Sharp metric obstructions for quasi-Einstein metrics

Details Sharp metric obstructions for quasi-Einstein metrics Sharp metric obstructions for quasi-Einstein metrics

2011-10-13

arxiv.org/abs/1110.3010v1

Mathematics

Differential Geometry

23 pages

Scientific paper

Using the tractor calculus to study conformally warped manifolds, we adapt results of Gover and Nurowski to give sharp metric obstructions to the existence of quasi-Einstein metrics on suitably generic manifolds. We do this by introducing an analogue of the curvature tractor, itself the tractor analogue of the curvature of the Fefferman-Graham ambient metric. We then use these obstructions to produce a tensorial invariant which is polynomial in the Riemann curvature and its divergence, and which gives the desired obstruction. In particular, this leads to a generalization to arbitrary dimensions of an algorithm due to Bartnik and Tod for finding static metrics. We also explore the consequences of this work for gradient Ricci solitons, finding an obstruction to their existence on suitably generic manifolds, and observing an interesting similarity between the nonnegativity of the curvature tractor and Hamilton's matrix Harnack inequality.

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