Conformal invariants

Let be any Riemannian manifold, consisting of a smooth manifold and a non-degenerate symmetric form on the tangent bundle of , not necessarily positive-definite. By definition, for any strictly positive smooth function the Riemannian manifold is conformally equivalent to (cf. also Conformal mapping), and a tensor (cf. also Tensor analysis) constructed from and its covariant derivatives is a conformal invariant if and only if for some fixed weight the tensor is independent of . The tensor is itself a trivial conformal invariant of weight , and the dimension of and signature of can be regarded as trivial conformal invariants, of weight . However, there are many non-trivial conformal invariants of Riemannian manifolds of dimension , and non-trivial scalar conformal invariants have been the subject of much recent work, sketched below. One can also extend the definition to include conformal invariants that are not tensors; these will not be considered below.

An -dimensional Riemannian manifold is flat in a neighbourhood of a point if there are coordinate functions , such that

on , where and is the signature of . A manifold is (locally) conformally flat if it is locally conformally equivalent to a flat manifold; the modifier "locally" is a tacit part of the definition, normally omitted. Clearly, conformally flat manifolds have no non-trivial conformal invariants.

For any smooth manifold , let be the ring of smooth real-valued functions (regarded as an algebra over ), let be the usual -module of -forms over , and for any , let denote the -fold tensor product over . In particular, the non-degenerate symmetric form of a Riemannian manifold will be regarded as a symmetric element of , as above. The conformal invariance condition is entirely local, so that one may as well assume that is itself an open set in . One finds that the signature is of little interest in the construction of conformal invariants, since strategically placed signs turn constructions for the strictly Riemannian case into corresponding constructions for the general case. Hence the existence of conformal invariants depends only on the dimension .

In the next few paragraphs the discussion of conformal invariants is organized by dimension ; at the end the discussion centres exclusively on recent work concerning scalar conformal invariants for the cases .

Dimension one.

Any -dimensional Riemannian manifold is trivially conformally flat, so that there are no non-trivial conformal invariants in dimension .

Dimension two.

If is a Riemannian manifold of dimension , let

in some neighbourhood of any point . The question of conformal flatness of breaks into two cases, as follows.

i) If the usual method of factoring into a product of two linear homogeneous factors leads to a product of linearly independent -forms, whose symmetric part is . Since , there are smooth functions , , , in a neighbourhood of such that and , so that . By setting and , one then has in a neighbourhood of ; hence is conformally flat.

ii) The case is the classical problem of finding isothermal coordinates for a Riemann surface, first solved by C.F. Gauss in a more restricted setting. More recent treatments of the same problem are given in [a9], [a10], [a4]; these results are easily adapted to the smooth case to show that any (smooth) Riemannian surface with a positive-definite (or negative-definite) metric is conformally flat. It follows from i) and ii) that there are no non-trivial conformal invariants in dimension .

Dimension at least three.

Some classical conformal invariants in dimensions are as follows (their constructions will be sketched later):

In 1899, E. Cotton [a7] assigned a tensor to any Riemannian manifold of any dimension ; it is conformally invariant of weight only in the special case , and J.A. Schouten [a12] showed that in this case is conformally flat if and only if .

In 1918, H. Weyl [a14] constructed a tensor for any Riemannian manifold of dimension , conformally invariant of weight for all dimensions although it vanishes identically for . Schouten [a12] showed that a Riemannian manifold of dimension is conformally flat if and only if , and is now known as the Weyl curvature tensor (cf. also Weyl tensor).

The remaining classical tensor was constructed by R. Bach [a1] in 1921; although exists in any dimension , it is conformally invariant, of weight , only for Riemannian manifolds of dimension , and in this dimension if and only if is conformally equivalent to an Einstein manifold (see below).

Algebraic background.

The primarily algebraic background needed to describe these three classical conformal invariants is also needed to sketch the more recent construction of the scalar conformal invariants, mentioned earlier. Let be any commutative ring with unit that is also an algebra over the real numbers; the ring will later be for a smooth manifold . Let be an -module, let , let , and assume that the natural homomorphism from to its double dual is an isomorphism ; the -module will later be the -module of -forms on , and will be the -module of smooth vector fields on . As before, for any let denote the -fold tensor product over , later the -module of contravariant tensors of degree over .

