View source for Conformal invariants

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c1201801.png" /> be any [[Riemannian manifold|Riemannian manifold]], consisting of a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c1201802.png" /> and a non-degenerate symmetric form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c1201803.png" /> on the tangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c1201804.png" />, not necessarily positive-definite. By definition, for any strictly positive smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c1201805.png" /> the Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c1201806.png" /> is conformally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c1201807.png" /> (cf. also [[Conformal mapping|Conformal mapping]]), and a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c1201808.png" /> (cf. also [[Tensor analysis|Tensor analysis]]) constructed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c1201809.png" /> and its covariant derivatives is a conformal invariant if and only if for some fixed weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018010.png" /> the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018011.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018012.png" />. The tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018013.png" /> is itself a trivial conformal invariant of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018014.png" />, and the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018015.png" /> and signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018016.png" /> can be regarded as trivial conformal invariants, of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018017.png" />. However, there are many non-trivial conformal invariants of Riemannian manifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018018.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018019.png" />-dimensional Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018020.png" /> is flat in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018021.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018022.png" /> if there are coordinate functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018024.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018025.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018026.png" /></td> </tr></table>

on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018029.png" /> is the signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018030.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018031.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018032.png" /> be the ring of smooth real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018033.png" /> (regarded as an algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018034.png" />), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018035.png" /> be the usual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018036.png" />-module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018037.png" />-forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018038.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018039.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018040.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018041.png" />-fold tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018042.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018043.png" />. In particular, the non-degenerate symmetric form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018044.png" /> of a Riemannian manifold will be regarded as a symmetric element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018045.png" />, as above. The conformal invariance condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018046.png" /> is entirely local, so that one may as well assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018047.png" /> is itself an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018048.png" />. One finds that the signature is of little interest in the construction of conformal invariants, since strategically placed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018049.png" /> signs turn constructions for the strictly Riemannian case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018050.png" /> into corresponding constructions for the general case. Hence the existence of conformal invariants depends only on the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018051.png" />.

In the next few paragraphs the discussion of conformal invariants is organized by dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018052.png" />; at the end the discussion centres exclusively on recent work concerning scalar conformal invariants for the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018053.png" />.

==Dimension one.==
Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018054.png" />-dimensional Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018055.png" /> is trivially conformally flat, so that there are no non-trivial conformal invariants in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018056.png" />.

==Dimension two.==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018057.png" /> is a Riemannian manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018058.png" />, let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018059.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018060.png" /></td> </tr></table>

in some neighbourhood of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018061.png" />. The question of conformal flatness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018062.png" /> breaks into two cases, as follows.

i) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018063.png" /> the usual method of factoring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018064.png" /> into a product of two linear homogeneous factors leads to a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018065.png" /> of linearly independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018066.png" />-forms, whose symmetric part is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018067.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018068.png" />, there are smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018072.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018073.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018075.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018076.png" />. By setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018078.png" />, one then has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018079.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018080.png" />; hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018081.png" /> is conformally flat.

ii) The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018082.png" /> is the classical problem of finding [[Isothermal coordinates|isothermal coordinates]] for a [[Riemann surface|Riemann surface]], first solved by C.F. Gauss in a more restricted setting. More recent treatments of the same problem are given in [[#References|[a9]]], [[#References|[a10]]], [[#References|[a4]]]; these results are easily adapted to the smooth case to show that any (smooth) Riemannian surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018083.png" /> with a positive-definite (or negative-definite) metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018084.png" /> is conformally flat. It follows from i) and ii) that there are no non-trivial conformal invariants in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018085.png" />.

==Dimension at least three.==
Some classical conformal invariants in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018086.png" /> are as follows (their constructions will be sketched later):

In 1899, E. Cotton [[#References|[a7]]] assigned a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018087.png" /> to any Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018088.png" /> of any dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018089.png" />; it is conformally invariant of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018090.png" /> only in the special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018091.png" />, and J.A. Schouten [[#References|[a12]]] showed that in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018092.png" /> is conformally flat if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018093.png" />.

