From: "G. A. Edgar" Subject: Re: Godel's Theorem Date: Sun, 28 May 2000 09:29:36 -0400 Newsgroups: sci.math Summary: [missing] In article <393089E0.9C3F7DBB@home.com>, Rajarshi Ray wrote: > In the same vein can it > be shown whether Geometry is less/more rich than Arithmetic? Tarski showed that the first-order theory of Euclidean geometry is decidable. So (in your sense) it is less rich than the first-order theory of the integers. -- Gerald A. Edgar edgar@math.ohio-state.edu ============================================================================== From: "W. Dale Hall" Subject: Re: What is differential geometry (esp. w.r.t computability theory)? Date: Wed, 29 Nov 2000 02:01:11 -0800 Newsgroups: sci.math lemma_one@my-deja.com wrote: > > I would like to know what differential geometry is, but can't find a > "reasonably intuitive definition." Perhaps one really doesn't exist, > which would be unfortunate since such definitions exist for analysis, > set theory, logic, number theory, and every other branch hitherto I have > encountered. The definitions I have seen immediately start talking about > "manifolds", and I sure haven't haven't found any "reasonably intuitive > definition" of these things. With mathworld.wolfram.com down, it is hard > for me to locate decent definitions of terms I don't know. > Here's a shot: a manifold (of dimension n) is a locally-(n-dimensional) Euclidean space, i.e., a space in which each location appears to be in an n-dimensional Euclidean space, if you only look nearby. Examples: any vector space over R is a manifold of dimension equal to the vector space dimension. The circle is a 1-dimensional manifold, the sphere and torus are 2-dimensional manifolds. If m > n, the pre-image f^(-1)(x) by a function f:R^m --> R^n, of any point x where the Jacobian (matrix of partials of the component functions , D_i (f_j)) has rank n, is a manifold of dimension m-n. Differential Geometry studies manifolds that have (1) suitably differentiable defining relations, and (2) some imposed geometric structure, such as a metric of some sort. The notions of curvature, geodesics, parallel transport along curves, and others, come from the field of Differential Geometry. > Perhaps this would help me: what background is good for studying > differential geometry? What kinds of people typically like to study it? > Analysis types? Dif. Eq. people? Logicians and set theorists? Number > theorists? Algebraists? > Differential Geometry is a branch of Analysis, but it has considerable overlap with topology. Algebraists may have interest in the field due to its applicability to the study of Lie groups. DE people may find applications of a number of PDE's in the field, but I don't know what logicians and set theorists would find of interest. > I saw an announcement recently that Robert I. Soare is giving a lecture > on Computability Theory and Differential Geometry. I know what > computability theory is. Are these two areas particularly related? > > Sent via Deja.com http://www.deja.com/ > Before you buy. I would guess that there may be some particularly intractible problems in the field, but don't know of any offhand. Dlae. ============================================================================== From: foltinek@math.utexas.edu (Kevin Foltinek) Subject: Re: What is differential geometry (esp. w.r.t computability theory)? Date: 29 Nov 2000 14:26:23 -0600 Newsgroups: sci.math In article <901tpc$14h$1@nnrp1.deja.com> lemma_one@my-deja.com writes: > I would like to know what differential geometry is, but can't find a > "reasonably intuitive definition." Reasonably intuitive definition: Differential geometry is the study of smooth things which don't depend on how you look at them. (The word "smooth" is important there - without it, you would be looking at topology, algebra, and lots of other things as well. I'll say more about what "smooth" means below.) There are two branches of the subject: local differential geometry and global differential geometry. Local differential geometry is the study of "invariants" which don't depend on the coordinate system you are using (or, in a sort of dual interpretation, the study of the coordinate transformations, or "symmetries", which don't change a certain thing). Global differential geometry relates the local geometry to much more topological properties. There's a theorem called the "Gauss-Bonnet theorem", which says that the integral of the curvature of a surface is some constant times the Euler characteristic. The curvature is something which comes directly out of local differential geometry; the Euler characteristic is strictly topological. The Gauss-Bonnet theorem is a result from global differential geometry. > The definitions I have seen immediately start talking about > "manifolds", and I sure haven't haven't found any "reasonably intuitive > definition" of these things. Very intuitive definition: a manifold is a set (actually a topological space) which, in a small enough neighbourhood, looks like R^n. (Note that I said R^n, not "Euclidean space"; a manifold does not automatically come with a distance function or anything like that.) If you're really paying attention, you'll notice that I just contradicted myself: first I mentioned a "small" neighbourhood, then I said that a manifold has no distance, hence no meaning of "small". I also didn't define "looks like". These two points are where the technical details start to become necessary. "Small neighbourhood" really means that "there is some open set"; the union of open sets is open, and the point of "small neighbourhood" is that if you take the union of too many open sets, you might lose the property that you're interested in (in this case, that it looks like R^n). Right away, we see the "local" nature