We see that the problem of determining when the metric G is locally conformally flat is equivalent to proving the local existence of solutions of the Beltrami system. For n = 2, it can be shown that the Beltrami system is equivalent to the Beltrami equation.
Sketch of the Procedure
If there are C1 coordinates such that the expression ofg is more frequent, we consider this to be a property of the retractions ofg, and not a property of itself. Therefore, artificial hypothesis that asks for certain regularity of the coordinates under which the metric is expressed (which is necessary when working with abstract manifolds) will appear.
Some Remarks about Notation
Moreover, it can be seen from the proof how the components Rlijk of the curvature tensor arise naturally as integrability conditions for the existence of coordinates in which the metric is flat. This calculation shows how the components of the curvature tensor arise naturally (well, as naturally as the calculations above) as something that must vanish for the metric to be flat.
Conformal Change of the Curvature Tensor
If the opposite-sign convention for curvature is selected, the Ricci tensor is defined by contracting on the second and fourth entries instead of the first and fourth. Let's see what the expression of gradient, divergence and Laplacian is in local coordinates.
Conformal Flatness for Regular Metrics. Weyl-Schouten Theorem
So we have seen that for the solution of system (17) we can focus on the solution of a simpler system that gives We saw in Theorem 6 that ifdim(M) = 3, if it is conformally flat, is equivalent to the condition that the Cotton tensor c vanishes, since these are integrability conditions of the system related to conformal flatness.
Consequences of the Weyl-Schouten Theorem
Note that this is not true in dimension ≥ 4, since in this case the Weyl tensor does not necessarily vanish. By the lemma above, r =k(g?g) =w+s?g and from the uniqueness of the decomposition we have that w= 0 and s=kg. We could make sense of the Weyl tensorwforC1 metric, for example, by considering distributional derivatives of the metric, and then we might wonder whether w= 0 implies that (M, g) is locally conformally flat.
Our strategy here will be to find suitable coordinates around any point such that the equation w(g) = 0 of the Weyl tensor, which is zero as a distribution, viewed as a PDE that g satisfies, is elliptic when expressed in these coordinates. Note that when s=k∈N, there can be confusion between functions with kcontinuous derivatives. not necessarily bounded), and works in the Holder space Ck because both sets are denoted by the symbol Ck. Therefore, we will try to write Ck+0 for the holder space, although we will probably forget and hope the context will clarify the ambiguity. Note that we allow the bound to become arbitrarily large as the order of the derivative increases, so in this space we do not have normk · kC∞+0.
Such a proof exploits the fact that every linear elliptic equation with constant coefficients aα admits a fundamental solution K(y) that is regular outside the origin 0 ∈ Rn and has a pole in 0 of some order. This fact is well known to de Laplace, and the construction of the fundamental solution to the general case is found in [17], Chapter III.
Fourier Analysis, Pseudodifferential Operators and Elliptic Regularity
It is easy to prove that with this topology the linear form :S →C is continuous if and only if there exist constants C and N such that for every ϕ∈ S it holds. So we have to live with it and define the product whenever it makes sense. To see that p(x, D)ϕ ∈ S, recall that for any multiindex β ∈Nn we can differentiate under the integral sign in (51) because the derivatives Dβx of both p(x, ξ) and e2πix·ξ have polynomial growth in ξ, so we have absolute convergence.
Note that the class of symbols C∗rSρ,δm(Rn) is defined in the same spirit as the class Sρ,δm, only in this case we have fewer derivatives in the variable x. Also, there exists a constant C such that for every function Cl evaluated u such that we have u ∈ C∗s+m(Rn;Cl). We have the corresponding version for linear matrix-valued pseudodifferential operators defined on an open set, taqkingδ = 1 in the previous Theorem 13.
As in Proposition 10, we see that for δ → 0 it holds that ˆPα converges to Qα uniformly in Rn, and furthermore ˆPα ∈C∗r(Rn;Ck×l). We say that F is coercive if there exists ah ∈ K such that it holds for every sequence uj ∈K that kujk → ∞. Since F is coercive, there exists sh∈K such that for every sequence uj ∈K such that kujk → ∞ holds.
Now we can use Proposition 17 and conclude that there exists ∇u∈Ksuch that for every∇v∈K we have.
Regularity of Weak Solutions
To take the limit we used that U is bounded, so Lp(U) ⊂L1(U), and by the bound of. Once this is seen we apply theorem 19 and we have that, given ϕ∈Cc∞(Ω), Z. 90) and applying the dominated convergence theorem to both sides of (90) we conclude. Note first that if ϕ∈W1,p(Ω) has compact support, then we have Z. Then we have forh small. 118).
The only new requirement we ask is that ∂Ω is regular to apply Sobolev embedding theorems. Inserting this into (133) we get that. 135) This shows that (134) can be interpreted as a linear elliptic equation satisfying u inV. Note that we previously saw that p(x, D) is uniformly elliptic of order 2 in V, being V and arbitrary open set compactly contained in G={∇u6= 0} ⊂Ω.
