### 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 language | English (US) |
---|---|

Pages (from-to) | 108-146 |

Number of pages | 39 |

Journal | Differential Geometry and its Application |

Volume | 65 |

DOIs | |

State | Published - Aug 1 2019 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology
- Computational Theory and Mathematics

### Cite this

*Differential Geometry and its Application*,

*65*, 108-146. https://doi.org/10.1016/j.difgeo.2019.03.008

}

*Differential Geometry and its Application*, vol. 65, pp. 108-146. https://doi.org/10.1016/j.difgeo.2019.03.008

**Jacobians with prescribed eigenvectors.** / Benfield, Michael; Jenssen, Helge Kristian; Kogan, Irina A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Jacobians with prescribed eigenvectors

AU - Benfield, Michael

AU - Jenssen, Helge Kristian

AU - Kogan, Irina A.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85064083886&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85064083886&partnerID=8YFLogxK

U2 - 10.1016/j.difgeo.2019.03.008

DO - 10.1016/j.difgeo.2019.03.008

M3 - Article

AN - SCOPUS:85064083886

VL - 65

SP - 108

EP - 146

JO - Differential Geometry and its Applications

JF - Differential Geometry and its Applications

SN - 0926-2245

ER -