In 1918, H. Weyl [[#References|[a14]]] constructed a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018094.png" /> for any Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018095.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018096.png" />, conformally invariant of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018097.png" /> for all dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018098.png" /> although it vanishes identically for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c12018099.png" />. Schouten [[#References|[a12]]] showed that a Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180100.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180101.png" /> is conformally flat if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180102.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180103.png" /> is now known as the Weyl curvature tensor (cf. also [[Weyl tensor|Weyl tensor]]).

The remaining classical tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180104.png" /> was constructed by R. Bach [[#References|[a1]]] in 1921; although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180105.png" /> exists in any dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180106.png" />, it is conformally invariant, of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180107.png" />, only for Riemannian manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180108.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180109.png" />, and in this dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180110.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180111.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180112.png" /> be any [[Commutative ring|commutative ring]] with unit that is also an [[Algebra|algebra]] over the real numbers; the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180113.png" /> will later be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180114.png" /> for a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180115.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180116.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180117.png" />-module, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180118.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180119.png" />, and assume that the natural homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180120.png" /> to its double dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180121.png" /> is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180122.png" />; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180123.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180124.png" /> will later be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180125.png" />-module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180126.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180127.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180128.png" /> will be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180129.png" />-module of smooth vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180130.png" />. As before, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180131.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180132.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180133.png" />-fold tensor product over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180134.png" />, later the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180135.png" />-module of contravariant tensors of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180136.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180137.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180138.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180139.png" />-module isomorphism that interchanges the two factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180140.png" />, an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180141.png" /> is symmetric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180142.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180143.png" /> be the submodule of symmetric elements; it consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180144.png" />-linear combinations of products of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180145.png" />. One can regard any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180146.png" /> as a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180147.png" />, so that there is an induced homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180148.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180149.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180150.png" />. The isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180151.png" /> permits one to regard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180152.png" /> as a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180153.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180154.png" /> is non-degenerate if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180155.png" /> is an isomorphism. In this case the inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180156.png" /> provides a unique element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180157.png" /> that can be regarded as a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180158.png" /> with values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180159.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180160.png" />. One easily verifies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180161.png" /> is itself non-degenerate.

For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180162.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180163.png" /> be an unordered pair of distinct elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180164.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180165.png" /> be non-degenerate. Then one can evaluate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180166.png" /> on the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180167.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180168.png" />th and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180169.png" />th factors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180170.png" /> to obtain a well-defined <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180171.png" />-linear [[Contraction(2)|contraction]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180172.png" />. The symmetry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180173.png" /> guarantees that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180174.png" /> does not require an ordering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180175.png" />. Similarly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180176.png" /> is any unordered subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180177.png" />, there is a well-defined <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180178.png" />-linear contraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180179.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180180.png" />.

An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180181.png" /> is alternating if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180182.png" />, and there is a submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180183.png" /> that consists of all such alternating elements. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180184.png" /> is the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180185.png" /> for a Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180186.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180187.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180188.png" />-module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180189.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180190.png" />, then the classical Riemannian curvature tensor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180191.png" /> (cf. also [[Curvature tensor|Curvature tensor]]; [[Riemann tensor|Riemann tensor]]) is a symmetric element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180192.png" />, for the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180193.png" />; a construction is sketched below. The corresponding [[Ricci curvature|Ricci curvature]] is the contraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180194.png" />, and the corresponding [[Scalar curvature|scalar curvature]] is the contraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180195.png" />. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180196.png" /> is of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180197.png" />, there is a nameless tensor

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180198.png" /></td> </tr></table>

that is used to construct all three classical conformal invariants.

The construction of the Weyl curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180199.png" /> uses a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180200.png" />-module homomorphism from the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180201.png" /> to the submodule of symmetric elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180202.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180203.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180204.png" /> be the isomorphism that permutes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180205.png" />th factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180206.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180207.png" /> to the left of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180208.png" /> factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180209.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180210.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180211.png" /> is cyclic in the usual sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180212.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180213.png" /> simply places the first factor into the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180214.png" />th slot; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180215.png" /> is the identity, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180216.png" /> interchanges the first two factors as before. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180217.png" />, set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180218.png" /></td> </tr></table>

By looking at the special cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180219.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180220.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180221.png" />, one obtains

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180222.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180223.png" /></td> </tr></table>

these cases induce the announced homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180224.png" />.

For any Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180225.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180226.png" />, the Weyl curvature tensor is the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180227.png" />, which is a non-trivial conformal invariant of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180228.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180229.png" />. Although the principal feature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180230.png" /> is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180231.png" /> if and only if the Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180232.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180233.png" /> is conformally flat, it also provides a basic tool for constructing other conformal invariants for manifolds of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180234.png" />. For example, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180235.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180236.png" /> be the tensor product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180237.png" /> copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180238.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180239.png" /> as unordered sets. Then the [[Contraction(2)|contraction]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180240.png" /></td> </tr></table>

is a non-trivial scalar conformal invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180241.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180242.png" /> for any Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180243.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180244.png" />.

The curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180245.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180246.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180247.png" />, and the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180248.png" /> assigned to any Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180249.png" /> are all constructed via the Levi-Civita connection associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180250.png" />, defined below, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180251.png" /> depends implicitly upon the Levi-Civita connection. The remaining classical conformal invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180252.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180253.png" />, for Riemannian manifolds of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180254.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180255.png" />, 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 [[#References|[a11]]].

===Levi-Civita connection.===
For any smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180256.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180257.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180258.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180259.png" />-forms as before, a connection (cf. also [[Connections on a manifold|Connections on a manifold]]) is a sequence of real linear homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180260.png" /> such that the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180261.png" /> covers the classical de Rham complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180262.png" /> (cf. also [[Differential form|Differential form]]); that is, the diagram

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180263.png" /></td> </tr></table>

commutes for the usual projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180264.png" /> from tensor products to exterior products over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180265.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180266.png" /> is the quotient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180267.png" /> by the two-sided ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180268.png" />. Furthermore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180269.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180270.png" /> is the permutation with parity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180271.png" /> that moves the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180272.png" />st factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180273.png" /> to the left of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180274.png" /> factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180275.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180276.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180277.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180278.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180279.png" />; the product rule is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180280.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180281.png" />. It follows that the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180282.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180283.png" /> also preserves products. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180284.png" /> is a Riemannian manifold, with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180285.png" /> as usual, there is a unique connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180286.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180287.png" />; this is the Levi-Civita connection associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180288.png" /> (cf. also [[Levi-Civita connection|Levi-Civita connection]]).

One useful property of any connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180289.png" /> for any smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180290.png" /> is that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180291.png" /> the composition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180292.png" /></td> </tr></table>

is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180293.png" />-linear, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180294.png" /> interchanges the first two factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180295.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180296.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180297.png" /> is the identity isomorphism; for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180298.png" /> the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180299.png" /> is the curvature operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180300.png" />. In particular, for any Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180301.png" /> and corresponding Levi-Civita connection, the tensor product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180302.png" /> and the identity isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180303.png" /> restricts to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180304.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180305.png" />, and the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180306.png" /> of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180307.png" /> itself is the Riemannian curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180308.png" />, lying in the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180309.png" />.

Even though the Levi-Civita connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180310.png" /> of a Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180311.png" /> is defined in part by the requirement that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180312.png" /> for the Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180313.png" />, observe that the definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180314.png" /> of the Riemannian curvature is obtained by applying the curvature operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180315.png" /> only to the first factor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180316.png" />. Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180317.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180318.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180319.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180320.png" /> all require the first two derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180321.png" />, in the obvious sense. The same remark applies to the Weyl curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180322.png" />.

===Cotton tensor.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180323.png" /> be any Riemannian manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180324.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180325.png" /> as before, let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180326.png" /></td> </tr></table>

be the Levi-Civita connection, which restricts to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180327.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180328.png" /> be the cyclic permutation of the factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180329.png" /> that moves the third factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180330.png" /> to the left of the first two factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180331.png" />. The Cotton tensor is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180332.png" /></td> </tr></table>

which visibly depends on third derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180333.png" />; this is equivalent to the original definition of E. Cotton [[#References|[a7]]], and it has the evident cyclic symmetry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180334.png" />. Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180335.png" /> is a conformal invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180336.png" /> is of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180337.png" />, and Schouten [[#References|[a12]]] showed in this case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180338.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180339.png" /> is conformally flat, as noted earlier.

===Closed oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180340.png" />-dimensional Riemannian manifolds.===
If one considers closed oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180341.png" />-dimensional Riemannian manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180342.png" />, the Chern–Simons invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180343.png" /> is shown in [[#References|[a6]]] to depend only on the conformal equivalence class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180344.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180345.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180346.png" /> is a critical value if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180347.png" /> is conformally flat. S.S. Chern [[#References|[a5]]] gave a simplified proof of this result by using the criterion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180348.png" /> of the preceding paragraph.

===Bach tensor.===
For any Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180349.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180350.png" />, the Bach tensor is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180351.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180352.png" /></td> </tr></table>

for the Levi-Civita connection

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180353.png" /></td> </tr></table>

one easily verifies that the Bach tensor is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180354.png" />. It is conformally invariant only in the special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180355.png" />, and in that case one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180356.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180357.png" /> is conformally equivalent to a Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180358.png" /> such that the Ricci curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180359.png" /> is a constant multiple of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180360.png" /> itself. Riemannian manifolds with the latter property are known as Einstein manifolds.

Recall that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180361.png" /> the contractions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180362.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180363.png" /></td> </tr></table>

of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180364.png" />-fold tensor product of the Weyl curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180365.png" /> are scalar conformal invariants of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180366.png" />, and observe that any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180367.png" />-linear combination of such contractions is also a scalar conformal invariant of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180368.png" />. Such scalar conformal invariants involve the Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180369.png" /> and its first and second order derivatives. However, the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180370.png" /> is not itself conformally invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180371.png" />, so that in general one cannot expect contractions of products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180372.png" /> to produce conformal invariants if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180373.png" />. The following observations suggest a reasonable modification of the construction.

First, observe that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180374.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180375.png" /> are Riemannian manifolds for which there is an embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180376.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180377.png" />, then any scalar conformal invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180378.png" /> restricts to the corresponding scalar conformal invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180379.png" />. Since the construction of conformal invariants is an entirely local question, it suffices to consider embeddings of open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180380.png" /> into open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180381.png" />, for example. The hypotheses can be weakened if the conformal equivalence class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180382.png" /> has a real-analytic representative with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180383.png" />. One can then assign a coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180384.png" /> and use power series about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180385.png" /> to describe the Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180386.png" /> of an embedding, knowing that only the restrictions of the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180387.png" /> to the submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180388.png" /> are of any interest, the inclusion being

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180389.png" /></td> </tr></table>

The second observation is a classical result, not directly related to conformal invariants. Given any Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180390.png" />, with Levi-Civita connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180391.png" /> and Riemannian curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180392.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180393.png" /> is an even number, then the contractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180394.png" /> involve derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180395.png" /> of order up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180396.png" />; furthermore, such contractions are visibly coordinate-free. Results in [[#References|[a15]]] imply that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180397.png" /> is locally real-analytic, then any coordinate-free polynomial combination of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180398.png" /> and the components of the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180399.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180400.png" />-linear combination of such contractions, which are known as Weyl invariants.

The third observation is that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180401.png" /> is a Ricci-flat Riemannian manifold, in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180402.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180403.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180404.png" />; in this case the Riemannian curvature tensor itself is a classical conformal invariant: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180405.png" />. Even though one cannot expect the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180406.png" /> nor contractions of products of such derivatives to be conformal invariants, the identifications <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180407.png" /> suggest that the contractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180408.png" /> may be of value in the Ricci-flat case, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180409.png" /> is an even number.

===General construction of scalar conformal invariants.===
The preceding observations lead to a general construction of scalar conformal invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180410.png" />, with a dimensional restriction that will be specified later. One first covers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180411.png" /> by sufficiently small coordinate neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180412.png" /> and writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180413.png" /> for each resulting Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180414.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180415.png" /> C. Fefferman and C.R. Graham [[#References|[a8]]] use a technique that appeared independently in [[#References|[a13]]] to introduce a codimension-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180416.png" /> embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180417.png" />, described later, and to devise a [[Cauchy problem|Cauchy problem]] whose solution provides a Ricci-flat manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180418.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180419.png" />. A further feature of the construction guarantees that any Weyl invariant in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180420.png" /> restricts to a conformal invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180421.png" />, of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180422.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180423.png" />-linear combinations of scalar conformal invariants of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180424.png" /> are also scalar conformal invariants of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180425.png" />, for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180426.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180427.png" /> of non-negative integers with an even sum one can use a smooth partition of unity subordinate to the covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180428.png" /> by the coordinate neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180429.png" /> to obtain a scalar conformal invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180430.png" /> itself, known as a Weyl conformal invariant.

