What do you think?
Rate this book
Graduate Texts in Mathematics #218
Introduction to Smooth Manifolds
This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.
This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard's theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.
Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.
This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard's theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.
Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.
724 pages, Hardcover
First published January 1, 2002
31 people are currently reading
286 people want to read
About the author
John M. Lee
17 books11 followersJohn M. (Jack) Lee is Professor Emeritus of Mathematics at the University of Washington.
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 11 of 11 reviews
July 25, 2020
As philosophers of mathematics are wont to comment on, geometry differs from arithmetic in that it involves spatial intuition, while, in contrast, the number theorist may proceed from a pure concept of number, entirely abstracted from the world in which we happen to live. For this reason, the reputation of geometry has suffered in certain quarters, and this reviewer does not think that Bertrand Russell’s logical atomism does, in fact, succeed in removing all intuitive content from the continuum—but that is a subject for another day. Suffice it for now to remark that we are born, as it were, with a rather sophisticated notion of the continuum which will never cease to propel further advances in the science of space. Topology, as it was formalized in the early days of the twentieth century by Hausdorff, encodes our spatial intuition into a powerful axiomatic scheme that has proved surprisingly versatile, with applications ranging from algebraic topology in low-dimensional spaces to functional analysis in infinitely many dimensions, where the choice of an appropriate topology (or often several) is crucial. Yet, the price one pays for all this is that the topological axioms turn out to be too general for the limited purpose of investigating the space of our experience, or mathematical constructs meant to model it. Hence, one wants to impose further conditions with which to characterize space, and the most convenient hit upon thus far are those defining what is known as a smooth manifold. That is why the theory of smooth manifolds is central to differential geometry and its numerous applications to physics.
John Lee, whose introduction to topological manifolds we have reviewed here earlier, has continued his pedagogical efforts with a welcome introduction to smooth manifolds, also in the Springer Verlag series of graduate texts in mathematics and now out in a second edition. At over seven hundred pages, it may appear daunting, but the student should remember that, as the saying goes, there is no royal road to learning and that painstaking application to mastering the fundamentals will be rewarded with clarity of understanding later on. The pacing over twenty-two chapters is quite leisurely. It starts out with the basic definitions and some standard examples of smooth structures and mappings between them. Smoothness means that functions should be continuously differentiable infinitely many times; it is not so much of a restriction as might at first appear, since in practice most of the time one deals with functions given in analytical form, which are always smooth. Rather, the condition of smoothness serves to render concepts from calculus unproblematical, in as much as derivatives will exist up to any order one may need; as such, it captures our elementary intuition about the nature of space. Lee does treat the case of manifolds with boundary or corners with care, in view of their use in the context of Stokes’ theorem in chapters to come.
Starting in chapter three on tangent vectors, Lee delivers what we have a right to expect from him, a good course on the elements of smooth manifold theory. He reworked his text completely for the second edition, and the fruit of his labors shows. The perspective is modern throughout. Lee adopts a definition of a tangent as a derivation in the global algebra of smooth functions, a departure from the more conventional approach to go first to germs of functions at a point. Presumably he does so in order to keep things simpler, but, it seems to this reviewer, he obscures somewhat thereby the strict locality of the concept of the tangent. However that may be, armed with the notion of tangent and differential of a map, he can proceed to investigate submersions, immersions, embeddings and submanifolds via the rank theorem (in both local and global forms). The discussion, although it seems to progress at a glacial pace, is very thorough and clear, as is the treatment of the subtle distinction between an embedding and the image of an embedding in the subspace topology. Equipped with all this technology, he arrives at a few results in the theory of manifolds that are more than quite trivial: Sard’s theorem on the vanishing in measure of the set of critical points of a smooth mapping and Whitney’s embedding and approximation theorems, which say respectively that every smooth manifold can be embedded in a Euclidean space of sufficiently high dimension and that continuous maps can be closely approximated by smooth ones.