of differential geometry. So what does it mean for a small open set to "look like" R^n? Well, since we have a topology, we could say that it is homeomorphic to an open set in R^n. But this isn't quite enough for the purposes of differential geometry; there's that differential word. We'd like to say that it's diffeomorphic to an open set in R^n. Unfortunately we can't, because we don't know what it means to differentiate a function defined on a topological space; so we do the next best thing, and say that if we go from R^n, through a homeomorphism to our manifold, then through another homeomorphism back to R^n, this composition from an open subset of R^n to another open subset of R^n must be a diffeomorphism. (In fact, it's not quite right that a manifold is a [...] which looks like R^n; it's actually necessary to say that a manifold is a [...] with a concrete method of *making* it look like R^n. Sometimes the same topological space can be made to look like R^n in two ways which are different in the differential sense.) You might want to think about the requirement that the composition I described above is a diffeomorphism. The homeomorphism to an open set in R^n is a coordinate system; the composition is a change of coordinates. Contained in the very definition of manifold is the idea that the differential structure (i.e., what it means to differentiate) is independent of coordinates; so there is no ambiguity about whether something is "smooth" or not. Now, put some additional structure onto this manifold. This structure could be something like a Riemannian metric, a control system, a PDE, a complex or almost-complex structure, etc. Usually (in practice, i.e., if you're an engineer) you do this by choosing coordinates and writing something down. It's fairly straightforward to figure out what the thing would look like in a different coordinate system, in other words, what would you write down in the other coordinates. In this sense, these two things that you wrote down are really (after changing coordinates) the same thing; the point of local differential geometry is to be able to extract the "minimal" set of information which allows you to construct the thing you wrote down. One point of global geometry is to understand if, and how, you can extend the thing to the entire manifold; or, what kinds of manifolds are required in order that such a thing exist globally. > Perhaps this would help me: what background is good for studying > differential geometry? This depends on what sort of differential geometry. Global is usually pretty heavy on analysis (the PDE sort of analysis); local is rather algebraic (rings, group representations, etc.). Global seems to be much more "popular" these days. > I saw an announcement recently that Robert I. Soare is giving a lecture > on Computability Theory and Differential Geometry. I know what > computability theory is. Are these two areas particularly related? Can't help you here. I don't think they are "particularly related" (in the sense that, say, algebraic geometry and complex analysis are), but one never knows. I would guess it to be much more likely that a question in one subject can be answered by (techniques from) the other subject. Kevin. ============================================================================== From: foltinek@math.utexas.edu (Kevin Foltinek) Subject: Re: What is differential geometry (esp. w.r.t computability theory)? Date: 01 Dec 2000 11:06:46 -0600 Newsgroups: sci.math In article <903is5$bca$1@nnrp1.deja.com> lemma_one@my-deja.com writes: > I doubt some mathematician > woke up one day and thought, "today I'm going to define a manifold as a > space in which... and then I'm going to create a branch of mathematics > called differential geometry where we study these objects." This is similar to what did actually happen. Look up Riemann's "Inaugural Lecture". (Spivak's five volume "Comprehensive Introduction to Differential Geometry" contains a translation and a small bit of historical information. I think it's in Volume 2 but I'm not sure because someone has borrowed my copy from me. And I think it's quite readable even if you don't already know differential geometry though of course you might not appreciate all the details.) Riemann's ideas in this lecture were, I think, generalizations of what Gauss had done. Gauss studied surfaces in space, motivated I think by his surveying work (in which it became clear that the sum of the angles in a triangle did not add up to 180 degrees, if your triangles are on a sphere). This work on surfaces in space was very naturally motivated by the fact that we live on one. Riemann's ideas involved (partly) trying to see what could be said about surfaces without using the fact that they sit in space, and (partly) extending them to higher dimensions. Both of these are obvious generalizations (obvious to Riemann, at least, and today they would be considered obvious by most). The major topic of Riemann's lecture was determining when a surface was flat, which he phrased in terms of trying to find a coordinate system in which (what is now called) the Riemannian metric was the Euclidean metric, and he found the obstruction which is now called the Riemann curvature tensor. I don't know where the modern ideas about an atlas of coordinate charts (the modern rigorous interpretation of "looks locally like R^n") originated, or who was involved in their formulation, but it is rather obvious that something like that needed to be done, in order to make rigorous the idea of "change of coordinates" on an object which is globally not the same as R^n; and clearly, if you want to generalize surfaces (very few of which look like R^n globally) in the two ways which Riemann did, you need to do something like this. Kevin.