Motivating definitions from differential geometry

Hi

I have always had an issue with understanding the definitions used in mathematics. I need examples before I can start using and reasoning with them. However, with tensor products, I have been completely stuck.

Stillwell's Elements of Algebra was that made abstract algebra "click" for me. However, in the case of algebra, I got the logic behind those definitions; It just took an extra "push" for me to start comfortably using them in proofs. Tensor products, however, I am not even 100% sure I get the underlying logic of.

What I am now hoping to find is the equivalent to Stillwells book for differential geometry. That book does the opposite of what most textbooks do; Textbooks start off with abstract definitions before presenting concrete examples. Stillwell goes through all major results using numbers and polynomials before showing how that logic can be generalized to groups, rings, fields and galois theory.

Of course, I'll try any book that can prepare me for the definitions in differential geometry. Bachman has a good book on differential forms which I am going through right now. However, I obviously need to learn a lot more. Which other books can help me prepare for differential geometry?

Answers and Replies

fresh_42
Mentor
I understand your point. I have similar habits and like to have an example in mind when I read a definition.

Tensors are a good example, as they are simply a linear construction like a matrix is. However, they are used by their roles as multi-linear functions. As a matrix is a tensor, too, let's take this example. In physics they are usually written as coordinates and basis vectors, in differential geometry the latter are ##\frac{\partial}{\partial x_i}## or ##dx_i##. That make them look as something different than a linear transformation, yet they still are. The language on the other hand is driven by this transformation, by what the written matrix aka tensor does. This way people talk about e.g. a curvature tensor, which we almost automatically associate with something curved, although it's still this linear thing. They apply the curvature tensor to other things like points or vectors, as a matrix is applied to vectors, or the family of Jordan matrices (e.g. = gradient) of a function is applied to (= evaluated at) a certain point to determine a single one, which let us further forget its linearity. And then they write the entire thing in coordinates again, which are structured as e.g. a matrix or in general a (multi-linear) tensor. It is this mixture that is confusing. It's due to the fact, that one and the same differential can play so many different roles. I once made the fun and listed a few. I found ten, and "slope" wasn't among them. (see at the beginning of https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/)

It's difficult to recommend a book you asked for, as in the end this depends on so much individual properties whether a certain book fits well or not, that the chances to tell you some which do not are high. I like
https://www.amazon.com/dp/0387903577/?tag=pfamazon01-20&tag=pfamazon01-20
which has really a lot of examples and graphics, but if it fits you?

The book is among the sources I listed at the end of
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/#toggle-id-1
IIRC each part of this little series of five articles has a list of sources which are basically the same, but I think the one at the end of part 5 was the most extended.

And as you mentioned tensors, maybe this article
https://www.physicsforums.com/insights/what-is-a-tensor/
and especially the example at the end can shed some light on it. But it's from the algebraic point of view, not the differential geometry one, so I'm not siúre whether it will help. However, it's short.

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Avatrin and ibkev
I'm not sure if this will be what you are looking for, but it certainly does the opposite of what most textbooks do:

https://mathphysicsbook.com

The idea behind the book is to avoid historical motivations, long proofs and derivations, and (here's where I may lose you) tools for practical calculations. Instead the focus is on the ideas behind the definitions, with lots of geometrical viewpoints and detailed illustrations.

For example, tensors originate from a tensor product defined by linearity under various operations, but then are often most profitably viewed as multilinear mappings or multi-dimensional arrays. Exterior forms originate from adding the rule that the product of a vector with itself vanishes, but then are often best viewed as multilinear mappings, anti-symmetric tensors, or anti-symmetric multi-dimensional arrays.

Differential forms then smoothly assign an exterior form to each point on a manifold. But...as multilinear mappings, what do they operate on? Vector fields...and what are vectors? Little arrows? To me at least, going at least once through what a tangent vector field actually is (a mapping between real functions on the manifold) is really helpful in avoiding reliance on simplified pictures which may not actually be correct.

Plenty of confusion also arises from hidden assumptions made in various texts. For example, there are at least two different ways to associate an exterior form with a multilinear mapping or an anti-symmetric array! And the inner product of those exterior forms is not the same as the inner product of the associated tensors.

