We study problems related to the metric of a Riemannian manifold with a particular focus on certain cohomogeneity one metrics. In Chapter 2 we study a set of cohomogeneity one Einstein metrics found by A. Dancer and M. Wang. We express these in terms of elementary functions and nd explicit sectional curvature formulae which are then used to determine sectional curvature asymptotics of the metrics. In Chapter 3 we construct a non-standard parametrix for the heat kernel on a product manifold with multiply warped Riemannian metric. The special feature of this parametrix is that it separates the contribution of the warping functions and the heat data on the factors; this cannot be achieved via the standard approach. In Chapter 4 we determine explicit formulae for the resolvent symbols associated with the Laplace Beltrami operator over a closed Riemannian manifold and apply these to motivate an alternative method for computing heat trace coecients. This method is entirely based on local computations and to illustrate this we recover geometric formulae for the heat coecients. Furthermore one can derive topological identities via this approach; to demonstrate this application we nd explicit formulae for the resolvent symbols associated with Laplace operators on a Riemann surface and recover the Riemann-Roch formula. In the nal chapter we report on an area of current research: we introduce a class of symbols for pseudodi erential operators on simple warped products which is closed under composition. We then extend the canonical trace to this setting, using a cut - o integral, and nd an explicit formula for the extension in terms of integrals over the factor.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:684435 |
| Date | January 2016 |
| Creators | Grieger, Elisabeth Sarah Francis |
| Contributors | Pushnitski, Alexander ; Scott, Simon Gareth |
| Publisher | King's College London (University of London) |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://kclpure.kcl.ac.uk/portal/en/theses/on-heat-kernel-methods-and-curvature-asymptotics-for-certain-cohomogeneity-one-riemannian-manifolds(40d2e3ab-3eb8-4141-bcc5-ca38e0705f65).html |
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