whats a good intro book to tensors and manifolds?
There are several threads here in which people have recommended sets of notes on relativity in which tensors are taught.
In one of those sets of notes, that I particularly liked, the author Sean Carroll recommended the book by Frank Warner, on Differentiable manifolds and lie groups, as a "standard". I kind of like Michael Spivak's little book Calculus on manifolds, and his much longer series on Differential Geometry, say the first volume for starters.
This is a mathematician talking, so I recommend getting some opinions from the physics experts too. Of course Carroll is presumably a physicist.
Warnes book is nice because it also has an introduction to "Hodge theory" as I recall. Others here have recommended Tensor analysis on manifolds by Bishop and Goldberg, because it is not only a good classic text, but it is available in paper for a song.
Look on the threads "What is a tensor", and Differential geometry lecture notes, and Math "Newb" Wants to know what a Tensor is, and others , for some free sites with downloadable material on tensors.
I would warn you of one thing. I myself am primarily educated in the mathematics of manifolds and tensor bundles as in Spivak's calculus on manifolds. As you can see from numerous exchanges I have had with physicists on this forum I have great difficulty understanding what they are talking about. Thus i would suggest that it is not enough to understand only the mathematical concepts of manifolds and tensors, but one should go further and see these concepts in use either in differential geometry, or in physics.
Their is a book I love and I think is very well suited as an intro : Geometry, Particles, and Fields (Graduate Texts in Contemporary Physics) by Bjorn Felsager
It is written by a high-school teacher, and oriented to physics application. However, it is quite rigourous enough (to me). It goes from the very beginning to advanced stuff in physics.
There are notes I highly recommended, because they are free on internet and starting at elementary level:
1) An Introduction To Tensors for Students of Physics and Engineering, by Joseph C. Kolecki
2) Quick Introduction to Tensor Analysis, by R.A. Sharipov
3) An Introduction to Tensor Analysis and Continuum Mechanics, by J.H. Heinbockel
And I would like to thank the authors of these notes.
THE book on tensor calculus is by Synge and Schild (Tensor Calculus) but I don't know whether it's still in print.
It's published by Dover
(enter "tensor" in the search box).
Although Synge and Schild is great, I would have to say that
Schouten's "Ricci Calculus" is THE book on tensor calculus...
see also Schouten's "Tensor Analysis for Physicists" (Dover).
Schaum's Outline of Tensor Calculus is excellent, though it uses the classical, rather than modern notation. If you want to learn to calculate functions on curved spaces this is a good place to go. Another great resource is MTW's Gravitation, though a book on relativity, has some insightful sections on tensors.Of course, all the other books mentioned in this thread are good choices.
A modest proposal:
I would suggest that one reason it is hard for physicists and mathematicians to communicate is that some physicists seem to continue to educate themselves via extremely old fashioned mathematics books, teaching a version of tensor calculus that is about 100 years old. I know Einstein used it, but the mathematics Einstein used is not sufficient to understand even today's physics.
Modern physicists like (the real, departed) Feynman, and now Witten, have been not only creating new exciting physics, but also new mathematics, and are also inspiring mathematicians to try to catch up with the innovations they are bringing to both subjects. I have been a lecturer at the International Center for Theoretical Physics in Trieste (on Riemann surfaces, and theta functions on abelian varieties) and so I know a little about what at least some of today's physicists want to learn. In particular it is important to learn how geometry is linked to, and illuminated by, modern topology and analysis, such as deRham cohomology, and modern theory of partial differential operators.
So I recommend, even if one learns the old versions of Ricci calculus, to also at least look at modern books like Spivak's differential geometry volumes 1 and 2. If that is too mathematical, I suggest trying Misner, Thorne and Wheeler, as recommended above. At some point one might want to look also at books such as the volume of proceedings "Lectures on Riemnann Surfaces" given at ICTP in 1987, pub. by World Scientific in 1989 (for background on string theory), and some works on algebraic geometry. Miles Reid's book is the easiest, and Shafarevich is very nice for a next course. To me many of the references recommended here, although excellent for what they are, nonetheless fall largely into the category: "primarily of historical interest".
Feyman's work, in particular the theory of "Feynman integrals" has apparently led recently to the exciting mathematical topic of quantum cohomology, growing out of "quantum gravity" and other topics in which modern physicists lead the way. It seems to me at least, that these leaders are not using 100 year old mathematics, they are using mathematics that has not even been perfected yet.
