In this note, we study some conformal invariants of a Riemannian manifold (Mn, g) equipped with a smooth measure m. In particular, we show that there is a natural definition of the Ricci and scalar curvatures associated to such a space, both of which are conformally invariant. We also adapt the methods of Fefferman and Graham [Fefferman, C. & Graham, C. R. (1985) Astérisque, Numero Hors Serie, 95-116] and Graham, Jenne, Mason, and Sparling [Graham, C. R., Jenne, R., Mason, L. J., & Sparling, G. A. J. (1992) J. London Math. Soc. 46, 557-565] to construct families of conformally covariant operators defined on these spaces. Certain variational problems in this setting are considered, including a generalization of the Einstein-Hilbert action.
|Original language||English (US)|
|Number of pages||6|
|Journal||Proceedings of the National Academy of Sciences of the United States of America|
|State||Published - Feb 21 2006|
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