Why study manifolds

There are many good texts for learning real analysis , some of which are mentioned in the answers to this question. It will be important that your real analysis education cover not only single-variable differentiation and integration, but also multivariable differentiation and integration as well. Where point-set topology is concerned, some of it will hopefully be covered when you learn real analysis.

Indeed, many analysis texts consider point-set topology to be a part of analysis. You will need to learn about metric spaces and topological spaces.

An excellent book for this is " Topology " Munkres. Finally, once you've gotten through all of this, I would say the text to use for manifold theory is " Introduction to Smooth Manifolds " John Lee. In fact, the book has an appendix at the end which gives a rapid treatment of each of these four subjects. So, if you absolutely cannot contain your curiosity, this might be worth looking into. Optional: One more helpful but not necessary pre-requisite is elementary differential geometry or classical differential geometry , which is a beautiful topic to learn once you've finished with multivariable calculus.

My recommended textbook is " Elementary Differential Geometry " Pressley. You should most likely have a background in linear algebra as you said, multivariable calculus, and real and complex analysis. For instance, Rudin begins a discussion of manifolds after discussing standard multivariable analysis in his Principles of Mathematical Analysis. I think the prerequisites you will need to study manifold theory depend on which aspect of manifold theory that you wish to study.

For example, a good knowledge of algebraic topology is more essential if you wish to study differential topology than if you wish to study differential geometry although you should eventually learn algebraic topology in some depth no matter which aspect of manifold theory you pursue. However, the prerequisites to study the standard theory of differentiable manifolds are generally speaking point-set topology, linear algebra and advanced multivariable calculus. A good knowledge of point-set topology and linear algebra implies that you have the mathematical maturity necessary to study manifold theory as well as the necessary knowledge; thus, it is important to carefully study these two subjects.

For example, you might wish to look at Topology: A First Course by James Munkres and Linear Algebra Done Right by Sheldon Axler which will provide you with more knowledge in these subjects than is strictly necessary in manifold theory but this knowledge will be essential in your study of other branches of mathematics.

Finally, I think Principles of Mathematical Analysis by Walter Rudin furnishes a solid knowledge of the elements of advanced calculus both single-variable and multivariable that will be necessary for manifold theory. In short, it would be a good idea to use manifold theory as a means to advance your knowledge of other essential branches of mathematics because the prerequisites for manifold theory are more fundamental in mathematics as a whole than manifold theory itself. I agree with analysisj 's response and up-voted it.

I just wanted to add that it might be useful to review some topology depending on how your real and complex analysis courses were taught. Some analysis courses use books that are light on the point set topology theory. In addition to linear algebra, analysis and topology as others have suggested, learning some classical differential geometry probably wouldn't be a bad idea either.

The very abstract definitions one encounters in the theory of manifolds are inspired by the principles of differential geometry much as point-set topology was inspired by analysis.

Seeing these concepts made tangible by concrete calculations will give more meaning to the more elaborate machinery of manifolds and differential forms. Before you can even learn the precise definition of manifolds and theorems about manifolds you should be familiar with topological notions that students typically learn in analysis.

But you should take a look at Chapter 2 of Munkres's Topology. Try to understand what you can. If you get too lost, consult an introductory analysis book like Rosenlicht's Introduction to Real Analysis and hop back and forth between the two.

If you want to do calculus on manifolds, you should certainly learn some linear algebra and multivariable calculus first. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. What math is necessary to learn manifolds? Ask Question. Asked 9 years, 10 months ago. Active 9 years, 10 months ago. Viewed 8k times. Amitesh Datta Kr0n42 Kr0n42 1 1 silver badge 3 3 bronze badges.

In fact, manifolds are quite far away from freshman calculus -- at least two solid years of study away, in my opinion -- and in between lies a bunch of things you will have to take anyway, especially multi-variable calculus, linear algebra and basic real analysis.

Really I think it's too soon to be worrying about this, and I don't mean this in the discouraging sense, I literally mean that you needn't worry about it yet If you want advice, here it is: first, enroll in a university or very good liberal arts college -- i.

Second, concentrate now on fully mastering the material up to and including BC calculus. Really having this down will serve you well, even unto your study of manifolds. Good luck! Clark, what's «BC Calculus»? The big idea is that we can also have "open ended" curves that extend out to infinity, which are natural mappings to a one dimensional line.

Figure 2: Circles, parabolas, hyperbolas and cubic curves are all 1D Manifolds. Note: the four different colours are all on separate axes and extend out to infinity if it has an open end source: Wikipedia. Let's now move onto 2D manifolds. The simplest one is a sphere. You can imagine each infinitesimal patch of the sphere locally resembles a 2D Euclidean plane. Similarly, any 2D surface including a plane that doesn't self-intersect is also a 2D manifold.

Figure 3 shows some examples. For these examples, you can imagine that each point on these manifolds locally resembles a 2D plane. This best analogy is Earth. We know that the Earth is round but when we stand in a field it looks flat. We can of course have higher dimension manifolds embedded in even larger dimension Euclidean spaces but you can't really visualize them.