The good news is that once all this is surfaced and explicitly laid out, everything at least has a chance of becoming much clearer, and more traditional textbooks can be gone through without so many unanswered questions hanging over things. Or at least that's the idea. Hopefully the book ends up being helpful for you.

mathwonk
Science Advisor
Homework Helper
2020 Award
I know only a little differential geometry, but if I wanted to study it with a view to understanding the constructions intuitively, I would consult works by Gauss (General Investigation of curved surfaces), Riemann (essay reproduced in Spivak vol. 2, chapter 4A), Spivak (chapters 3 and 4 of volume 2), Nikulin /Shafarevich (Geometry and Groups), and David Henderson (Differential Geometry, a geometric introduction).

The main thing seems to me to be grasping curvature, first intuitively, then how it is expressed as a tensor.

Avatrin,

I've always had a beef with how riemannian geometry and differential geometry was taught at my college, and many other colleges, for the same reason. You should be aquainted with classical differential geometry in 3D before going into the abstract formulation because otherwise it becomes dry formality. You can try Kreyszig's book

https://www.amazon.com/dp/0486667219/?tag=pfamazon01-20&tag=pfamazon01-20

As for understanding tensors, you can try the following book, but I'll try to give you the intuition after.

https://www.amazon.com/dp/0486638332/?tag=pfamazon01-20&tag=pfamazon01-20

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Tensors:

Let's say we are working in the usual three dimentional real space, R^3. If you want to represent a vector, you have a canonical choice for the basis, (1,0,0), (0,1,0), and (0,0,1). In this canonical basis you can represent all vectors in a standard way. If you were in R^n, you would also have this advantage of a canonical choice of basis.

Let's now say that you are working in a some vector space V with n dimensions. The problem here is that you might not have a canonical choice for a basis. So you don't have a canonical choice to represent vectors in V (that is what happens with the tangent space of a differentiable manifold). But any vector w in V is a specific element. Its representation might change if you change the basis, but the element is still the same.

Now let's say you have a linear transformation T from R^3 to R^3. Since you have a canonical choice of basis for R^3, you have a canonical choice for the matrix that represents T.

But if you have a linear transformation S from V to V, where V is a vector space of dimension n, you might not have a canonical choice of basis and therefore, not a canonical choice of matrix that represents S. But whatever matrix you choose to represent the linear transformation S, the transformation takes the same vectors w to the same vectors S(w).

Let's say we have a bilinear transformation B from VxV to R. You can represent the bilinear transformation by a matrix, but you can run into the same problem. You might not have a canonical representation for B because V might not have a canonical choice for a basis.

The same happens with multilinear transformations from Vx...xV to R. The best known multilinear transformation is the determinant. The geometric significance of the determinant is that det(v1,v2,...,vn) is the n-dimensional volume of the n-dimensional solid determined by the vectors v1,v2,...v,n. The determinant is also used for integration when you change variables in the integral.

So, in differential geometry you deal with tangent spaces which do not have a canonical choice for a basis. And this is because for each point x in a manifold M there are many ways to map the neighbourhood of x to a region of R^n. So you end up with no standard choice for a basis of a tangent space to the manifold. Each vector is still the same, but its representation changes with the choice of basis for the tangent space. The vector is independent of the coordinates chosen to represent it.

A tensor is a multilinear transformation between vector spaces that is independent of the coordinates chosen to represent it. Therefore they are useful in differential geometry because you do not have a standard canonical choice for basis of tangent spaces.

Multilinear tranformations are useful to calculate volumes and, particularly, volume elements which are used for integration. They are useful to calculate inner products in vector spaces V, which means they end up being useful to calculate distances in manifolds, regardless of coordinates. Multilinear transformations are also useful to calculate different kinds of curvatures.

So tensors are used to calculate many useful things in differential geometry independently of coordinate representation. And, since differential geometry deals with differential manifolds that can have many charts for each neighbourhood of a point, tensors become necessary in differential geometry.

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Avatrin