (=-=[8]-=-) called a submanifold M of a rank r symmetric space U/U0 a curved flat if TpM is tangent to an r-dimensional flat of U/U0 at p for each p ∈ M. They noted that the equation for curved flats is an inte... |

26 | Geometries and symmetries of soliton equations and integrable elliptic systems. to appear
- Terng
(Show Context)
Citation Context ...sed loop group factorization to construct infinitely many families of explicit solutions ([14]), and noted that many these systems occur as the GaussCodazzi equations for submanifolds in space forms (=-=[12, 15]-=-). The main goals of this paper are: (i) review some of these known results, (ii) use techniques from soliton theory to construct infinitely many integral submanifolds and conservation laws for the ex... |

17 |
Hamiltonian hierarchies on semi-simple Lie algebras
- Sattinger
- 1984
(Show Context)
Citation Context ...sition 2.1 we see that ∑ i (aiλ + [ai,v])dxi is a flat Gvalued connection on Rr for all λ ∈ C. Hence for each fixed t, v(· · · ,t) is a solution of the U/U0-system. □ The following is well-known (cf. =-=[11, 13]-=-):CURVED FLATS AND CONSERVATION LAWS 15 Theorem 5.5. (1) Qb,j(x,t) is a polynomial in u,∂xv, · · · ,∂ j−1 x v, (2) Qb,j satisfies the following recursive formula (3) Qb,0 = b, Qb,1 = [b,v]. Proof. A ... |

10 |
Submanifold geometry of real Grassmannian systems, The Memoirs, vol 155
- Brück, Du, et al.
(Show Context)
Citation Context ...,vxi ] = [[ai,v],[aj,v]], 1 ≤ i ̸= j ≤ r. (1.2) These systems occur naturally in submanifold geometry. For example, the Gauss-Codazzi equations for isometric immersions of space forms in space forms (=-=[9, 12, 2]-=-), for isothermic surfaces in R n ([7, 2]), and for flat Lagrangian submanifolds in C n or in CP n ([15]). The U/U0-system also arises naturally from soliton theory (cf. [12]). In fact, given 1 ≤ i ≤ ... |

10 | Isometric immersions of space forms and soliton theory
- Ferus, Pedit
- 1996
(Show Context)
Citation Context ...,vxi ] = [[ai,v],[aj,v]], 1 ≤ i ̸= j ≤ r. (1.2) These systems occur naturally in submanifold geometry. For example, the Gauss-Codazzi equations for isometric immersions of space forms in space forms (=-=[9, 12, 2]-=-), for isothermic surfaces in R n ([7, 2]), and for flat Lagrangian submanifolds in C n or in CP n ([15]). The U/U0-system also arises naturally from soliton theory (cf. [12]). In fact, given 1 ≤ i ≤ ... |

4 |
Lectures given at MSRI “Integrable system seminar”, 2003, unpublished notes
- Bryant
(Show Context)
Citation Context ...M of a rank r symmetric space U/U0 a curved flat if TpM is tangent to an r-dimensional flat of U/U0 at p for each p ∈ M. They noted that the equation for curved flats is an integrable system. Bryant (=-=[6]-=-) used the involution σ to construct a natural ∗ Research supported in part by NSF Grant DMS-0306446. † Research supported in part by Postdoctoral fellowship of MSRI. 12 CHUU-LIAN TERNG ∗ AND ERXIAO ... |

1 |
Lectures given at MSRI summer school
- Bryant
- 1999
(Show Context)
Citation Context ...Thus vr n (I) ⊂ vo n (I) ⊂ vn(I) ⊂ Grn(TM). An integral submanifold is called regular if all of its tangent spaces are regular integral elements. We state the following two theorems that are given in =-=[5]-=-: Theorem 3.1. (Cartan-Kähler Theorem) Suppose (M, I) is a real analytic EDS and that N ⊂ M is a connected real analytic regular n-dimensional integral submanifold of I with r(N) ≥ 0. Let R ⊂ M be a r... |