Stuggling with the abstraction of manifolds

[dyn]

Hi. I have a physics background but I am trying to get to grips with differential geometry and struggling with the abstract nature of it. I have a few questions if anyone can help ?
Is a smooth manifold the same as a differentiable manifold ? Does it have to be infinitely differentiable ? Is 3-D Euclidean space infinitely differentiable ?
Is a vector field in 3-D just the gradient operator ?
I have some notes that say a vector field maps a manifold to itself by differentiating a 1-parameter family of maps. I thought a vector field was basically a tangent vector at every point and so was a different manifold ?

 

[Matterwave]

A smooth manifold is an infinitely differentiable manifold (sometimes called a C-infinite manifold). 3-D Euclidean space is smooth. A vector field is not "just the gradient operator", there is just 1 gradient operator in 3-D Euclidean space, but there are many vector fields. I think you are confused with the 1-1 correspondence between vectors and directional derivatives. A vector field defines a directional derivative at every point on the manifold, the directions can change (smoothly) and so can the magnitude.

A vector field is indeed a tangent vector at every point, I don't know what you mean by "so was a different manifold", a vector field is not a manifold. The tangent space + the base manifold make a fiber bundle manifold called the tangent bundle, maybe that is what you're thinking of?

A vector field defines a 1 parameter family of mappings of a manifold onto itself. This is because a vector field defines a set of integral curves (curves which are tangent to the vector field at every point), and the 1 parameter family of mappings can be defined as moving a set parameter along those integral curves of the vector field. This is called a Lie dragging of the manifold.

 

[WWGD]

When you talk about a manifold being k-times differentiable, you (usually, at least ) mean that the chart maps are k-times differentiable. This would imply that Euclidean k-space (use, e.g., identity charts, together with the result that differentiability is independent of the choice of charts ) is infinitely-differentiable as a manifold.

Like Matterwave says, manifold tangent spaces are spanned by gradient operators; these tangent vectors are/form vector spaces (of dimension equal to that of the manifold in which they live), and the three standard gradients are bases, just like, e.g., the vectors {(1,0,..0), (0,1,0,..0),..,(0,0,..,0,1)} are a basis for Euclidean space $\mathbb R^n$. This means that every vector field in a 3-manifold is a linear combination of these gradient vectors (of course, you may choose a different basis; see, e.g., the frame bundle for more on this.)

And also as Matterwave said (helps to have repetition in slightly-different terms to have ideas sink-in ), vector fields do give rise to integral curves and to flows, which are diffeomorphisms of the manifold to itself. Picard's theorem guarantees the (local) existence of integral curves associated to a vector field V, i.e., curves C(t) with C'(t)= V(t) where the tangent vector field to the curve agrees with the value of the vector field at the same parameter point ( see, e.g., Hopf-Rinow for results re the global existence of integral curves). Note that this is not true in higher-dimensional cases, i.e., there may not be n-submanifolds of a manifold M whose tangent bundles agree with an assignment of n-planes at each point ( an integral curve is a 1-submanifold whose tangent space agrees with a given vector field). This last falls under the issue of integrability of manifolds, re Frobenius ' theorem.

 

[homeomorphic]

My advice if you find it too abstract is to spend some time reading about the 3-dimensional case, which is basically just a souped-up version of calculus 3, almost. There are a lot of books on the geometry of curves and surfaces. My favorite is O'Neill.

 

[dyn]

On another thread I posted that I have a book called Elementary Differential Geometry by Pressley which covers the 3-D case but it contains none of the stuff relevant to the notes I have such as manifolds and differential forms. I am looking for an intro book that covers the relevant stuff but I need one with worked examples so I can try and see what is going on.
In 3-D Euclidean space the manifold is just x , y ,z so how is this infinitely differentiable ? If I differentiate once I get constants , differentiate again and it becomes zero ?

