Abstract Nonsense

Differential Geometry of Curves and Surfaces

Point of Post: In this post we motivate the upcoming discussion of the differential geometry of curves and surfaces.

Motivation

So, we begin discussing another topic, “Differential Geometry of Curves and Surfaces”. The first questions is (as it always should be)–what and why? Namely, what is the differential geometry of curves and surfaces and why is it interesting/important. The first of these questions is easily enough answered. Differential geometry of curves and surfaces (just called differential geometry when there is no confusion, or even just geometry) is the study of surfaces, mostly living in $\mathbb{R}^3$, as well as their lower dimensional counterparts, curves. In particular, we’d like to study the aspects of curves and surfaces which are preserved under the kinds of transformations that don’t ‘stretch’. But, let’s try not to just relegate the motivation to ‘studying maps that don’t stretch’, let’s really talk about what we are trying to preserve. Ostensibly topology and differential geometry seem quite similar–they are both studying ‘geometric objects’ and the properties of these objects that are invariant under certain ‘admissible’ transformations. So, what precisely is the difference–what makes geometry..geometry? The key distinction to make between geometry and topology is the local vs. global phenomenon. Namely, one could say that a topologist is concerned with ‘global’ properties of geometric objects and geometers are concerned with ‘local’ properties. Take for example the classic example of the sphere and doughnut. Everyone and their kid sister  knows that the sphere and the doughnut are not ‘homeomorphic’, but that they are ‘locally homeomorphic’. A reasonable way to explain what this means is that locally the two objects look alike in the sense that if you pick your two favorite points, one on the coffee cup and one on the doughnut, and cut out an extremely tiny piece of the object around each point, the cut-outs can be made to look exactly alike with the right amount of topologically admissible massaging (i.e. bending and stretching but no tearing of holes). That said, the entire objects themselves aren’t homeomorphic because one has ‘a hole’ and the other doesn’t. Geometers are much more discerning. You give two small little neighborhoods on each of the objects to the geometer, and they’d be able to tell you immediately that the surface of a doughnut and a sphere are not ‘isomorphic’ (whatever that means).  To beat a dead horse, geometers could tell you that two spaces are different by putting them on each and locally measuring things (what they measure, we shall soon see). Hopefully this gives one a very, very, very vague idea of what geometry is about.

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So, why is it interesting? As is mathematicians plight, to convince ninety-nine percent of the population as to why a subject is interesting one must appeal to either physics or shakey real-world parlor tricks. So, let’s get that out of the way. For the first of these, one cannot do modern physics without knowing relativity (or at least I am told). Notions of curving spacetime and the like are entirely within the realm of differential geometry. Trying to build turns in roads such that a car going ___ miles and hour will not skid off of it, differential geometry. The list goes on and on. So, what about the second? What ‘parlor tricks’ (i.e. applications of mathematical theorems to the real world, that assume a lot about the situation) can we product with differential geometry? Well, have you ever tried gift wrapping a basketball? If so, or if you have watched someone else do so, you know that it’s not an easy task. Go ahead and try if you haven’t. If you can’t, don’t feel bad…you can’t do it. Literally, differential geometry will show that one cannot wrap a piece of paper around a ball without ripping it at some point–cool huh?

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So, why would a mathematician find geometry (at this level) interesting? Besides the obvious fact that all math is useful (you’d be shocked how often I hear professional mathematicians say “Oh, I was doing ___ in subject X and couldn’t possibly solve it until I remembered this random theorem from subject Y!” there is one other very particularly useful reason for studying geometry. Namely, geometry of curves and surfaces provides excellent motivation and intuitive backing for more advanced differential geometry and differential topology. And, to be more down to earth, it’s just plain cool.

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