If is the -module isomorphism that interchanges the two factors , an element is symmetric if . Let be the submodule of symmetric elements; it consists of -linear combinations of products of the form . One can regard any as a homomorphism , so that there is an induced homomorphism such that for any . The isomorphism permits one to regard as a homomorphism , and is non-degenerate if is an isomorphism. In this case the inverse provides a unique element that can be regarded as a homomorphism with values for any . One easily verifies that is itself non-degenerate.

For any , let be an unordered pair of distinct elements in and let be non-degenerate. Then one can evaluate on the tensor product of the th and th factors of to obtain a well-defined -linear contraction . The symmetry of guarantees that does not require an ordering of . Similarly, if is any unordered subset of , there is a well-defined -linear contraction , where .

An element is alternating if , and there is a submodule that consists of all such alternating elements. If is the ring for a Riemannian manifold , and if is the -module of -forms on , then the classical Riemannian curvature tensor of (cf. also Curvature tensor; Riemann tensor) is a symmetric element , for the submodule ; a construction is sketched below. The corresponding Ricci curvature is the contraction , and the corresponding scalar curvature is the contraction . In case is of dimension , there is a nameless tensor

that is used to construct all three classical conformal invariants.

The construction of the Weyl curvature tensor uses a -module homomorphism from the submodule to the submodule of symmetric elements in . If , let be the isomorphism that permutes the th factor in to the left of the first factors in , so that is cyclic in the usual sense that , and simply places the first factor into the th slot; in particular, is the identity, and interchanges the first two factors as before. For any , set

By looking at the special cases , for any and , one obtains

these cases induce the announced homomorphism .

For any Riemannian manifold of dimension , the Weyl curvature tensor is the difference , which is a non-trivial conformal invariant of weight whenever . Although the principal feature of is that if and only if the Riemannian manifold of dimension is conformally flat, it also provides a basic tool for constructing other conformal invariants for manifolds of dimensions . For example, for any , let be the tensor product of copies of , and let as unordered sets. Then the contraction

is a non-trivial scalar conformal invariant of weight for any Riemannian manifold of dimension .

The curvatures , , , and the tensor assigned to any Riemannian manifold are all constructed via the Levi-Civita connection associated to , defined below, so that depends implicitly upon the Levi-Civita connection. The remaining classical conformal invariants and , for Riemannian manifolds of dimensions and , respectively, as well as most of the scalar conformal invariants that will be introduced below, will be constructed explicitly via a version of the Levi-Civita connection that is sketched in the next two paragraphs; more details of this version appear in [a11].

Levi-Civita connection.

For any smooth manifold with -module of -forms as before, a connection (cf. also Connections on a manifold) is a sequence of real linear homomorphisms such that the complex covers the classical de Rham complex (cf. also Differential form); that is, the diagram

commutes for the usual projections from tensor products to exterior products over , where is the quotient of by the two-sided ideal generated by . Furthermore, if and if is the permutation with parity that moves the st factor to the left of the first factors , then

for any and ; the product rule is

for . It follows that the covering of also preserves products. If is a Riemannian manifold, with metric as usual, there is a unique connection such that ; this is the Levi-Civita connection associated to (cf. also Levi-Civita connection).

One useful property of any connection for any smooth manifold is that for any the composition

is -linear, where interchanges the first two factors of and is the identity isomorphism; for any the homomorphism is the curvature operator . In particular, for any Riemannian manifold and corresponding Levi-Civita connection, the tensor product of and the identity isomorphism restricts to a -linear mapping , and the image of the metric itself is the Riemannian curvature tensor , lying in the submodule .

Even though the Levi-Civita connection of a Riemannian manifold is defined in part by the requirement that for the Riemannian metric , observe that the definition of the Riemannian curvature is obtained by applying the curvature operator only to the first factor of . Consequently, , , , and all require the first two derivatives of , in the obvious sense. The same remark applies to the Weyl curvature tensor .

Cotton tensor.

Let be any Riemannian manifold of dimension , with as before, let

be the Levi-Civita connection, which restricts to , and let be the cyclic permutation of the factors that moves the third factor to the left of the first two factors . The Cotton tensor is

which visibly depends on third derivatives of ; this is equivalent to the original definition of E. Cotton [a7], and it has the evident cyclic symmetry . Furthermore, is a conformal invariant if is of dimension , and Schouten [a12] showed in this case that if and only if is conformally flat, as noted earlier.

Closed oriented -dimensional Riemannian manifolds.

If one considers closed oriented -dimensional Riemannian manifolds , the Chern–Simons invariant is shown in [a6] to depend only on the conformal equivalence class of , and is a critical value if and only if is conformally flat. S.S. Chern [a5] gave a simplified proof of this result by using the criterion of the preceding paragraph.