Note that uϕ isC∗r+1 close to ϕ(x0) regardless of the regularity of the coordinate systemϕ, but if we want to have that regularity as a function defined in M, we must require the coordinatesϕ to beC∗r+1. Then we have aC∗r+1 diffeomorphismF : Ω0→ V ⊂Rn, being Ω0 ⊂Ω a smaller neighborhood of ϕ(x0), such that the coordinates of F are p-harmonic functions for the Riemannian manifold (Ω0, gϕ).
Some properties of p-harmonic coordinates
Suppose that the metric is given by the class Cr when expressed in some C1 coordinate system ϕ near a point x0. It states that when a metric is conformally flat, the coordinate diagram that makes the metric a multiple of the identity must necessarily be n-harmonic. But since the metric g2 is a multiple of the identity in standard coordinates, we know from Proposition 25 that the standard coordinates are 2-harmonic (ie, harmonic).
The following proposition shows that n-harmonic coordinates are invariant under a conformal change of metric. First, since gϕ Cr with r > 1, we know that harmonic coordinates exist and that the metric has maximum regularity in such coordinates. From this theorem we see that if the Ricci tensor is smooth in the harmonic coordinates φ and the metric is C2,α, it follows that g is also smooth in the coordinates φ.
In arbitrary coordinates, the theorem only says that we get two derivatives on the metric when we express it in harmonic coordinates, but with respect to the regularity of the metric in the original coordinates, not to the regularity of the Ricci tensor. Since the Ricci tensor is invariant under isometries, Ric(g1) = 0, but the metric g1 is not smooth, not even C3.
Definition of Tensors with Low Regular Metrics
First, we explicitly write the expressions of the (3,1) curvature tensor R, the Ricci tensor Ric, the Schouten tensor s, the Weyl tensor w and the (3,1)-Weyl tensor W. It is important to note that with this definition the curvature Rabcd has all the symmetries of the Riemann curvature tensor. To see this, look at the expression of Rabcd in (156), and note that we can expand, multiply and differentiate the expression of the Christoffel symbols in terms of the metric as in the usual case, i.e.
The symmetries of Rabcdderive from its formal expression in terms of the metric, so they are the same as in the smooth case. We claim that the conformal behavior of the weekly defined Weyl tensor remains the same, i.e. Wabcd0 =λWabcd. To see this note that. 158) since by Proposition 27 we can apply Leibnitz's rule to each of the sumands and also apply the distributive law.
We will see that we can apply the Leibnitz rule for the derivatives of the metric up to and including order 3. Briefly, to analyze how the Cotton tensor changes we are interested in the Sab0 components of the Schouten tensor, which depend on the second derivatives of the metric via the Ricci tensor R0ab and the scalar curvatureR0rsg0rsgab0 (i.e. via the Ricci tensor).
Ellipticity of the Ricci Tensor in Harmonic Coordinates
Assume that in some coordinate systemφ the expression of the metric gφ Cr for some r > 1. Assume that in some coordinate systemφ the expression of the metric gφ Cr for some >1. Suppose that in some coordinate system φ the expression of the metric gij(φ) is Cr for some r > 1 and that the curvature tensorRabcd(φ) vanishes (as a distribution) in the φcoordinates.
AsRabcd(φ) = 0 oggij(φ) isCr in the φcoordinates, then in all harmonic coordinatesϕ we have atgij(ϕ) isC∗r, so we can also understand the curvature tensor in the ϕcoordinates. Since Rabcd(ϕ) = 0, we see from the Cassian version of the test case for regular metrics that another coordinate systemφ∗ exists, such that gij(φ∗) =δij iφ∗ coordinates. 1) Note that the components Rij of the Ricci tensor depend on all metrics g and not only on the component gij, so the Ricci tensor is a differential operator acting on g, but it is not linear. However, the ricci tensor algebraically depends on the derivative of the metric up to order 2, and if we look at its expression, we have.
That said, to establish the regularity of the Ricci tensor, consider the main symbol of the now linear operator Tg(h) =dgRic(h), which is P2(x, ξ) :=B1(x)ξ2, where ξ2 represents products of two of the components of ξ∈Rn. And the crucial point is that the main symbol of the Ricci tensor considered as a linear operator through this trick is again B1(x)ξ2.
Ellipticity of the Weyl Tensor in n-Harmonic Coordinates
Considering the expression of Sab given in 157, we want to see the expression of the Weyl tensor. This formula will be decisive for the ellipticity of the weyl tensor in n-harmonic coordinates. Then we will have Γl= 0 for all l, and then the proof of the ellipticity of the Weyl tensor Wϕ in the ϕ coordinates will be trivial.
Assume that the expression of the Weyl tensor Wabcd suffices that Wabcd ∈C∗s for some > r−2 in some system ϕ of n-harmonic coordinates. From Theorem 18, we know that n-harmonic coordinates exist near x0 and, furthermore, the expression of the metric C∗r is in any n-harmonic coordinate system. Moreover, the principal symbol of the differential operator Wabcd(˜g) coincides with the principal symbol of its linearization σ˜g(Wabcd(˜g))(x, ξ).
Now, since the Weyl tensor term Wφ vanishes near x0, we must also necessarily have Wη = 0 due to tensoriality. However, this can also be derived directly from the classical proof of the Weyl-Schouten theorem.
