Acta Applicandae Mathematicae

, Volume 110, Issue 1, pp 73–107 | Cite as

Curve Flows and Solitonic Hierarchies Generated by Einstein Metrics

  • Sergiu I. VacaruEmail author


We investigate bi-Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non-stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric-induced nonlinear connection (N-connection) structure. On spacetime manifolds, we consider a nonholonomic splitting of dimensions and define a new class of liner connections which are ‘N-adapted’, metric compatible and uniquely defined by the metric structure. We prove that for such a linear connection, one yields couples of generalized sine-Gordon equations when the corresponding geometric curve flows result in solitonic hierarchies described in explicit form by nonholonomic wave map equations and mKdV analogs of the Schrödinger map equation. All geometric constructions can be re-defined for the Levi-Civita connection but with “noholonomic mixing” of solitonic interactions. Finally, we speculate why certain methods and results from the geometry of nonholonmic manifolds and solitonic equations have general importance in various directions of modern mathematics, geometric mechanics, fundamental theories in physics and applications, and briefly analyze possible nonlinear wave configurations for modeling gravitational interactions by effective continuous media effects.


Curve flow (Semi) Riemannian spaces Nonholonomic manifold Nonlinear connection Bi-Hamiltonian Solitonic equations 

Mathematics Subject Classification (2000)

37K05 37K10 37K25 35Q53 53B20 53B40 53C21 53C60 


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.The Fields Institute for Research in Mathematical ScienceTorontoCanada
  2. 2.Faculty of MathematicsUniversity “Al. I. Cuza” IaşiIaşiRomania

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