Bach tensor.

For any Riemannian manifold of dimension , the Bach tensor is

for the Levi-Civita connection

one easily verifies that the Bach tensor is an element of . It is conformally invariant only in the special case , and in that case one has if and only if is conformally equivalent to a Riemannian manifold such that the Ricci curvature is a constant multiple of the metric itself. Riemannian manifolds with the latter property are known as Einstein manifolds.

Recall that for any the contractions

of the -fold tensor product of the Weyl curvature tensor are scalar conformal invariants of weight , and observe that any -linear combination of such contractions is also a scalar conformal invariant of weight . Such scalar conformal invariants involve the Riemannian metric and its first and second order derivatives. However, the derivative is not itself conformally invariant if , so that in general one cannot expect contractions of products to produce conformal invariants if . The following observations suggest a reasonable modification of the construction.

First, observe that if and are Riemannian manifolds for which there is an embedding with , then any scalar conformal invariant of restricts to the corresponding scalar conformal invariant of . Since the construction of conformal invariants is an entirely local question, it suffices to consider embeddings of open sets into open sets , for example. The hypotheses can be weakened if the conformal equivalence class of has a real-analytic representative with coordinates . One can then assign a coordinate and use power series about to describe the Riemannian metric of an embedding, knowing that only the restrictions of the derivatives to the submanifold are of any interest, the inclusion being

The second observation is a classical result, not directly related to conformal invariants. Given any Riemannian manifold , with Levi-Civita connection and Riemannian curvature , if is an even number, then the contractions involve derivatives of of order up to ; furthermore, such contractions are visibly coordinate-free. Results in [a15] imply that if is locally real-analytic, then any coordinate-free polynomial combination of and the components of the derivatives is a -linear combination of such contractions, which are known as Weyl invariants.

The third observation is that if is a Ricci-flat Riemannian manifold, in the sense that , then so that ; in this case the Riemannian curvature tensor itself is a classical conformal invariant: . Even though one cannot expect the derivatives nor contractions of products of such derivatives to be conformal invariants, the identifications suggest that the contractions may be of value in the Ricci-flat case, whenever is an even number.

General construction of scalar conformal invariants.

The preceding observations lead to a general construction of scalar conformal invariants of , with a dimensional restriction that will be specified later. One first covers by sufficiently small coordinate neighbourhoods and writes for each resulting Riemannian manifold . For each C. Fefferman and C.R. Graham [a8] use a technique that appeared independently in [a13] to introduce a codimension- embedding , described later, and to devise a Cauchy problem whose solution provides a Ricci-flat manifold with . A further feature of the construction guarantees that any Weyl invariant in restricts to a conformal invariant of , of weight . Since -linear combinations of scalar conformal invariants of weight are also scalar conformal invariants of weight , for any fixed -tuple of non-negative integers with an even sum one can use a smooth partition of unity subordinate to the covering of by the coordinate neighbourhoods to obtain a scalar conformal invariant of itself, known as a Weyl conformal invariant.

T.N. Bailey, M.G. Eastwood and Graham [a2] completed the proof of the following Fefferman–Graham conjecture [a8], which depends upon the parity of : If is a Riemannian manifold of odd dimension , then every scalar conformal invariant of is a Weyl conformal invariant. If is a Riemannian manifold of even dimension , then the preceding statement is true only for scalar conformal invariants of weight , and there is a conformally invariant element in of weight that serves as an obstruction to finding a formal power series solution of the Cauchy problems used to construct the ambient manifolds ; the obstruction vanishes if is conformally equivalent to an Einstein manifold; if the obstruction is the Bach conformal invariant .

There are some exceptional scalar conformal invariants for even dimensions and weight , first observed in [a2]; the catalogue of all such exceptional invariants was completed in [a3].

Fefferman–Graham method.

This method, introduced in [a8], allows one to construct the codimension- embeddings of the Riemannian manifolds , and to formulate the Cauchy problems whose solutions turn each ambient space into a Ricci-flat manifold with the desired properties.

One starts with the fibration over in which the fibre over each consists of positive multiples of the metric at ; one may as well suppose that . The multiplicative group of real numbers acts on the fibres by mapping into , and this permits one to regard the fibration as a fibre bundle with structure group (cf. also Principal fibre bundle). Clearly, any section of the fibre bundle can be regarded as a Riemannian manifold that is conformally equivalent to .