From here, Lee launches into a nice treatment of the basics of Lie groups and Lie algebras as another application of the circle of ideas surrounding smoothness. Then, he treats flows of vector fields in a fair amount of detail before going on to vector bundles in general and the cotangent and tensor bundles in particular. The whole apparatus of differential forms and the exterior derivative, which can be confusing to beginners but is in fact vital to a mature grasp of geometry, is built up in preparation for Stokes’ theorem (which generalizes integration by parts). Lee’s explanation of how to define integration on a curved manifold is well done. Another treat is a very thorough treatment of orientations, not often encountered in texts at this level. This part of the text is finished off with a somewhat rushed exposition of de Rham cohomology, the Mayer-Vietoris theorem (which renders cohomology effectively computable) and degree theory, followed by an exhaustive and rather technical proof of the de Rham theorem, which establishes a surprising isomorphism between the de Rham and singular cohomology groups, thus showing the topological invariance of the former. The reader will appreciate being inducted into these higher mysteries, or at least being given a glimpse into them although a text of such expansive scope cannot afford to dwell on them. Those wanting more could look to Warner’s old textbook, which covers these things from the more advanced sheaf-theoretic point of view.
A word about the homework exercises strewn liberally throughout the text, some 397 in all over 22 chapters: as always, working a good number of them is indispensable, as they do force one to get acquainted with what one has just read and to make use of it. Very few are strictly computational; most can be seen as filling in a gap in the exposition or elaborating a straightforward extension of the material. As in his previous book on topological manifolds, a fair number are included to point the way to more advanced topics, which means that they are not in themselves very hard but intended merely to introduce a sophisticated way of looking at what one already knows. The level of the majority of the problems is reasonable for the textbook’s audience, but one could be disappointed not to come across very many that are very challenging. Lee’s statement in the preface that most of them should take a typical student hours to solve seems like rather an overestimate, at least for most of the book; as mentioned below, in the few chapters at the end several do in fact rise to the level of difficulty that could take hours (but not days!).
The last four chapters are worth the wait, as Lee gets at last to some significant applications of the ideas he has so slowly built up in the preceding chapters: distributions and foliations, the exponential map and Lie group actions and their quotients, symplectic and contact structures and the beginnings of the theory of first-order partial differential equations. The coverage of these topics is complete; for instance, the Frobenius theorem is studied both from the local and from the global point of view and in terms of distributions in the tangent bundle or differential ideals in the graded algebra of differential forms. Numerous alternate ways of characterizing involutivity are presented. Likewise, the exponential map from a Lie algebra to its Lie group, its relation to one-parameter subgroups, the closed subgroup theorem, the Lie correspondence, the adjoint representation and the connection between normal subgroups and ideals in the Lie algebra are all minutely explained.
Chapter 21 completes the text’s full treatment of introductory Lie group theory with proper Lie group actions, covering and homogeneous spaces and the quotient manifold theorem, with applications to Lie theory. The author’s point of view on Lie groups is concerned more with their geometry than with their algebra; there is nothing here on the rich structure and classification of Lie algebras or on the theory of group representations, also a very rich and extensive subject. For that one could suggest the elementary texts by Brian Hall and Bröcker and tom Dieck. But obviously Lee’s sturdy presentation must be the starting point.
The concluding chapter on symplectic manifolds barely goes beyond the Darboux theorem (which puts a symplectic manifold into a canonical form), its analogue for contact structures and the basic ideas of Hamiltonian vector fields up to Noether’s theorem. Nevertheless, the final section on non-linear first-order partial differential equations (focusing on the Hamilton-Jacobi equation from classical mechanics) does round out the chapter nicely and illustrates what the concepts are good for. Needless to say, none of the really deep results on symplectic geometry can be conveyed here nor is Noether’s theorem connecting symmetries and conserved quantities put to any use in physics, where it is so important to the modern perspective on classical dynamics. The Hamiltonian formulation of classical dynamics is, after all, what makes symplectic geometry into an interesting subject in the first place and, over the course of the twentieth century, it has been elaborated into a high art, with such gems as the Kolmogorov-Arnold-Moser and Nekhoroshev theorems. But Lee is not a physicist, either by training or by inclination! The problem sets at the close of these last four chapters get longer and more challenging than in the rest of the book. If one persists, one will gain the satisfaction of seeing how all the apparatus of smooth manifold theory can lead to interesting mathematics, though, of course, one barely scratches the surface of entire mature disciplines here.
