Available at: https://digitalcommons.calpoly.edu/theses/3005
Date of Award
6-2025
Degree Name
MS in Mathematics
Department/Program
Mathematics
College
College of Science and Mathematics
Advisor
Patrick Orson
Advisor Department
Mathematics
Advisor College
College of Science and Mathematics
Abstract
Let $M$ be a closed, connected, smooth manifold. What are the symmetries of $M$? From a geometric viewpoint, the symmetries of $M$ are precisely its isometries, the self maps which preserve lengths and angles. In the smooth category, the symmetries of $M$ are its diffeomorphisms, the self maps which are smooth and have a smooth inverse. Does expanding our notion of symmetries to include diffeomorphisms result in ``more'' symmetries in a meaningful sense? As a formal conjecture, the claim is that the isometry group of $M$ is a deformation retract of the diffeomorphism group of $M$. If $M$ is the $n$-sphere, this is the Smale conjecture, and it holds only in dimension less than four. In this thesis, we restrict $M$ to be a surface, and we prove the Earle-Eells theorem: if $M$ is neither the sphere, torus, projective plane, nor Klein bottle, then the isometry group of $M$ is a deformation retract of the diffeomorphism group of $M$. We follow Gramain's 1973 proof via Hatcher's 2014 exposition, rather than the original proofs of Earle and Eells in 1969 and 1970.