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The Ricci flow on two-dimensional almost-Riemannian manifolds

Ridder, Luuk de (2021) The Ricci flow on two-dimensional almost-Riemannian manifolds. Bachelor's Thesis, Applied Mathematics.

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Abstract

The Ricci flow has shown to be a powerful tool in the study of Riemannian geometry. In this setting, we provide explicit proofs for the isometric invariance of the Levi-Civita connection, Riemann curvature tensor and, subsequently, the Ricci curvature. The latter allows one to show that the Ricci flow is invariant under the infinite dimensional group of diffeomorphisms. This causes the Ricci flow, characterized as a heat-type non-linear partial differential equation, to be weakly parabolic. In general, existence and uniqueness theorems only apply to strongly parabolic equations. With DeTurck’s trick, we show that on a closed Riemannian manifold of any dimension, existence and uniqueness of short-time solutions to the Ricci flow can still be obtained. In addition, we investigate the evolution of the Ricci flow on two-dimensional almost-Riemannian structures (2-ARS) on compact, oriented and connected smooth manifolds. These are generalized Riemannian structures on surfaces for which an orthonormal frame is obtained from a pair of vector fields that satisfy the Hörmander condition. The vector fields can become collinear at certain points, which as a collection define a singular set. If one removes the singular set from the connected manifold, one obtains two regular Riemannian structures on both parts of the surface. However, these are non-complete Riemannian surfaces with boundary, giving rise to difficulties regarding the evolution of Ricci flow on these surfaces.

Item Type: Thesis (Bachelor's Thesis)
Supervisor name: Seri, M. and Veen, R.I. van der
Degree programme: Applied Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 20 Jul 2021 12:54
Last Modified: 20 Jul 2021 12:54
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/25347

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