Summary: a solid account of the basic theory of smooth manifolds, this sprawling tome covers just about everything a beginning graduate student needs to know at an appropriate level. Lee’s exposition and technique of proof are models of lucidity; one will note many a short aside that really helps the reader to appreciate why a concept matters or why it is formalized in just the way it is. As for overall style, perhaps one could say that Lee takes after Grothendieck. Unlike what is the case with many, who value concrete calculations, what appeals to Lee’s mind seems to be more the structural aspects of the subject. This makes for comparatively few explicit derivations but a great deal of conceptual clarity throughout. Most of the problems reflect this emphasis as well. Those willing to invest the time to go through the whole lengthy text patiently will be amply rewarded with deep insight into the foundations of differentiable manifolds and afterwards ready for doctoral-level research.
John Lee, whose introduction to topological manifolds we have reviewed here earlier, has continued his pedagogical efforts with a welcome introduction to smooth manifolds, also in the Springer Verlag series of graduate texts in mathematics and now out in a second edition. At over seven hundred pages, it may appear daunting, but the student should remember that, as the saying goes, there is no royal road to learning and that painstaking application to mastering the fundamentals will be rewarded with clarity of understanding later on. The pacing over twenty-two chapters is quite leisurely. It starts out with the basic definitions and some standard examples of smooth structures and mappings between them. Smoothness means that functions should be continuously differentiable infinitely many times; it is not so much of a restriction as might at first appear, since in practice most of the time one deals with functions given in analytical form, which are always smooth. Rather, the condition of smoothness serves to render concepts from calculus unproblematical, in as much as derivatives will exist up to any order one may need; as such, it captures our elementary intuition about the nature of space. Lee does treat the case of manifolds with boundary or corners with care, in view of their use in the context of Stokes’ theorem in chapters to come.
Starting in chapter three on tangent vectors, Lee delivers what we have a right to expect from him, a good course on the elements of smooth manifold theory. He reworked his text completely for the second edition, and the fruit of his labors shows. The perspective is modern throughout. Lee adopts a definition of a tangent as a derivation in the global algebra of smooth functions, a departure from the more conventional approach to go first to germs of functions at a point. Presumably he does so in order to keep things simpler, but, it seems to this reviewer, he obscures somewhat thereby the strict locality of the concept of the tangent. However that may be, armed with the notion of tangent and differential of a map, he can proceed to investigate submersions, immersions, embeddings and submanifolds via the rank theorem (in both local and global forms). The discussion, although it seems to progress at a glacial pace, is very thorough and clear, as is the treatment of the subtle distinction between an embedding and the image of an embedding in the subspace topology. Equipped with all this technology, he arrives at a few results in the theory of manifolds that are more than quite trivial: Sard’s theorem on the vanishing in measure of the set of critical points of a smooth mapping and Whitney’s embedding and approximation theorems, which say respectively that every smooth manifold can be embedded in a Euclidean space of sufficiently high dimension and that continuous maps can be closely approximated by smooth ones.