Let be the corresponding principal fibre bundle, and observe that since , the pullback of the metric over needs at least one additional term to serve as a Riemannian metric over . It is useful to replace by another -bundle with , and to try to construct a (non-degenerate) metric on such that

1) the restriction is ;

2) the group elements map into over all of ;

3) is Ricci-flat, with the consequence noted earlier. There is an implicit additional assumption, that the conformal equivalence class containing is real-analytic in the sense that there is a representative of the conformal class of for which one can choose coordinates in such that , for coefficients that are real-analytic functions of ; one may as well assume that itself has this property.

The Fefferman–Graham method [a8] leads to a metric of the form

that satisfies 1)–3) for all (), for real-analytic functions of that satisfy the initial condition 1), as formal power series about ; convergence is obtained in some neighbourhood of . Observe that the metric trivially satisfies the homogeneity condition 2) over all of . The Riemannian curvature is itself conformally invariant by the consequence of condition 3), and the homogeneity condition implies that any Weyl invariant restricts over the section of to a Weyl conformal invariant in , as required.

References

[a1]	R. Bach, "Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs" Math. Z. , 9 (1921) pp. 110–135 (Also: Jahrbuch 48, 1035) MR1544454 Zbl 48.1035.01
[a2]	T.N. Bailey, M.G. Eastwood, C.R. Graham, "Invariant theory for conformal and CR geometry" Ann. of Math. , 139 : 2 (1994) pp. 491–552 MR1283869 Zbl 0814.53017
[a3]	T.N. Bailey, A.R. Gover, "Exceptional invariants in the parabolic invariant theory of conformal geometry" Proc. Amer. Math. Soc. , 123 (1995) pp. 2535–2543 MR1243161 Zbl 0844.53008
[a4]	S.S. Chern, "An elementary proof of the existence of isothermal parameters on a surface" Proc. Amer. Math. Soc. , 6 (1955) pp. 771–782 MR0074856 Zbl 0066.15402
[a5]	S.S. Chern, "On a conformal invariant of three-dimensional manifolds" , Aspects of Math. and its Appl. , North-Holland (1986) pp. 245–252 MR0849555 Zbl 0589.53011
[a6]	S.S. Chern, J. Simons, "Characteristic forms and geometric invariants" Ann. of Math. , 99 (1974) pp. 48–69 MR0353327 Zbl 0283.53036 Zbl 0591.53050
[a7]	E. Cotton, "Sur les variétes à trois dimensions" Ann. Fac. Sci. Toulouse , 1 (1899) pp. 385–438 (Also: Jahrbuch 30, 538-539) MR1508211 Zbl 30.0538.01
[a8]	C. Fefferman, C.R. Graham, "Conformal invariants" , The Mathematical Heritage of Élie Cartan (Lyon, 1984) , Astérisque (1985) pp. 95–116 MR0837196 Zbl 0602.53007
[a9]	A. Korn, "Zwei Anwendungen der Methode der sukzessiven Anwendungen" Schwarz Festschrift (1914) pp. 215–229 (Also: Jahrbuch 45, 568) Zbl 45.0568.01
[a10]	L. Lichtenstein, "Zur Theorie der konformen Abbildungen nichtanalytischer, singularitätenfreier Flächenstücke auf ebene Gebiete" Bull. Internat. Acad. Sci. Gracovie, Cl. Sci. Math. Nat. Ser. A. (1916) pp. 192–217 (Also: Jahrbuch 46, 547)
[a11]	H. Osborn, "Affine connection complexes" Acta Applic. Math. (to appear) MR1741659 Zbl 0956.53014
[a12]	J.A. Schouten, "Über die konforme Abbildung -dimensionaler Mannigfaltigkeiten mit quadratischer Maß bestimmung auf eine Mannigfaltigkeit mit euklidischer Maß bestimmung" Math. Z. , 11 (1921) pp. 58–88 (Also: Jahrbuch 48, 857-858)
[a13]	J.A. Schouten, J. Haantjes, "Beitgräge zur allgemeinen (gekrümmten) konformen Differentialgeometrie I–II" Math. Ann. , 112/113 (1936) pp. 594–629; 568–583
[a14]	H. Weyl, "Reine Infinitesimalgeometrie" Math. Z. , 2 (1918) pp. 384–411 (Also: Jahrbuch 46, 1301) MR1544327 Zbl 46.1301.01
[a15]	H. Weyl, "The classical groups" , Princeton Univ. Press (1939) (Reprint: 1946) MR0000255 Zbl 0020.20601 Zbl 65.0058.02