Modern tensor notation is in my opinion superior, the problem is that many texts that teach it use abstraction as an excuse to avoid doing any actual calculations - in some cases modern methods step around tedious calculations in a profound and brilliant manner, but in the wrong hands modern tensor notation is obscure gibberish. A case in point is Darling's book - an excellent book overall, and one I recommend, but at times it is degenerates into storm of pretentious machinery. At one point he defines two objects, then proves that they are actually the same object to begin with. What is the point of that? At another point he gives that wrong formula for the wedge product of two forms, and error he would have found easily if he had actually tried to calculate anything with it. If there was a book using modern notation that was not divorced from the nuts and bolts of doing actual calculations I would recommend that, but unfortunately there are none.
P.S. DeRham's book on cohomology is excellent - it is strangely obscure. Another excellent book on algebraic geometry is "An Invitation to Algebraic Geometry" by Karen Smith et. al.
You make an excellent point. It is certainly true that many authors of modern math books dwell excessively on the theory and omit useful calculations, in every subject. Now that you have raised this as a key virtue lacked by most modern treatments, maybe someone will recommend a book unknown to us which does have it. In the meantime perhaps your argument implies that physicists need to read both types of books, modern and classical.
I have not read Darling, but can suggest a reason for giving two different definitions of the same object in some cases. Each definition may have different advantages, perhaps one is intuitively more appealing while the other admits easier calculation. Or one was used historically, while the other is of more modern acceptance. Just a guess. As I recall, Spivak for example discusses the curvature tensor in progressive degrees of abstraction, starting from Riemann's original version, continuing up through modern incarnations.
What I think this comes down to is the dichotomy between calculating a quantity and understanding the meaning of that quantity. I claim that understanding allows calculation, but not vice versa.
For instance, on page 14, of his nice notes on GR, Sean Carroll gives the transformation law, (1.51) in his numbering, for tensors and then says: "Indeed a number of books like to define tensors as collections of numbers transforming according to (1.51). While this is operationally useful, it tends to obscure the deeper meaning of tensors as geometric entities with a life independent of any chosen coordinate system." On page 15 he describes the scalar or dot product as a familiar example of a tensor of type (0,2).
I am going to go out on a limb here and try to make a trivial calculation, beginning from a conceptual definition of a tensor of type (0,2) as a bilinear map from pairs of tangent vectors to numbers. I.e. I will try to derive the transformation law from the conceptual meaning.
A simple example of such a tensor is a scalar product, i.e. a symmetric, bilinear mapping from pairs of tangent vectors to scalars. Such a thing is often denoted by brackets (or a dot) taking the pair of tangent vectors v,w to the number <v,w>. Now if f:M-->N is a differentiable mapping from one manifold M to another manifold N, such as a coordinate change, then one can pull back a scalar product from N to M using the derivative of f.
I.e. if u,z are two tangent vectors at a point p of M, then applying the derivative of f to them takes them to 2 tangent vectors at the image point f(p) in N, where we can apply <,> to them. I.e. if <,> is the scalar product on N, then the pulled back scalar product f*(<,>) acts on u,z by the obvious, only possible law: f*(<u,z>) = <f'(u),f'(z)>, where f' is the derivative of f, given as a matrix of partials of f with respect to local coordinates in M and N. For example we could denote this matrix as f' = [dyi/dxj].
Now suppose we express the scalar product in N as a matrix, i.e. in local coordinates as A = [akl], sorry about the lack of subscripts. Imagine k and l are subscripts on a.
Then if we want to express the pulled back scalar product as a matrix, we just see what it does to the vectors u,z as follows:
f*(<u,z>) = <f'(u),f'(z)> = [f'(u)]* [A] [f'(z)] = * [f']* [A] [f'] [z], where now everything is thought of as a matrix, and star means transpose of the matrix.
Well since the matrix of partials f' is just [dyi/dxj], and A is [akl], we just multiply out the matrices to get the matrix of the pulled back scalar product as [f*(<,>)]
= [f']* [A] [f'] = the matrix whose i,j entry is akl (dyk/dxi)(dyl/dxj), summed over k,l.
Now this is exactly the transformation law Carroll calls (1.51) on page 14 of his notes and everyone else also calls the transformation law for a tensor of type or rank (0,2) in the various web sources given here and above.
