(pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. In practice...

Space forms Space group Ricci-flat manifold Conformally flat manifold Affine manifold Peter Scott, The geometries of 3-manifolds. (errata), Bull. London Math...

Euclidean space is (globally) conformally flat in this sense. A locally conformally flat manifold is locally conformal to a Möbius geometry, meaning that...

condition for the Riemannian manifold being conformally flat. Any 2-dimensional (smooth) Riemannian manifold is conformally flat, a consequence of the existence...

development of conformally flat n-manifolds, in which the model-space is the n-sphere. The development of a conformally flat manifold is a conformal local diffeomorphism...

geometry, Ricci-flatness is a condition on the curvature of a Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical...

holds true for Dirac operators on conformally flat manifolds and conformal manifolds which are simultaneously spin manifolds. The Cayley transform or stereographic...

tensor of g {\displaystyle g} . Einstein manifolds with k = 0 {\displaystyle k=0} are called Ricci-flat manifolds. In local coordinates the condition that...

Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as Ricci flatness, yielding...

it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were first given this...

In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric g {\displaystyle g} (also called a...

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a...

class of conformally invariant differential operators known as the GJMS operators. A related construction is the tractor bundle. The model flat geometry...

Yau proved the positive mass theorem in the class of locally conformally flat manifolds.[SY88] In 2017, Schoen and Yau published a preprint claiming to...

closed orientable Riemannian 2-manifolds into elliptic/parabolic/hyperbolic cases. Each such manifold has a conformally equivalent Riemannian metric with...

group, with equality holding if and only if the open set is locally conformally flat. Monge 1850, pp. 609–616, a note contributed by Liouville as editor...

An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that...

pseudo-Riemannian manifold of dimension n: If n ≥ 4 then the manifold is conformally flat if and only if its Weyl tensor is zero. If n = 3 then the manifold is conformally...

for n = 3 is necessary and sufficient condition for the manifold to be locally conformally flat. By contrast, in dimensions n ≥ 4, the vanishing of the...

2 {\displaystyle G_{2}} -manifold are 7-dimensional, Ricci-flat, orientable spin manifolds. In addition, any compact manifold with holonomy equal to G...

-dimensional Riemannian manifold is contained in S U ( n ) {\displaystyle SU(n)} , then the manifold is a Ricci-flat Kähler manifold (Kobayashi & Nomizu 1996...

hyperbolic space Conformal map is a map which preserves angles. Conformally flat a manifold M is conformally flat if it is locally conformally equivalent to...

"The Higher-Dimensional Chern–Gauss–Bonnet Formula for Singular Conformally Flat Manifolds". The Journal of Geometric Analysis. 29 (2): 1043–1074. doi:10...

geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of single variable calculus, vector calculus,...

S2CID 118833790. Zbl 0551.58011. Schoen, R.; Yau, S.-T. (1988). "Conformally flat manifolds, Kleinian groups and scalar curvature". Inventiones Mathematicae...

surfaces, one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that...

structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold. Lorentzian manifolds can be classified according...

covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space....

1090/s0002-9904-1969-12155-5. MR 0233306. Kulkarni RS (1972). "Conformally Flat Manifolds". Proceedings of the National Academy of Sciences. 69 (9): 2675–2676...

and adds the boundary of the manifold, thus making it compact. For a given metric g i j {\displaystyle g_{ij}} , a conformal compactification scales the...