From here, Lee launches into a nice treatment of the basics of Lie groups and Lie algebras as another application of the circle of ideas surrounding smoothness. Then, he treats flows of vector fields in a fair amount of detail before going on to vector bundles in general and the cotangent and tensor bundles in particular. The whole apparatus of differential forms and the exterior derivative, which can be confusing to beginners but is in fact vital to a mature grasp of geometry, is built up in preparation for Stokes’ theorem (which generalizes integration by parts). Lee’s explanation of how to define integration on a curved manifold is well done. Another treat is a very thorough treatment of orientations, not often encountered in texts at this level. This part of the text is finished off with a somewhat rushed exposition of de Rham cohomology, the Mayer-Vietoris theorem (which renders cohomology effectively computable) and degree theory, followed by an exhaustive and rather technical proof of the de Rham theorem, which establishes a surprising isomorphism between the de Rham and singular cohomology groups, thus showing the topological invariance of the former. The reader will appreciate being inducted into these higher mysteries, or at least being given a glimpse into them although a text of such expansive scope cannot afford to dwell on them. Those wanting more could look to Warner’s old textbook, which covers these things from the more advanced sheaf-theoretic point of view.
A word about the homework exercises strewn liberally throughout the text, some 397 in all over 22 chapters: as always, working a good number of them is indispensable, as they do force one to get acquainted with what one has just read and to make use of it. Very few are strictly computational; most can be seen as filling in a gap in the exposition or elaborating a straightforward extension of the material. As in his previous book on topological manifolds, a fair number are included to point the way to more advanced topics, which means that they are not in themselves very hard but intended merely to introduce a sophisticated way of looking at what one already knows. The level of the majority of the problems is reasonable for the textbook’s audience, but one could be disappointed not to come across very many that are very challenging. Lee’s statement in the preface that most of them should take a typical student hours to solve seems like rather an overestimate, at least for most of the book; as mentioned below, in the few chapters at the end several do in fact rise to the level of difficulty that could take hours (but not days!).
The last four chapters are worth the wait, as Lee gets at last to some significant applications of the ideas he has so slowly built up in the preceding chapters: distributions and foliations, the exponential map and Lie group actions and their quotients, symplectic and contact structures and the beginnings of the theory of first-order partial differential equations. The coverage of these topics is complete; for instance, the Frobenius theorem is studied both from the local and from the global point of view and in terms of distributions in the tangent bundle or differential ideals in the graded algebra of differential forms. Numerous alternate ways of characterizing involutivity are presented. Likewise, the exponential map from a Lie algebra to its Lie group, its relation to one-parameter subgroups, the closed subgroup theorem, the Lie correspondence, the adjoint representation and the connection between normal subgroups and ideals in the Lie algebra are all minutely explained.
Chapter 21 completes the text’s full treatment of introductory Lie group theory with proper Lie group actions, covering and homogeneous spaces and the quotient manifold theorem, with applications to Lie theory. The author’s point of view on Lie groups is concerned more with their geometry than with their algebra; there is nothing here on the rich structure and classification of Lie algebras or on the theory of group representations, also a very rich and extensive subject. For that one could suggest the elementary texts by Brian Hall and Bröcker and tom Dieck. But obviously Lee’s sturdy presentation must be the starting point.
The concluding chapter on symplectic manifolds barely goes beyond the Darboux theorem (which puts a symplectic manifold into a canonical form), its analogue for contact structures and the basic ideas of Hamiltonian vector fields up to Noether’s theorem. Nevertheless, the final section on non-linear first-order partial differential equations (focusing on the Hamilton-Jacobi equation from classical mechanics) does round out the chapter nicely and illustrates what the concepts are good for. Needless to say, none of the really deep results on symplectic geometry can be conveyed here nor is Noether’s theorem connecting symmetries and conserved quantities put to any use in physics, where it is so important to the modern perspective on classical dynamics. The Hamiltonian formulation of classical dynamics is, after all, what makes symplectic geometry into an interesting subject in the first place and, over the course of the twentieth century, it has been elaborated into a high art, with such gems as the Kolmogorov-Arnold-Moser and Nekhoroshev theorems. But Lee is not a physicist, either by training or by inclination! The problem sets at the close of these last four chapters get longer and more challenging than in the rest of the book. If one persists, one will gain the satisfaction of seeing how all the apparatus of smooth manifold theory can lead to interesting mathematics, though, of course, one barely scratches the surface of entire mature disciplines here.
