proof of criterion for conformal mapping of Riemannian spaces
The key observation is that the angle between curves and which intersect at a point is determined by the tangent vectors to these two curves (which we shall term and ) and the metric at that point, like so:
Moreover, given any tangent vector at a point, there will exist at least one curve to which it is the tangent. Also, the tangent vector to the image of a curve under a map is the pushforward of the tangent to the original curve under the map; for instance, the tangent to at is . Hence, the mapping is conformal if and only if
for all tangent vectors and to the manifold . Now, by elementary algebra, the above equation is equivalent to the requirement that there exist a scalar such that, for all and , it is the case that or, in other words, for some scalar field .
|Title||proof of criterion for conformal mapping of Riemannian spaces|
|Date of creation||2013-03-22 16:22:03|
|Last modified on||2013-03-22 16:22:03|
|Last modified by||rspuzio (6075)|