Abstract math is rarely easy to reason about in higher dimensions. Hopefully after seeing all these examples, you've developed some intuition around manifolds. In the next section, we'll head back to the math with some differential geometry. Now that we have some intuition, let's take a first look at the formal definition of topological manifolds , which I took from [1]:.

This definition is hard to understand especially because a Hausdorff space is never defined. That's not too important because we're not going to go into the topological formalities, the most important parts are the new terminology, which thankfully have an intuitive interpretation.

Let's take a look at Figure 4, which should clear up some of the ideas. Finally, if we have a bunch of charts whose domains exactly spans the entire manifold, then this is called an atlas. The best analogy for all of this is really just geography. I've never really studied geography beyond grade school but I'm guessing you have similar terminology such as charts, coordinate systems, and atlases.

The ideas are, on the surface, similar. However, I'd probably still stick with Figure 4, which is much more accurate. Wrapping your head around manifolds can be sometimes be hard because of all the symbols. The key thing to remember is that manifolds are all about mappings. Mapping from the manifold to a local coordinate system in Euclidean space using a chart; mapping from one local coordinate system to another coordinate system; and later on we'll also see mapping a curve or function on a manifold to a local coordinate too.

Sometimes we'll do one "hop" e. And since most of our mappings are we can "hop" back and forth as we please to get the mapping we want. So make sure you are comfortable with how to do these "hops" which are nothing more than simple function compositions. The most important one for our conversation being transition maps that are infinitely differentiable, which we call smooth manifolds.

The motivation here is that once we have smooth manifolds, we can do a bunch of nice things like calculus. Remember, once we have smooth mappings to lower dimensional Euclidean space, things are a lot easier to analyze.

Performing analysis on a manifold embedded in a high dimensional space could be a major pain in the butt, but analysis in a lower-dimensional Euclidean space is easy relatively! It requires a single chart that is just the identity function, which also makes up its atlas. We'll see below that many of concepts we've been learned in Euclidean space have analogues when discussing manifolds.

Further, we'll need more than one chart mapping because a chart can only work on an open set the analogue to an open interval, i. We can also find other charts to map the unit circle. Let's take a look at another construction using standard Euclidean coordinates and a stereographic projection. Figure 5 shows a picture of this construction. We can define two charts by taking either the "north" or "south" pole of the circle, finding any other point on the circle and projecting the line segment onto the x-axis.

The "north" pole point is visualized in blue, while the "south" pole point is visualized in burgundy. Note: the local coordinates for the charts are different. This defines a mapping for every point on the circle except the "north" pole. Figure 6 shows a visualization never mind the different notation, I used a drawing from Wikipedia instead of trying to make my own :p.

This chart can cover every point except the starting point. Using two charts each with a point e. Pick an arbitrary focal point on the sphere not on the hyperplane you are projecting to e. Pick a plane that intersects at the "equator" relative to the focal point. From this, we can derive similar formulas using the same similar triangle argument as the previous example.

The symmetric equations can also be found for the "south" pole. To actually calculate things like distance on a manifold, we have to introduce a few concepts. For a 2D manifold embedded in 3D , this would be a plane. Figure 7 shows a visualization of this on a manifold.

It can however look like this when it is embedded in a higher dimension space like it is here for visualization purposes e. Manifolds don't need to even be embedded in a higher dimensional space recall that they are defined just as special sets with a mapping to Euclidean space so we should be careful with some of these visualizations.

However, it's always good to have an intuition. Let's try to formalize this idea in two steps: the first step is a bit more intuitive, the second step is a deeper look to allow us to perform more operations. So far so good, this is just repeating what we had in Figure 4.

Basically, this just defines a curve that runs along our manifold. Now we want to imagine we're walking along this curve in the local coordinates i. Thus, the velocity is just the instantaneous rate of change of our position vector with respect to time.

Tangent vectors as velocities only tell half the story though because we have a tangent vector specified in a local coordinate system but what is its basis? Recall a vector has its coordinates an ordered list of scalars that correspond to particular basis vectors. So understanding how the tangent spaces relate between different points and potentially charts on a manifold is important. Note the introduction of partial derivatives and summations in the third line, which is just an application of the multi-variable calculus chain rule.

We can see that by introducing this test function and doing our little trick we get the same velocity as Equation 7 but with its corresponding basis vectors. Okay the next part is going to be a bit strange but bear with me. We're going to take the basis and re-write it like so:.

But it's important to remember that when we're using this notation, implicitly there is a chart behind it. We know it's some test function that we used, but it was arbitrary.

And in fact, it's so arbitrary we're going to get rid of it! Remember, a vector space doesn't need to be our usual Euclidean vectors, they can be anything that satisfy the vector space properties, including differential operators! A bit mind bending if you're not used to these abstract definitions. Now that we have a basis for our tangent vectors, we want to understand how to change basis between them.

Now we want to look at how we can convert from a tangent space in one chart to another. After some wrangling with the notation, we can see the change of basis is basically just an application of the chain rule.