Diffeomorphisms conformal on distributions

Kamil Niedzia/lomski¹

¹ Department of Mathematics and Computer Science University of /Lódź Banacha 22 90-238 /Lódź, Poland

Annales Polonici Mathematici, Tome 95 (2009) no. 2, pp. 115-124

Résumé

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 $2\dim D\dim M+1$.

Détail

DOI : 10.4064/ap95-2-2

Keywords: local diffeomorphism between riemannian manifolds define eigenvalues eigenvalues self adjoint positive definite operator *df where * denotes operator adjoint conformal distribution dim lambda geq dim d dim where lambda denotes eigenspace corresponding coefficient conformality lambda moreover has distinct eigenvalues there locally distribution conformal only dim dim