Notice too, if you can imagine my subscripts, that this satisfies the summation convention for subscripts. But I am not dependent on that because I know what it means, so i don't care whether I can see the subscripts or not, whereas someone dependent on seeing where the indices are may not be able to follow this.
Anyone who knows conceptually what a tensor is would immediately realize that a homogeneous polynomial of degree d in the entries of a tangent vector, is a (symmetric) tensor of type (0,d), and that the components of the tensor are merely the coefficients of the polynomial (written as a non commutative polynomial, i.e. with a separate coefficient for xy and for yx). It follows of course that they transform via a d dimensional matrix of size n, where n is the dimension of the manifold, i.e. by a collection of n^d numbers.
Subscript enthusiasts write this as a symbol like T, with d subscripts.
That is an extremely cumbersome way to discuss tensors in my opinion, and leaves me at the mercy of the type setter, whereas knowing what they mean always bails me out eventually.
I actually wrote a graduate algebra book, including linear and multilinear alkgebra once, and I discovered to my amusement that I could actually write down tensor products as matrices, and so on, just from the definitions, although I had never needed to do so before in my professional life.
peace and love,
In 1996, at the end of a chapter on tensors in my graduate algebra notes, after writing out a calculation of the tensor product of two matrices, I wrote the following extremely naive remarks:
"The complexity of this sort of calculation may be responsible for the fearsome reputation which "tensor analysis" once enjoyed. In ancient times, books on the topic were filled with lengthy formulas laden with indices. Learning the subject meant memorizing rules for manipulating those indices. Nowadays, confronted with the statement that such and such quantity is "a tensor", I hope we will understand this to mean simply the quantity has certain linearity properties with respect to each of its components. Of course skill in their use will still require an ability to calculate. In this regard, note that we are usually able to recover explicit calculations from our abstract approach, provided we always know exactly what the maps are that yield our isomorphisms. When we know the maps, a choice of bases gives us a calculation. Thus we must resist the tendency to remember only that certain modules are isomorphic, without knowing what the isomorphisms are. Fortunately the maps are virtually always the simplest ones we can think of."
I have just perused Spivak's volume 2, chapters 4,5, and 6, and want to recommend them extremely highly to everyone here who is interested in tensors and the relation between their classical and modern incarnations. I.e. this is not just a modern treatment, and not just a classical treatment, but he gives both treatments, with relations clearly drawn betwen the two.
I.e. forget volume I, which may seem like a mathematician's indulgence, with its abstract definitions and modern treatments of manifolds. Go straight to the good stuff in volume II. The chapter headings are already enlightening:
1. is on curves,
2. is called "what they knew about surfaces before Gauss" and is only 8 pages long.
3. is on Gauss's theory of surfaces, and I mean Gauss's own version. Spivak presents Gauss's own work, Disquisitiones generales circa superficies curvas, and explains how to make sense of the original: "how to read Gauss", then explains how to state the results in modern terms.
chapter 4. is a translation of Riemann's inaugural lecture on manifolds, including his generalization to n dimensions of Gauss' theory of curvature of surfaces. Basically, certain combinations of partial derivatives, now called Christoffel symbols, represent the coefficients of the "error term" needed to force the partial derivatives of a tensor of type (0,1), with respect to given local coordinates, to be themselves again a tensor. Then a certain combination of these Christoffel symbols defines the curvature tensor.
In chapter 5 Spivak presents the classical Ricci calculus, subtitled "the debauch of indices", and proves computationally the "test case" that a manifold with zero curvature tensor is locally isometric to (flat) euclidean space.
Then in chapter 6, Spivak presents the modern approach to a "connection", as simply a way of differentiating pairs of vector fields, linear over the functions in one of the variables, and obeying the Leibniz rule in the other variable.
He then relates an abstract connection to a "classical connection", i.e. an expression analogous to the Christoffel symbols, but not necessarily arising from a metric, hence not necessarily symmetric. Nonetheless any connection leads again to a curvature tensor, which now is simply a certain commutator expression of derivatives.
Spivak then reproves the basic test case, much more easily, using modern concepts. He continues to pursue the evolution of the concept of a connection through several modern versions, reproving the "test case" in all seven times, each time revealing more geometric content, as modern conceptual tools permit this.
