A Riemannian manifold $(M,g)$ is locally conformally flat if it is locally conformal to $\mathbb{R}^n$ with the flat metric. I learn that Weyl tensor of a locally conformally flat manifold must vanish. I would like to ask: Is there any example of manifold $M$ such that it cannot be equipped with a metric $g$ with $(M,g)$ being locally flat? Is there any topological restriction on locally confomrally flat manifolds? Is there any classification theorem for locally conformally flat manifolds?
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
6
5
|
||||||
|
|
12
|
The simplest example is $S^n$, it is locally conformally flat with the standard metric, and is not flat for obvious reasons. While flat manifolds are precisely quotients of $\mathbb R^n$ by discreet group of isometries, one should not expect to have a classification of conformally flat manifolds in higher dimensions. For example, already in dimension 4 it was proven by Kapovich in M. Kapovich. Conformally flat metrics on 4-manifolds. J. Differential Geom. 66 (2004), no. 2, 289–301, that arbitrary finitely presented group can be a subgroup of a fundamental group of a conformally flat manifold. The article of Kapovich is and from its introduction you will learn a lot on the question. 4-dimensional manifolds with LCF structure have zero signature, in dimension 3 it is known that some manfiolds don't admit conformally flat structure, first example was constructed in W. Goldman, Conformally flat manifolds with nilpotent holonomy, Transactions of AMS 278 (1983). One more remark -- all hyperbolic manifolds (of constant negative sectional curvature) are all conformally flat. A connected sum of two conformally flat manifolds is conformally flat and so this already gives you a large collection of examples. |
|||
|
|
You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
|
9
|
A simple obstruction is this: No compact (without boundary), simply-connected $n$-manifold that is not diffeomorphic to the $n$-sphere carries a locally conformally flat structure. The reason is that the developing map construction shows that any locally conformally flat metric $g$ on a simply-connected $M^n$ is, up to a conformal factor, the pullback of the standard metric on $S^n$ under some immersion $\phi:M^n\to S^n$. If $M$ is also compact without boundary, then $\phi$ is a covering map and, hence, a diffeomorphism. |
||
|
|
