Diffeomorphisms conformal on distributions

Volume 95 / 2009

Kamil Niedzia/lomski Annales Polonici Mathematici 95 (2009), 115-124 MSC: Primary 53A30; Secondary 53B20. DOI: 10.4064/ap95-2-2


Let $f:M\to N$ be a local diffeomorphism between Riemannian manifolds. We define the eigenvalues of $f$ to be the eigenvalues of the self-adjoint, positive definite operator $df^*df:TM\to TM$, where $df^*$ denotes the operator adjoint to $df$. We show that if $f$ is conformal on a distribution $D$, then $\dim V_{\lambda}\geq 2\dim D-\dim M$, where $V_{\lambda}$ denotes the eigenspace corresponding to the coefficient of conformality $\lambda$ of $f$. Moreover, if $f$ has distinct eigenvalues, then there is locally a distribution $D$ such that $f$ is conformal on $D$ if and only if $2\dim D<\dim M+1$.


  • Kamil Niedzia/lomskiDepartment of Mathematics and Computer Science
    University of /Lódź
    Banacha 22
    90-238 /Lódź, Poland

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