Summary: a solid account of the basic theory of smooth manifolds, this sprawling tome covers just about everything a beginning graduate student needs to know at an appropriate level. Lee’s exposition and technique of proof are models of lucidity; one will note many a short aside that really helps the reader to appreciate why a concept matters or why it is formalized in just the way it is. As for overall style, perhaps one could say that Lee takes after Grothendieck. Unlike what is the case with many, who value concrete calculations, what appeals to Lee’s mind seems to be more the structural aspects of the subject. This makes for comparatively few explicit derivations but a great deal of conceptual clarity throughout. Most of the problems reflect this emphasis as well. Those willing to invest the time to go through the whole lengthy text patiently will be amply rewarded with deep insight into the foundations of differentiable manifolds and afterwards ready for doctoral-level research.
August 18, 2012
Introduction to Smooth Manifolds from John Lee is one of the best introduction books I ever read. I read most of this book, except for the appendices at the end and proofs of some corollaries. This book covers a couple of subjects:
(*) First the theory of smooth manifolds in general (ch1, 2, 3, 4, 5 and 6), smooth maps, (co)tangent spaces, (co)vector fields and vector bundles. These first few chapters contain a lot of examples. These six chapters can be used as a first introduction course to smooth manifolds. Basically only tools from topology, linear algebra and calculus are used here.
(*) The second major part is about sub-manifolds (ch7, 8, 9 and 10), theorems like the Inverse Function, Implicit Function and Rank theorem are covered. Here the definition of Lie groups is introduced. Also Whitney's approximation theorems are discussed.
(*) The third major part is about integration on manifolds (ch11, 12, 13 and 14). These tools are constructed from scratch. There is a lot of attention to Tensors, Riemannian manifolds, differential forms and orientation on manifolds. In ch14 Stokes theorem is proved, a nice milestone in Differential Geometry.
(*) Chapters 15 and 16 can be viewed as a appendix of the third major part. In these sections the Rham theory is developed, homology and cohomology with the Rham Theorem as main result.
(*) The last part (ch17, 18, 19, 20) explores the circle of ideas surrounding integral curves and flows of vector fields. Also a lot of new theory for Lie groups and Lie algebras is introduced and developed in this section.
This book is very well written. It contains a lot of information (it's 628 pages long!) but is very well organized. Every new concept starts with a good motivation, mostly based on the "ordinary" case in R^2, R^3 or R^n and is then developed in more generality. In this way it is good didactic book and in my opinion suitable for one or more courses on Differential Geometry. Since everything is well organized and it contains a lot of information it can also be used as a reference. I encourage everyone who has some interests in differential geometry to buy this book and at least read some sections of it. 5/5 stars!
(*) First the theory of smooth manifolds in general (ch1, 2, 3, 4, 5 and 6), smooth maps, (co)tangent spaces, (co)vector fields and vector bundles. These first few chapters contain a lot of examples. These six chapters can be used as a first introduction course to smooth manifolds. Basically only tools from topology, linear algebra and calculus are used here.
(*) The second major part is about sub-manifolds (ch7, 8, 9 and 10), theorems like the Inverse Function, Implicit Function and Rank theorem are covered. Here the definition of Lie groups is introduced. Also Whitney's approximation theorems are discussed.
(*) The third major part is about integration on manifolds (ch11, 12, 13 and 14). These tools are constructed from scratch. There is a lot of attention to Tensors, Riemannian manifolds, differential forms and orientation on manifolds. In ch14 Stokes theorem is proved, a nice milestone in Differential Geometry.
(*) Chapters 15 and 16 can be viewed as a appendix of the third major part. In these sections the Rham theory is developed, homology and cohomology with the Rham Theorem as main result.
(*) The last part (ch17, 18, 19, 20) explores the circle of ideas surrounding integral curves and flows of vector fields. Also a lot of new theory for Lie groups and Lie algebras is introduced and developed in this section.