Thus Spivak presents a thoroughly classical treatment of connections and curvature, Ricci calculus, and Christoffel symbols, then shows how these concepts are viewed nowadays in simpler more conceptual terms.
This seems to me the ideal place to learn to speak all these languages. In fact back in 1970, I once claimed that although I did not know what they meant, I did know that nowadays "Christoffel symbols" had become a triviality. Some people laughed at me for saying this, at which point I bet them I could prove my statement by learning what they meant in 5 minutes and then explain it to their satisfaction. I grabbed Spivak, opened volume 2, and persuaded them in a few minutes.
This is truly a great book, which made a uniquely valuable contribution to understanding differential geometry.
To recommend Spivak again, I read this book (volume II) in one day almost 30 years ago, and have never consulted it again (except for the bet above) until yesterday for about 30 minutes. So it is not going to eat up a lot a lot of your time to give it the once over. It is so well written you can learn something from it very quickly. Although obviously in such a short time I did not come anywhere near mastering anything, still I feel I did learn something.
a used copy is available for $20 from http://www.ericweisstein.com/encyclopedias/books/DifferentialGeometry.html
and a new copy for about $40 from "publish or perish".
Well after perusing the website of the publisher, I see the first and second editions are no longer available, and I am slightly disappointed to note that apparently the cover art has changed, and there are no longer strange animals on the front of voilume 5 waving flags and and marching in the name of "The generalized Gauss Bonnet theorem and what it means for mankind". You can never have too much nonsense in amthematics. The chapter on p.d.e. called "and now a word from our sponsor" remains however.
Which volume has the information on covariant derivatives? I am going to get my hands on a copy - the copy at the local tech school is the edition with the horrendeous typography, almost unreadable, which has kept me from a serious reading.
well I am going to guess it is volume 2. volume 1 is a treatise on general manifold theory, then vol 2 is the evolution of differentiual geometry since riemann and gauss, focussing on the curvature tensor.
i do not know what a covariant derivative is, but the modern version of a connection is called there a koszul connection, and is simply a way of pairing two vector fields X,Y and getting another one called delta sub X (Y). it is linear in X over the ring of smooth functions, and obeys the leibniz rule in Y for multiplication by smooth functions. this is the modern version of christoffel symbols.
I will just guess that this is a covariant derivative, but i can find out later.
still i presume all the machinery of differential geometry is in that volume 2, because vol 3 is sort of classical examples like equations of hypersurfaces "codazzi equations etc"
and I forget what vol 4 is, but 5 seems to be fancy stuff like gauss bonnet a la chern in n dimensions. we can probably see the contents of the chapters on the publish or perish website, i'll look and see.
I also recall that volumes 1 and 2 were the actual content of the copurse mike taught, and 3,4,5 were added later, so surely he taught covariant derivatives in the course. but they might be already in vol 1 at the end. i only have vols 1 and 2 but they are at the office.
ok i just looked on the publish or eprish website and here is the table of contents for volume 2 and you can see the words covariant derivatives in chapter 7, cartans theory of moving frames:
