Conformal Geometry from the Riemannian Viewpoint

Complete locally conformally flat manifolds of negative curvature

“…Recall that a warped product I × f F with one-dimensional base is locally conformally flat if and only if the fiber is a space of constant sectional curvature. Local conformal flatness is independent of the warping function f [Lafontaine 1988], in opposition to the case of higher-dimensional base just considered. In what remains of this section we look at the local structure of a locally conformally flat multiply warped space with one-dimensional base.…”

“…Here represents the Kulkarni-Nomizu product (see [Lafontaine 1988], for example). A nonflat locally decomposable Riemannian manifold is locally conformally flat if and only if it is locally equivalent to the product N (c)×‫ޒ‬ of an interval and a space of constant sectional curvature, or to the product…”

“…A compact simply connected locally conformally flat manifold must be a Euclidean sphere [Kuiper 1949;Schoen and Yau 1988]. Locally symmetric manifolds which are locally conformally flat are either of constant sectional curvature or locally isometric to a product of two spaces of constant opposite sectional curvature [Lafontaine 1988;Yau 1973]. Complete locally conformally flat manifolds with nonnegative Ricci curvature have been studied by several authors; Zhu [1994] showed that their universal cover is in the conformal class of ‫ޓ‬ n , ‫ޒ‬ n or ‫ޒ‬ × ‫ޓ‬ n−1 , where ‫ޓ‬ n and ‫ޓ‬ n−1 are spheres of constant sectional curvature.…”

“…Locally conformally flat multiply warped spaces are investigated in Section 3. Our approach relies on the fact that any multiply warped space is in the conformal class of a suitable product, a fact previously observed for warped product metrics [Lafontaine 1988], which has several implications on the geometry of the fibers and the base of the multiply warped space. A local description of locally conformally flat spaces with the underlying structure of a multiply warped product is then obtained from the fact that any warping function must define a global conformal transformation on the base which makes it of constant sectional curvature.…”

Convex-cocompactness of Kleinian groups and conformally flat manifolds with positive scalar curvature

“…When n = 3, this means g N is flat. When n ≥ 4, because of the conformal flatness, g N is again flat (see [12]). Therefore Ω(Γ)/Γ must be covered by a flat torus.…”