This book is very well written. It contains a lot of information (it's 628 pages long!) but is very well organized. Every new concept starts with a good motivation, mostly based on the "ordinary" case in R^2, R^3 or R^n and is then developed in more generality. In this way it is good didactic book and in my opinion suitable for one or more courses on Differential Geometry. Since everything is well organized and it contains a lot of information it can also be used as a reference. I encourage everyone who has some interests in differential geometry to buy this book and at least read some sections of it. 5/5 stars!
Want to read
February 26, 2024chapter 7
Theorem 7.7 (existence of a universal covering group)
G connected lie group => exists unique simply connected lie group G'(universal covering group of G) that admits a smooth covering map pi:G'->G (homomorphism lie group)
chapter 8
The set of all smooth vector fields on M is a module over the ring C^infinty(M)
F:M->N and X vector in M then dF(p)(X(p))Y(F(p)) is a vector in N (if F is nice)
(X F-related Y)
Prop 8.16. X, Y F-related <-> X(foF)=(Yf)oF for all f smooth
Example 8.17 F:R->R^2 given by F(t)=(cos(t),sin(t)) then d/dt is F-related with Y=xpartial_y-ypartial_x.
Proof. (d/dt (foF))(t)=d/dt_t (foF(t))=d/dt|_t(f(cos(t),sin(t))=f_x(F(t))(-sin(t))+f_y(F(t)) cos(t)
((x partial_y-y partial_x)(f)oF)(t)=(x partial_y-y partial_x)|_F(t)f(x,y)=cos(t)f_y(F(t))-sin(t)f_x(F(t))
prop 8.26 (coordinate formule for the lie bracket)
[X,Y]=(XY^j-YX^j)partial_x_j
Chapter 14 (differential forms)
Differential forms on manifolds
Lambda^{k}T*M:=union_{p in M} Lamda^k(T_p*M)
A section of Lambda^kT*M is called a differential k-form (in short, k-form)
Appendix
Theorem c.15 (Taylor theorem)
f(x)=P_k(x)+R_k(x) where P_k(x)=f(a)+sum 1/m! sum_{I: |I|=m} derivative_I f(a)(x-a)^I
(multi-index notation)
Multi linear algebra
V_1, V_2, W vector spaces, a map F:V_1xV_2->W is multilinear if it is linear of each variable
F:V_1x...xV_k->R and G:W_1x...xW_l->R then F plus G: V_1x...xV_kxW_1x...xW_l->R by
F plus G(v_1,..., v_k, w_1,..., w_l)=F(v_1,...,v_k)G(w_1,...,w_l) (tensor product of F and G)
Covariant and contravariant tensors on a vector space
A covariant k-tensor on V is an element of the k-fold tensor product V^*plus ... plus V^*
typically think of a real valued multilinear function of k elements of V:
alpha: Vx...xV->R (k copies) (here k is the rank of alpha)
0-tensor: only numbers
F:M->N
dF(p):T_pM->T_{F(p)}N defined by dF(p)(v)(f)=v(foF)
Theorem 7.7 (existence of a universal covering group)
G connected lie group => exists unique simply connected lie group G'(universal covering group of G) that admits a smooth covering map pi:G'->G (homomorphism lie group)
chapter 8
The set of all smooth vector fields on M is a module over the ring C^infinty(M)
F:M->N and X vector in M then dF(p)(X(p))Y(F(p)) is a vector in N (if F is nice)
(X F-related Y)
Prop 8.16. X, Y F-related <-> X(foF)=(Yf)oF for all f smooth
Example 8.17 F:R->R^2 given by F(t)=(cos(t),sin(t)) then d/dt is F-related with Y=xpartial_y-ypartial_x.
