Jacobians with prescribed eigenvectors

Michael Benfield, Helge Kristian Jenssen, Irina A. Kogan

Research output: Contribution to journalArticle

Abstract

Let Ω⊂R n be open and let R be a partial frame on Ω; that is, a set of m linearly independent vector fields prescribed on Ω (m≤n). We consider the issue of describing the set of all maps F:Ω→R n with the property that each of the given vector fields is an eigenvector of the Jacobian matrix of F. By introducing a coordinate independent definition of the Jacobian, we obtain an intrinsic formulation of the problem, which leads to an overdetermined PDE system, whose compatibility conditions can be expressed in an intrinsic, coordinate independent manner. To analyze this system we formulate and prove a generalization of the classical Frobenius integrability theorems. The size and structure of the solution set of this system depends on the properties of the partial frame; in particular, whether or not it is in involution. A particularly nice subclass of involutive partial frames, called rich frames, can be completely analyzed. The involutive, non-rich case is somewhat harder to handle. We provide a complete answer in the case of m=3 and arbitrary n, as well as some general results for arbitrary m. The non-involutive case is far more challenging, and we only obtain a comprehensive analysis in the case n=3, m=2. Finally, we provide explicit examples illustrating the various possibilities.

Original languageEnglish (US)
Pages (from-to)108-146
Number of pages39
JournalDifferential Geometry and its Application
Volume65
DOIs
StatePublished - Aug 2019

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology
  • Computational Theory and Mathematics

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