 

[WWGD]

On another thread I posted that I have a book called Elementary Differential Geometry by Pressley which covers the 3-D case but it contains none of the stuff relevant to the notes I have such as manifolds and differential forms. I am looking for an intro book that covers the relevant stuff but I need one with worked examples so I can try and see what is going on.
In 3-D Euclidean space the manifold is just x , y ,z so how is this infinitely differentiable ? If I differentiate once I get constants , differentiate again and it becomes zero ?

You need to understand basic aspects of charts, and specifically of chart overlaps, to understand the meaning of a C^k -manifold. The charts describe an homeorphism from a subset of your manifold into R^n , for an n-dimensional manifolds. If two subsets intersect, you want the local
homeomorphisms to satisfy certain properties under composition. Do you know how this works?

And I don't want to force Mathematical formalism on you, but you need to be a bit more clear when you say things like " the manifold is just (x,y,z)" ; I do know what you mean, but it would be a good idea to use more standard language until you learn the theory. Yes, the chart maps (and the overlaps) are the identity, and the identity is infinitely-differentiable.

 

[homeomorphic]

On another thread I posted that I have a book called Elementary Differential Geometry by Pressley which covers the 3-D case but it contains none of the stuff relevant to the notes I have such as manifolds and differential forms. I am looking for an intro book that covers the relevant stuff but I need one with worked examples so I can try and see what is going on.

O'Neill sounds like a good fit, then. It explains what differential forms are and manifolds, and even has worked examples (I think that's not so critical, personally, but it has them). But you have to be patient before you get to the point where it connects up to the stuff you are trying to learn. It's going to take some effort for it to pay off, but if you study surfaces, after a while, you will be able to make the connection. If you want things to be less abstract, it will take a while to get there.

 

[dyn]

Thanks. What is the name of the O'Neill book ? As regards infinitely differentiable , if f(x) = x is infinitely differentiable then what isn't infinitely differentiable ?

 

[Matterwave]

Thanks. What is the name of the O'Neill book ? As regards infinitely differentiable , if f(x) = x is infinitely differentiable then what isn't infinitely differentiable ?

For example $f(x)=|x|$ is not even once differentiable at $x=0$. Non-differentiable functions tend to have kinks in them.

 

[homeomorphic]

What is the name of the O'Neill book ?

Elementary Differential Geometry. It's beneficial to start with curves, although you probably won't see the relevance to what you want to learn at first. One thing I like to do is to use my thumb, index finger, and middle finger to make an orthogonal basis for R^3 to convince myself of various facts involving frames intuitively, without computation (for example, you can immediately write the Frenet formulas from this description, if you think hard enough). The formal derivation typically involves this product rule trick that you'll see again and again, but that sometimes hides the intuition of how it works. For curves, my index finger represents the unit tangent vector to the curve, the middle finger will be the derivative of that, which will be perpendicular (the normal), and the binormal vector will be the cross product of those. That's the Frenet frame. Getting a feel for this sort of thing will help later on with other frame fields that you'll see in the context of surfaces, so you'll get an idea of how connection forms and shape operators and stuff like that work. Something like Frenet frames might not seem immediately relevant, but it will be eventually. You can try to skip things, but if you want it to seem less abstract, I don't think there's any royal road. You can either try to work with limited motivation or you can spend a lot of time on the motivation or some combination of the two.

There's also a really helpful chapter in Visual Complex Analysis that introduces differential geometry of surfaces because of its relation to hyperbolic geometry, which is intimately connected to complex analysis. It explains things like intrinsic geometry versus extrinsic geometry very well and gives a bit of an overview of the subject without getting into too many details. Hilbert has a strikingly similar chapter on differential geometry in his book with Cohn-Vossen, Geometry and the Imagination.

 

[homeomorphic]

If you do need to try to learn the more advanced stuff more quickly, I would suggest trying to set n = 2 for your manifold and ask what happens, see if you can visualize it, compute examples with surfaces, like a 2-sphere and so on. Another thing you can try to do is talk to someone who knows the subject. You're already doing that here, but we don't have chalkboards and so on, which can make it harder.

 

[homeomorphic]

Oh, and skip the chapter about classifying space curves up to isometry. Not necessary to continue.

 

[dyn]

Thanks for all your replies