The Schwarzian derivative and conformal mapping of Riemannian manifolds
If \((M,g)\) is a Riemannian manifold, \(\dim M\geq 2\), then, for a smooth function \(\varphi: M\to \mathbb{R}\), one defines the Schwarzian tensor \(B_ g(\varphi)=\text{Hess}(\varphi)-d\varphi \otimes d\varphi- (1/n)[\Delta\varphi- \|\text{grad } \varphi\|^ 2]g\). If \(f: (M,g)\to (M',g')\) is a local conformal diffeomorphism, the Schwarzian of \(f\) is \(S(f)=S_ g(f)=B_ g(\varphi)\), where \(\widehat g=e^{2\varphi} g=f^* g'\), \(\varphi=\log\| df\|\), and \(S(f\circ h)=S(h)+h^*S(f)\). A conformal diffeomorphism \(f\) is a Möbius transformation if \(S_ g(f)=0\). A Möbius change of metric is \(\widehat{g}=e^{2\varphi}g\), such that \(B_ g(\varphi)=0\). The Möbius metric \(\widehat{g}\) of \(\mathbb{R}^ n\) (and \(S^ n\)) is determined. If \(u=e^{-\varphi}\), then \(B_ g(\varphi)=0\) becomes a linear equation \(\text{Hess}(u)=(1/n)(\Delta u)g\); \(U(M)\) is the space of solutions of this equation. There exists a natural representation of \(\text{Möb}(M)\) into \(\Aut(U(M))\). If \((M,g)\) is an Einstein manifold, then a solution of \(B_ g(\varphi)=0\) gives a conformal metric \(\widehat g=e^{2\varphi}g\), which is also Einstein. Theorem 5.5 gives a list of complete manifolds admitting a conformal, nonhomothetic Einstein metric.
scientific article; zbMATH DE number 78553
scientific article