View source for Conformal connection

A [[Differential-geometric structure|differential-geometric structure]] on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247301.png" />, a special form of a connection on a manifold when the smooth fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247302.png" /> with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247303.png" /> has as its typical fibre the conformal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247304.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247305.png" />. The structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247306.png" /> attaches to each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247307.png" /> a copy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247308.png" /> of the conformal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c0247309.png" />, which is identified (up to a conformal transformation preserving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473010.png" /> and all directions at it) with the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473011.png" />, extended by a point at infinity. The conformal connection as a connection in this space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473012.png" /> associates with each smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473013.png" /> with origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473014.png" /> and each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473015.png" /> of it, a conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473016.png" /> such that a certain condition is satisfied (see below for the condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473017.png" />). Suppose that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473018.png" /> is described by a frame consisting of two points (vertices) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473019.png" /> mutually-orthogonal hypersurfaces passing through them. Such a frame is interpreted in the pseudo-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473020.png" /> as an equivalence class of bases satisfying the conditions

<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/c024/c024730/c02473021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

with respect to the equivalence

<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/c024/c024730/c02473022.png" /></td> </tr></table>

Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473023.png" /> is covered by coordinate regions and that in each domain a smooth field of frames in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473024.png" /> is fixed, such that the vertex defined by the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473025.png" /> is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473026.png" />. The condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473027.png" /> is as follows: As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473028.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473029.png" /> is displaced along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473030.png" /> towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473032.png" /> must converge to the identity mapping, and the principal part of its deviation from the latter must be defined, relative to the field of the frame in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473033.png" />, by a matrix 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/c024/c024730/c02473034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</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/c024/c024730/c02473035.png" /></td> </tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473036.png" /> linear differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473039.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473041.png" />, of type

<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/c024/c024730/c02473042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

In other words, the image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473043.png" /> of the frame at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473044.png" /> must be defined by the vectors

<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/c024/c024730/c02473045.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473046.png" /> is the tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473047.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473048.png" /> and

<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/c024/c024730/c02473049.png" /></td> </tr></table>

Under a transformation of the frame of the field at an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473050.png" /> according to the formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473052.png" />, preserving condition (1), that is, under a passage to an arbitrary element of the principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473053.png" /> of conformal frames in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473054.png" />, the forms (3) are replaced by the following <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473055.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473056.png" />:

<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/c024/c024730/c02473057.png" /></td> </tr></table>

that also form a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473058.png" /> of the form (2). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473059.png" />-forms

<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/c024/c024730/c02473060.png" /></td> </tr></table>

form a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473061.png" /> of the same structure as (2) and are expressed by the formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473062.png" /> in terms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473063.png" />, which in view of (3) are linear combinations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473064.png" /> and hence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473065.png" />. For elements of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473066.png" /> one has the structure equations of a conformal connection (where for simplicity the primes are omitted):

<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/c024/c024730/c02473067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4a)</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/c024/c024730/c02473068.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4b)</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/c024/c024730/c02473069.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4c)</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/c024/c024730/c02473070.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4d)</td></tr></table>

Here the right-hand sides are semi-basic, that is, they are linear combinations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473071.png" /> only; they form a system of torsion-curvature forms of the conformal connection and are transformed according to the rules

<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/c024/c024730/c02473072.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/c024/c024730/c02473073.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/c024/c024730/c02473074.png" /></td> </tr></table>

The equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473075.png" /> have an invariant sense and determine a conformal connection of zero torsion. 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/c024/c024730/c02473076.png" /></td> </tr></table>

Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473077.png" />:

<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/c024/c024730/c02473078.png" /></td> </tr></table>

and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473079.png" />:

<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/c024/c024730/c02473080.png" /></td> </tr></table>

The invariant identities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473082.png" /> determine the special class of so-called (Cartan) normal conformal connections.

The forms (3), forming a matrix of type (2), uniquely determine the conformal connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473083.png" />: The image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473084.png" /> of the frame at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473085.png" /> is defined by the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473086.png" /> of the system

<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/c024/c024730/c02473087.png" /></td> </tr></table>

with initial conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473088.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473089.png" /> are the equations of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473090.png" /> in some coordinate neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473091.png" /> of it with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473092.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473093.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473096.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473098.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c02473099.png" /> satisfying equations (4a)–(4d) with right-hand sides expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730100.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730101.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730102.png" />) are linearly independent, determine a conformal connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730103.png" /> in the above sense.

Conformal connections provide a convenient apparatus for the study of conformal mappings of Riemannian spaces. A conformal connection reduces to the [[Levi-Civita connection|Levi-Civita connection]] of some Riemannian space if there exists local fields of frames on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730104.png" /> with respect to which

<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/c024/c024730/c024730105.png" /></td> </tr></table>

For the curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730106.png" /> of this connection, defined by the equation

<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/c024/c024730/c024730107.png" /></td> </tr></table>

one has

<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/c024/c024730/c024730108.png" /></td> </tr></table>

Conversely, for each Levi-Civita connection of a Riemannian space there exists a unique normal conformal connection from which it is obtained in the above way. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730110.png" /> is expressed in terms of the Ricci tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730111.png" /> and the scalar curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730112.png" /> by the formula

<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/c024/c024730/c024730113.png" /></td> </tr></table>

The corresponding tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730114.png" /> is called the conformal curvature tensor of the Levi-Civita connection. Two Riemannian spaces are conformally equivalent if their Levi-Civita connections have the same normal conformal connections. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730115.png" />, a Riemannian space is conformally Euclidean if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730116.png" /> for it.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,   "Les espaces à connexion conforme"  ''Ann. Soc. Polon. Math.'' , '''2'''  (1923)  pp. 171–221</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Ogiue,   "Theory of conformal connections"  ''Kodai Math. Sem. Reports'' , '''19'''  (1967)  pp. 193–224</TD></TR></table>



====Comments====
Except when stated otherwise, Greek indices run from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730117.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730118.png" /> and Latin indices run from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730119.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024730/c024730120.png" /> in the article above.

For the notion of principal part (of a bundle mapping) cf. the editorial comments to [[Connections on a manifold|Connections on a manifold]].