Last Friday I had a short conversation with a fellow graduate student about mathematics in physics, particularly about topology. He was curious as to whether I had taken any topology courses as an undergraduate, after the professor and I had been chatting a bit about differential geometry (I have hence been referred to as “the math guy”…a title that is not altogether disagreeable). I told him that I had not, as I preferred to study space (geometry) instead of shape (topology). I remarked, however, that they are of course intimately connected, particularly when examining structure (algebra). He said that the professor had told him that quantum mechanics was all about topology, but I think that may be a bit of an overstatement. Quantum mechanics is really an application of linear algebra, and though topological effects are evident and the field can certainly be applied in many cases, I would say that QM’s dependence on topology is really through the language of linear algebra, and not directly through homeomorphisms or homotopy. Of course, he asked how I could know if I hadn’t taken a topology class. Though my favorite subject was differential geometry (calculus is the study of change, so differential geometry studies how a space changes as you move along\/through it), any math major will pick up bits of other math subjects by virtue of his\/her studies and conversations with fellow students\/professors. At that point I was reminded of a beautiful theorem that related differential geometry and topology: the Gauss-Bonnet Theorem.<\/p>\n
The Gauss-Bonnet Theorem states:<\/p>\\[\\int_M K\\;dA+\\int_{\\partial M}k_g\\;ds=2\\pi\\chi(M)\\]\n
where \\(M\\) is a compact two-dimensional manifold (locally Euclidean surface), \\(K\\) is the Gaussian curvature, \\(\\partial M\\) is the boundary of the manifold, \\(k_g\\) is the geodesic curvature of the boundary, and \\(\\chi(M)\\) is the Euler characteristic of the manifold. Here the curvatures, a topic from differential geometry, are directly related to the Euler characteristic, a topological invariant. The theorem can be generalized to an arbitrary even-dimensional space, but it is easiest to see the profundity when we consider a closed manifold, where the boundary term is zero.<\/p>\n
For a closed (orientable) surface, the Euler characteristic is given by \\(\\chi(M)=2(1-g)\\), where \\(g\\) is the genus<\/em> of the shape, i.e., the number of holes. So a sphere has genus 0, a torus (doughnut) has genus 1, a two-person inner-tube has genus 2, etc. In general, the Euler characteristic of a polyhedron is \\(\\chi=F-E+V\\), where \\(F\\) is the number of faces, \\(E\\) the number of edges, and \\(V\\) the number of vertices. It is amazing enough that this quantity is an invariant: for a cube, we have \\(F-E+V=6-12+8=2\\); in fact \\(\\chi=2\\) for ANY convex polyhedron. What is even more amazing is that you can partition your curved surface into some number of faces, edges, and vertices using any curves (not even just geodesics), and you will get the same number no matter how you divide it up, no matter how many faces\/edges. And this number will depend only on how many holes the surface has. The fact that the total Gaussian curvature is \\(2\\pi\\) times an integer is not a priori obvious, and I find this relation to be quite amazing. For example, a torus has genus 1, so its Euler characteristic \\(\\chi(M)=2(1-g)=0\\), which by the Gauss-Bonnet Theorem means that the total Gaussian curvature is zero. This means that for every area curving outward, there must be some area curving inward to cancel in the integral. Hence the curving on the interior of the hole exactly cancels the curving on the rest of the surface!<\/p>\n\n\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"pagelayer_contact_templates":[],"_pagelayer_content":"","footnotes":""},"categories":[7],"tags":[],"class_list":["post-60","post","type-post","status-publish","format-standard","hentry","category-mathematics"],"_links":{"self":[{"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/posts\/60","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/comments?post=60"}],"version-history":[{"count":1,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/posts\/60\/revisions"}],"predecessor-version":[{"id":154,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/posts\/60\/revisions\/154"}],"wp:attachment":[{"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/media?parent=60"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/categories?post=60"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/quantum-immortal.net\/blog\/wp-json\/wp\/v2\/tags?post=60"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}