Weyl–Schouten on a model space

For an arbitrary smooth Riemannian metric on a finite-dimensional real normed space, vanishing Weyl and Cotton tensors near a point give a positive smooth conformal factor making the metric flat near that point. The exponent φ can have any prescribed value and gradient at the point. This uses the local solutions of the conformal flattening equation from TauCeti.Manifold.exists_eventually_hessian_eq_model_space and the curvature decomposition into Weyl and Schouten tensors.

Consequently a smooth metric on a model space of dimension at least four is locally conformally flat if and only if its Weyl tensor vanishes. In dimension three the corresponding criterion is vanishing Cotton tensor: Weyl vanishes identically in that dimension. In higher dimensions the Cotton condition follows from the divergence identity div W = (dim M - 3) C.

These statements concern arbitrary variable metrics on the model space, rather than only the Euclidean metric. Local conformal flatness here means existence of a flat conformal rescaling; no Euclidean coordinate isometry is assumed.

References

• A. L. Besse, Einstein Manifolds, Springer (1987), Section 1.G.
• J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), Section 2.

theorem Bundle.ContMDiffRiemannianMetric.exists_eventually_curvatureTensor_rescale_exp_eq_zero_model_space {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] (g : ContMDiffRiemannianMetric (modelWithCornersSelf ℝ E) (↑⊤) E fun (x : E) => TangentSpace (modelWithCornersSelf ℝ E) x) (hn : 3 ≤ Module.finrank ℝ E) {x : E} (hW : ∀ᶠ (y : E) in nhds x, (CovariantDerivative.leviCivitaConnection (modelWithCornersSelf ℝ E) E).weylTensor y = 0) (hC : ∀ᶠ (y : E) in nhds x, (CovariantDerivative.leviCivitaConnection (modelWithCornersSelf ℝ E) E).cottonTensor y = 0) (c : ℝ) (ξ : TangentSpace (modelWithCornersSelf ℝ E) x) : ∃ (φ : E → ℝ) (hφ : ContMDiff (modelWithCornersSelf ℝ E) (modelWithCornersSelf ℝ ℝ) (↑⊤) φ), φ x = c ∧ TauCeti.Manifold.mgradient (modelWithCornersSelf ℝ E) φ x = ξ ∧ ∀ᶠ (y : E) in nhds x, (CovariantDerivative.leviCivitaConnection (modelWithCornersSelf ℝ E) E).curvatureTensor y = 0

Where Weyl and Cotton vanish, a smooth metric on a model space has a flat conformal rescaling e^(2φ) g near the point, with arbitrary prescribed value and gradient of φ.

theorem Bundle.ContMDiffRiemannianMetric.isLocallyConformallyFlat_iff_weylTensor_eq_zero_and_cottonTensor_eq_zero_model_space {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] (g : ContMDiffRiemannianMetric (modelWithCornersSelf ℝ E) (↑⊤) E fun (x : E) => TangentSpace (modelWithCornersSelf ℝ E) x) (hn : 3 ≤ Module.finrank ℝ E) : g.IsLocallyConformallyFlat ↔ (∀ (x : E), (CovariantDerivative.leviCivitaConnection (modelWithCornersSelf ℝ E) E).weylTensor x = 0) ∧ ∀ (x : E), (CovariantDerivative.leviCivitaConnection (modelWithCornersSelf ℝ E) E).cottonTensor x = 0

A smooth metric on a model space of dimension at least three is locally conformally flat exactly when its Weyl and Cotton tensors both vanish.

theorem Bundle.ContMDiffRiemannianMetric.isLocallyConformallyFlat_iff_weylTensor_eq_zero_model_space {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] (g : ContMDiffRiemannianMetric (modelWithCornersSelf ℝ E) (↑⊤) E fun (x : E) => TangentSpace (modelWithCornersSelf ℝ E) x) (hn : 4 ≤ Module.finrank ℝ E) : g.IsLocallyConformallyFlat ↔ ∀ (x : E), (CovariantDerivative.leviCivitaConnection (modelWithCornersSelf ℝ E) E).weylTensor x = 0

Weyl–Schouten in dimension at least four, on a model space. An arbitrary smooth Riemannian metric is locally conformally flat if and only if its Weyl tensor vanishes.

theorem Bundle.ContMDiffRiemannianMetric.isLocallyConformallyFlat_iff_cottonTensor_eq_zero_model_space {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] (g : ContMDiffRiemannianMetric (modelWithCornersSelf ℝ E) (↑⊤) E fun (x : E) => TangentSpace (modelWithCornersSelf ℝ E) x) (hn : Module.finrank ℝ E = 3) : g.IsLocallyConformallyFlat ↔ ∀ (x : E), (CovariantDerivative.leviCivitaConnection (modelWithCornersSelf ℝ E) E).cottonTensor x = 0

Weyl–Schouten in dimension three, on a model space. An arbitrary smooth Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes.