View source for Conformal structure

A conformal structure on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c0248201.png" /> is a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c0248202.png" /> of pairwise-homothetic Euclidean metrics on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c0248203.png" />. Any Euclidean metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c0248204.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c0248205.png" /> defines a conformal structure,

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called the conformal structure induced by the Euclidean metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c0248207.png" />. An automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c0248208.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c0248209.png" /> is called an automorphism of the conformal structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482010.png" /> if the induced transformation on the space of bilinear forms preserves the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482011.png" />. The group of automorphisms of a conformal structure is isomorphic to the linear conformal group

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which is the direct product of the multiplicative group of positive numbers and the orthogonal group.

A conformal structure on a manifold is a field of conformal structures on the tangent spaces, that is, a subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482013.png" /> of the bundle of symmetric bilinear forms on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482014.png" /> whose fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482015.png" /> are conformal structures on the corresponding tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482016.png" />. The bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482017.png" /> is topologically trivial and any section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482018.png" /> of it (giving rise to a Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482019.png" />) uniquely defines a conformal structure according to the formula

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The section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482021.png" /> is called a Riemannian metric subordinate to the conformal structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482022.png" />. Any other section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482023.png" /> of the bundle has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482025.png" /> is a positive function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482026.png" />, that is, the Riemannian metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482028.png" /> are conformally equivalent. Therefore a conformal structure can also be defined as a class of conformally-equivalent Riemannian metrics. A conformal structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482029.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482030.png" /> can be identified with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482031.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482033.png" /> consisting of all frames on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482034.png" /> that are orthonormal with respect to at least one Riemannian metric that is subordinate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482035.png" />. The main properties of a conformal structure are determined by the fact that a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482036.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482037.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482038.png" />-structure of order two: Its first extension is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482039.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482041.png" />, while the second extension is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482042.png" />-structure (a field of frames) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482043.png" />. Hence, in particular, it follows that the group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482044.png" /> (which is the same as the group of conformal transformations of any Riemannian metric subordinate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482045.png" />) is a Lie group of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482046.png" />, while the isotropy representation of its stationary subgroup in the tangent space of the second order is faithful.

As a rule, the group of automorphisms of a conformal structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482047.png" /> is the same as the group of motions of some Riemannian metric subordinate to it. The only exceptions are the standard conformal structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482048.png" /> on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482049.png" /> and on the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482050.png" />, generated by the standard Riemannian metrics. A conformal structure on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482051.png" /> is called locally flat if it is locally equivalent to the standard conformal structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482052.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482053.png" />, that is, if there exists in a neighbourhood of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482054.png" /> a flat Riemannian metric subordinate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482055.png" />. In order that a conformal structure be locally flat it is necessary and sufficient that the Weyl conformal curvature tensor of some (and therefore of any) Riemannian metric subordinate to it be zero. Examples of locally flat conformal structures are the standard conformal structures in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482056.png" />, on the spheres, in a Lobachevskii space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482057.png" />, and also in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482059.png" /> generated by the standard metrics. All locally flat conformal structures on simply-connected manifolds with a transitive group of automorphisms are accounted for in this way. The standard conformal structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482060.png" /> on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482061.png" /> is the only conformal structure having a maximal (in the sense of dimension) group of automorphisms. The sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482062.png" /> endowed with the conformal structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482063.png" /> is called a conformal space.

The notion of a conformal structure is closely related to that of a [[Conformal connection|conformal connection]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482064.png" />: such a connection always defines a conformal structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024820/c02482065.png" />; on the other hand, a conformal connection is the connection in the reduced principal fibre bundle that is defined by the given conformal structure.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kobayashi,   "Transformation groups and differential geometry" , Springer  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Sternberg,   "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.N. Kimel'field,   "Homogeneous regions of the conformal sphere"  ''Math. Notes'' , '''8'''  (1970)  pp. 653–656  ''Mat. Zametki'' , '''8''' :  3  (1970)  pp. 321–328</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D.V. Alekseevskii,   "Groups of conformal transformations of Riemannian spaces"  ''Math. USSR Sb.'' , '''18'''  (1973)  pp. 285–301  ''Mat. Sb.'' , '''89''' :  2  (1972)  pp. 280–296</TD></TR></table>