T.N. Bailey, M.G. Eastwood and Graham [[#References|[a2]]] completed the proof of the following Fefferman–Graham conjecture [[#References|[a8]]], which depends upon the parity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180431.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180432.png" /> is a Riemannian manifold of odd dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180433.png" />, then every scalar conformal invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180434.png" /> is a Weyl conformal invariant. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180435.png" /> is a Riemannian manifold of even dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180436.png" />, then the preceding statement is true only for scalar conformal invariants of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180437.png" />, and there is a conformally invariant element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180438.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180439.png" /> that serves as an obstruction to finding a formal power series solution of the Cauchy problems used to construct the ambient manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180440.png" />; the obstruction vanishes if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180441.png" /> is conformally equivalent to an Einstein manifold; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180442.png" /> the obstruction is the Bach conformal invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180443.png" />.

There are some exceptional scalar conformal invariants for even dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180444.png" /> and weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180445.png" />, first observed in [[#References|[a2]]]; the catalogue of all such exceptional invariants was completed in [[#References|[a3]]].

===Fefferman–Graham method.===
This method, introduced in [[#References|[a8]]], allows one to construct the codimension-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180446.png" /> embeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180447.png" /> of the Riemannian manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180448.png" />, and to formulate the Cauchy problems whose solutions turn each ambient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180449.png" /> into a Ricci-flat manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180450.png" /> with the desired properties.

One starts with the [[Fibration|fibration]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180451.png" /> in which the fibre over each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180452.png" /> consists of positive multiples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180453.png" /> of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180454.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180455.png" />; one may as well suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180456.png" />. The multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180457.png" /> of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180458.png" /> acts on the fibres by mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180459.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180460.png" />, and this permits one to regard the fibration as a fibre bundle with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180461.png" /> (cf. also [[Principal fibre bundle|Principal fibre bundle]]). Clearly, any section of the fibre bundle can be regarded as a Riemannian manifold that is conformally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180462.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180463.png" /> be the corresponding [[Principal fibre bundle|principal fibre bundle]], and observe that since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180464.png" />, the pullback <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180465.png" /> of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180466.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180467.png" /> needs at least one additional term to serve as a Riemannian metric over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180468.png" />. It is useful to replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180469.png" /> by another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180470.png" />-bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180471.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180472.png" />, and to try to construct a (non-degenerate) metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180473.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180474.png" /> such that

1) the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180475.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180476.png" />;

2) the group elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180477.png" /> map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180478.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180479.png" /> over all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180480.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180481.png" /> is Ricci-flat, with the consequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180482.png" /> noted earlier. There is an implicit additional assumption, that the conformal equivalence class containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180483.png" /> is real-analytic in the sense that there is a representative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180484.png" /> of the conformal class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180485.png" /> for which one can choose coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180486.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180487.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180488.png" />, for coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180489.png" /> that are real-analytic functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180490.png" />; one may as well assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180491.png" /> itself has this property.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180492.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180493.png" /></td> </tr></table>

that satisfies 1)–3) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180494.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180495.png" />), for real-analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180496.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180497.png" /> that satisfy the initial condition 1), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180498.png" /> as formal power series about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180499.png" />; convergence is obtained in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180500.png" />. Observe that the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180501.png" /> trivially satisfies the homogeneity condition 2) over all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180502.png" />. The Riemannian curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180503.png" /> is itself conformally invariant by the consequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180504.png" /> of condition 3), and the homogeneity condition implies that any Weyl invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180505.png" /> restricts over the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180506.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180507.png" /> to a Weyl conformal invariant in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120180/c120180508.png" />, as required.

====References====
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