*1. Curves in the Plane and in Space
******* Curvature of plane curves. Convex curves. Curvature and torsion of
******* space curves. The Serret-Frenet formulas. The natural from on a Lie group.
******* Classification of plane curves under the group of special affine motions.
******* Classification of curves in Euclidean n-space.
*2. What they knew about Surfaces before Gauss
******* Euler's Theorem. Meusnier's Theorem.
*3. The Curvature of Surfaces in Space
******* A. HOW TO READ GAUSS
********* B. GAUSS' THEORY OF SURFACES
*********** The Gauss map. Gaussian curvature. The Weingarten map; the first and
*********** second fundamental forms. The Theorema Egregium. Geodesics
*********** on a surface. The metric in geodesic polar coordinates. The integral
*********** of the curvature over a geodesic triangle.
*********** Addendum. The formula of Bertrand and Puiseux; Diquet's formula.
*4. The Curvature of Higher Dimensional Manifolds
******* A. AN INAUGURAL LECTURE
*********** "On the Hypotheses which lie at the Foundations of Geometry"
******* B. WHAT DID RIEMANN SAY?
*********** The form of the metric in Riemannian normal coordinates.
******* C. A PRIZE ESSAY
******* D. THE BIRTH OF THE RIEMANN CURVATURE TENSOR
*********** Necessary conditions for a metric to be flat. The Riemann curvature
*********** tensor. Sectional curvature. The Test Case; first version.
*********** Addendum. Finsler metrics.
*5. The Absolute Differential Calculus (The Ricci Calculus)
******* Covariant derivatives. Ricci's Lemma. Ricci's identities. The curvature tensor.
******* The Test Case; second version. Classical connections. The torsion tensor.
******* Geodesics. Bianchi's identities.
*6. The Dell Operator
******* Kozul connections. Covariant derivatives. Parallel translation.*
******* The torsion tensor. The Levi-Civita connection. The curvature tensor.
******* The Test Case; third version. Bianchi's identities. Geodesics.*
******* The First Variation Formula.
******* Addenda.*Connections with the same geodesics. Riemann's
******* invariant definition of the curvature tensor.
*7. The Repère Mobile (The Moving Frame)
******* Moving frames. The structural equations of Euclidean space.
******* The structural equations of a Riemannian manifold. The Test Case;
******* fourth version. Adapted frames. The structural equations in polar
******* coordinates. The Test Case; fifth version. The Test Case; sixth version.
******* "The curvature determines the metric".* The 2-dimensional case.
******* Cartan connections. Covariant derivatives and the torsion and curvature
******* tensors. Bianchi's identities.*
******* Addenda. Manifolds of constant curvature: Schur's Theorem;*
******* The form of the metric in normal coordinates. Conformally equivalent manifolds.
******* É. Cartan's treatment of normal coordinates.
*8. Connections in Principal Bundles
******* Principal bundles. Lie groups acting on manifolds. A new definition of
******* Cartan connections. Ehresmann connections. Lifts. Parallel translation
******* and covariant derivatives. The covariant differential and the curvature
******* form. The dual form and the torsion form. The structural equations.
******* The torsion and curvature tensors. The Test Case; seventh version.
******* Bianchi's identities.*
******* Addenda. The tangent bundle of F(M). Complete connections.
******* Connections in vector bundles. Flat connections
a very nice clean discussion is in milnor's book, characteristic classes, joint with stasheff, in appendix C, p.289, "curvature, connections, and characteristic classes".
he defines a "connection" on a bundle E over a manifold, as a C linear map from smooth sections of the given bundle E, to smooth sections of the tensor product of E with the cotangent bundle of the manifold. It is required that the map obey the leibniz rule. i.e. it takes the product of a function and a section to the product of the function times the image of the section, plus the tensor product of the section with the esterior derivative of the function.
Then, he calls the image of the section under this mapping, the covariant derivative of the section. so covariant derivative is just another name for a connection. you get a wonderfully clear explanation in about 15 pages from milnor, of several basic points, and the link with classical connections.
he then explains curvature and proves the gauss bonnet theorem connecting curvature with the euler characteristic. (Recall the polyhedral version of gauss bonnet: if we have a polyhedral surface and at each vertex define the curvature to be 2<pi> minus the sum of the angles of the polygons at that vertex, then the sum total over the curvatures over the polyhedron equals the euler characteristic, times 2<pi>.)
Being educated as a physicist, I understand many people who complain about "bourbaki" style of writing math textbooks, and I would not recommend to read the books by F. Warner and M. Spivak as a first introductory reading in modern geometry. (Spivak is only good to understand the historical line of development, but you have to have some background and being familiar with modern terminology for that.) In my opinion more or less suitable book, written by mathematicians for physicists and engineers, is
Dubrovin, Novikov, Fomenko, Modern Geometry v. 1,2,3.
This is three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics.
Topics of 1st volume starts from curves and surfaces and include tensors and their differential calculus, vector fields, differential forms, the calculus of variations in one and several dimensions, and even the foundations of Lie algebra. So, the first volume would be enough for start. I looked in 2 and 3 v. and think its close to the front of modern geometry and definitly prepares for the reading more special books...
The material of books is explained in simple and concrete language that is in terminology acceptable to physicists. There are some exercises, but should be more to get practical skills. If I will find the special problem book on modern geometry to accompanying this textbook, it would be excellent pair for any beginner.
gvk, i suggest you post your review of fomenko et al, at amazon.com for possibly wider distribution.
I have not seen this book but am very favorably impressed with writing of most russian texts.
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