Reddit reviews Introduction to Smooth Manifolds (Graduate Texts in Mathematics)

We need to make a few definitions.

A group is a set G together with a pair of functions: composition GxG -> G and inverse G -> G, satisfying certain properties, as I'm sure you know.

A topological group is a group G which is also a topological space and such that the composition and inverse functions are continuous. It makes sense to ask if a topological group for example is connected. Every group is a topological group with the discrete topology, but in general there is no way to assign an interesting (whatever that means) topology to a group. The topology is extra information that comes with a topological group.

A Lie group is more than a topological group. A Lie group is a group G that is also a smooth manifold and such that the composition and inverse are smooth functions (between manifolds).

In the same way that O(n) is the set of matrices which fix the standard Euclidean metric on R^n, the Lorentz group O(3,1) is the set of invertible 4x4 matrices which fix the Minkowski metric on R^4. The Lorentz group inherits a natural topology from the set of all 4x4 matrices which is homeomorphic to R^16. It is some more work to show that the Lorentz group in fact smooth, that is, a Lie group.

It is easy to see the Lorentz group is not connected: it contains orientation preserving (det 1) matrices and orientation reversing (det -1) matrices. All elements are invertible (det nonzero), so the preimage of R+ and R- under the determinant are disjoint connected components of the Lorentz group.

There are lots of references. Munkres Topology has a section on topological groups. Stillwell's Naive Lie Theory seems like a great undergraduate introduction to basic Lie groups, although he restricts to matrix Lie groups and does not discuss manifolds. To really make sense of Lie theory, you also need to understand smooth manifolds. Lee's excellent Introduction to Smooth Manifolds is an outstanding introduction to both. There are lots of other good books out there, but this should be enough to get you started.

>is there really no link to the role of this one-form dx and the role of the differential dx?

The differential dx is the one-form "dx", they're the same thing. The differential means nothing. In integration, there's really no need to have the "dx" and when you first do integration in Real Analysis it is usually omitted. If you do measure theory, then you may see d(mu), and this is just to represent the measure against which you're going the integration. It's a bookkeeping device. You can think of "Inta^(b) f(x) dx" as being analogous to "Sumi=a^(b) si". Limits of sums are analogous to limits of integrals, the summands are analogous to the integrand and "dx" is analogous to "i=", it's the same thing just in a different location.

In general, if M is an n-manifold, then it's space of n-forms is one dimensional. This means that it is equivalent to all things of the form w=f(x)dx1dx2...dxn (where these are wedge products). We can then view the integral as a linear function from n-forms to the real numbers. If we want to find Int(w), then we can cut up the manifold into flat pieces using Partitions of Unity, integrate the function f(x) over each of these patches using standard analysis, and then sum it all up.

If we have a line integral of a vector field on M, say the integral of (f(x,y),g(x,y)) along some curve C, then we usually write this as "IntC(f,g)·ds" and usually, we write ds=s'(t)dt so the integral is equal to "Int0^(1)(f,g)·s'(t)dt". What we have a function s:[0,1]->M and a 1-form w=fdx+gdy and we're using Pullbacks to pull the 1-form w on M into a 1-form s^()(w)=(f,g)·s'(t)dt on the manifold [0,1]. We then use standard integration (since this is a 1-form on a 1-dimensional manifold) to integrate.

Something like a curve being embedded into a manifold, like above, is called a 1-Simplex and we can view the integral as pairing k-forms with k-simplexes and returning a real number, via integration of pullbacks. Stokes Theorem, which generalizes the divergence theorem, Green's Theorem, and the Fundamental Theorem of Calculus, is a specific statement about this kind of pairing. Generally, we can learn about a k-form (aka vector field) by how it integrates along these simplexes. Things like the Maxwell Equations are specific statements about what we can learn about these k-forms via integration. We can use Stokes Theorem to then, instead, treat them as statements about k-forms themselves rather than having to use integrals. The fact that if F is the electromagnetic force, then there is a 1-form A so that F=dA already takes care of half of Maxwell's equations.

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As for the d operator, if we have a 0-form f(x,y) (aka smooth function), then how are we going to get a 1-form? This is a 1-dimensional thing going into a 2-dimensional thing. What we do is see how f(x,y) interacts with both basis elements and see that we should get fxdx+fydy. This definition does not depend on the basis, so this means that for every 0-form f, we get a natural 1-form df. If we have a 1-form (now in 3D), w=Adx+Bdy+Cdz, where A,B,C are any three smooth functions (they don't have to be the respective partial derivatives of a single function), then how can I get a 2-form? The basis for the 2-forms is dxdy, dxdz and dydz (pretend these are wedges). I can play the same game, see how all the components compare to larger ones. This means I wedge w by each dx,dy,dz and reduce things, so wdx is Axdxdx+ Bxdydx+Cxdzdx = -Bxdxdy-Cxdxdz. Doing this kind of things for all the ones gives the standard formula for dw. We're essentially just combining all the possible wedges and seeing what we get. Following these, we'll always get zero after two successive applications. This is, essentially, because of combinatorics and the fact that partial derivatives commute. In the end, it doesn't depend on basis, so it's natural. The differential operator is just applying derivatives to differential forms in all possible combinations, adding them together and reducing the wedges.

Most importantly, the function d:Tk^() -> Tk+1^(*) so that d^(2)=0, df is the above function and d behaves well under the wedge product. These are the things that matter.

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How I see it, visualizations are a crutch. They're good for a little, but you can't run unless you give them up. Not being able to do Differential Geometry intuitively without having to visualize and interpret everything will eventually become taxing. If, however, the manipulation of the symbols becomes your intuition, then you'll be able to do much more. Visualization is good in Calc 3, but this should be seen as the time to get a feel for the symbols. Differential Geoemtry is glorified Calc 3, but everything is much more abstract and making it concrete will just give you Calc 3 in the end, rather than Differential Geometry. The physical interpretation of Maxwell's equations is elevated to statements in Differential Geometry. These are a lot more powerful, and the definitions are essentially a guidebook to recovering the physical interpretations when you actually need to compute things. I find the best way to gain an intuition for purely symbolic stuff is to use it, accept it for what it is and just go. Occasionally, take a step back, follow the definitions back to the familiar so that you can see how what you do abstractly actually is in line with what you already know.

Of course, I'm a number theorist, I'm pretty biased against physical interpretations. So maybe I'm not completely fair there.

As for references, I've heard that Lee and Spivak are good.

EDIT: As for your edit, I mean that for every smooth function there is an associated 1-form. If f is a smooth function, and D is an element of the tangent space, then D(f) is a real number. We can then view f as a map of tangent vectors D -> D(f). This means that f can be viewed as an element in the cotangent space. The associated cotangent vector is df.

I love Jack Lee's series on manifolds:

Introduction to Topological Manifolds

Introduction to Smooth Manifolds

I've heard Munkres' Topology is fantastic as an introduction to general topology, but never read it myself.

what I meant is this one

That's quite the opinion, my mathematician friend. I'm sure it's not all that off-putting from a physicist's perspective. The reason I called it insightful was strictly for its geometric description of contravariance and covariance (with respect to an orthonormal basis). The diagram it provides (on page six) is one of the most enlightening ones I've ever had presented to me, for it really clarified in my head why the metric tensor in the Euclidean plane can be taken as the Kronecker delta. Sometimes it's nice to just have something to hang your hat on so that you can move on with your own research.

Though if any physicists are looking for a nice introduction to differential geometry, which is the landscape for these concepts, I highly recommend John Lee's Intro to Smooth Manifolds. I agree that it's enlightening and serves one well to have a firm understanding of geometry.

edit: justified some comments