Abstract

The classical Mobius group on the extended plane plays an important role in geometric function theory of Rieman surfaces. If we can generalize the Mobius group to any complete Riemannian manifold, we expect to get some information about the geometry of manifolds. B. Osgood and D. Stowe did this generalization. So we can study the geometry of a manifold by studying its Mobius group.

We have proved that if $M$ is a compact, oriented, Riemannian $n$-dimensional manifold with $n$ $>$ 2 and without boundary, and if the Ricci curvature of $M$ is non-positive, then Mob($M$) = Hty($M$), where Hty($M$) is the homothety group of $M$. The method we used to prove this is to construct a new vector field on $M$ which is related to the solution of the linear equation corresponding to the equation $B(\varphi)$ = 0, where $B$ is an operator used to define a generalized Schwarzian derivative. Using this new vector field and integration by parts, we reduced the solutions of that linear equation to that associated to the Laplace-Beltrami equation. Then by the maximum principle, any solution must be constant.

We have also proved the following result: A closed, oriented, Riemannian manifold $M$ with constant scalar curvature satisfies either Mob($M$) = Hty($M$) or $M$ is isometric to some sphere. As a consequence of this and the Yamabe Theorem, we show, for any closed oriented Riemannian manifold ($M,g$), there exists a metric $g\sp\prime$ such that either Mob($M,g\sp\prime$) = Hty($M,g\sp\prime$) or ($M,g\sp\prime$) is isometric to some sphere. In the course of the proof of this, we will use the conformal geometric method to show that if $f$ is a Mobius transformation on $M$ which has constant scalar curvature, the conformal factor $u$ is a constant function, then this result and the well-known Yamabe theorem give the proof of our result.

Finally, we also construct an example which shows that a non-homogeneous equation does not always have a positive solution on complete manifolds.