The Yamabe Invariant in Two Dimensions
In the case that, (so that M is a closed surface) the Einstein–Hilbert functional is given by
where is the Gauss curvature of g. However, by the Gauss–Bonnet theorem, the integral of the Gauss curvature is given by, where is the Euler characteristic of M. In particular, this number does not depend on the choice of metric. Therefore, for surfaces, we conclude that
For example, the 2-sphere has Yamabe invariant equal to, and the 2-torus has Yamabe invariant equal to zero.
Read more about this topic: Yamabe Invariant
Famous quotes containing the word dimensions:
“It seems to me that we do not know nearly enough about ourselves; that we do not often enough wonder if our lives, or some events and times in our lives, may not be analogues or metaphors or echoes of evolvements and happenings going on in other people?—or animals?—even forests or oceans or rocks?—in this world of ours or, even, in worlds or dimensions elsewhere.”
—Doris Lessing (b. 1919)