Proof. (d/dt (foF))(t)=d/dt_t (foF(t))=d/dt|_t(f(cos(t),sin(t))=f_x(F(t))(-sin(t))+f_y(F(t)) cos(t)
((x partial_y-y partial_x)(f)oF)(t)=(x partial_y-y partial_x)|_F(t)f(x,y)=cos(t)f_y(F(t))-sin(t)f_x(F(t))
prop 8.26 (coordinate formule for the lie bracket)
[X,Y]=(XY^j-YX^j)partial_x_j
Chapter 14 (differential forms)
Differential forms on manifolds
Lambda^{k}T*M:=union_{p in M} Lamda^k(T_p*M)
A section of Lambda^kT*M is called a differential k-form (in short, k-form)
Appendix
Theorem c.15 (Taylor theorem)
f(x)=P_k(x)+R_k(x) where P_k(x)=f(a)+sum 1/m! sum_{I: |I|=m} derivative_I f(a)(x-a)^I
(multi-index notation)
Multi linear algebra
V_1, V_2, W vector spaces, a map F:V_1xV_2->W is multilinear if it is linear of each variable
F:V_1x...xV_k->R and G:W_1x...xW_l->R then F plus G: V_1x...xV_kxW_1x...xW_l->R by
F plus G(v_1,..., v_k, w_1,..., w_l)=F(v_1,...,v_k)G(w_1,...,w_l) (tensor product of F and G)
Covariant and contravariant tensors on a vector space
A covariant k-tensor on V is an element of the k-fold tensor product V^*plus ... plus V^*
typically think of a real valued multilinear function of k elements of V:
alpha: Vx...xV->R (k copies) (here k is the rank of alpha)
0-tensor: only numbers
F:M->N
dF(p):T_pM->T_{F(p)}N defined by dF(p)(v)(f)=v(foF)
April 4, 2021
I used this book as a supplement to the suggested reading for a graduate course on Differential Topology.
Best decision I made this semester.
Lee's treatment of the fundamental concepts of smooth manifolds (which are a common foundation for both differential topology and differential geometry) is unparalleled. In particular, I beg you to read the end of Chapter 3, in which Lee discusses the *MANY* different ways in which tangent spaces have been defined over the decades. That discussion is essential reading, and will enable you to move around between different texts on the subject with ease.
Buy this book.
Best decision I made this semester.
Lee's treatment of the fundamental concepts of smooth manifolds (which are a common foundation for both differential topology and differential geometry) is unparalleled. In particular, I beg you to read the end of Chapter 3, in which Lee discusses the *MANY* different ways in which tangent spaces have been defined over the decades. That discussion is essential reading, and will enable you to move around between different texts on the subject with ease.
Buy this book.
March 13, 2017
The advantage of a long-winded approach emphasizing a geometric way of thinking is to place intuition first and formalism second. Give the reader an intuitive view of the subject--get them to "see" the right picture in as many words necessary (and I do expect that it helps to have a strong ability to visualize when learning the subject, at least using this approach). Within this approach (and especially when grounding everything in well-selected examples), the book is quite successful, chapters 3 and 16 on tangent vectors and integration on manifolds particularly so.
February 20, 2021
It's hard to judge differential geometry books at this point because I'm familiar with the material, but overall I think it's one of the best self-study books I've found. Very good at motivating the definitions.
Update: I still think it's good, although I think some of the proofs are hard to follow and I had to just do them myself. He also does an annoying thing where he first defines a concept on R^n (fine), but then doesn't clearly mark when he's extending the definition to arbitrary smooth manifolds. Please just give me a \begin{definition}.
Update: I still think it's good, although I think some of the proofs are hard to follow and I had to just do them myself. He also does an annoying thing where he first defines a concept on R^n (fine), but then doesn't clearly mark when he's extending the definition to arbitrary smooth manifolds. Please just give me a \begin{definition}.
September 1, 2025
9.4/10 Very good textbook. This is the Huckleberry Finn to ITM's Tom Sawyer. Lee's writing is always so clear and thorough, but still never feels too detailed. The exercises and problems are great, the pictures are helpful, and the subject matter is very cool. I was a bit worried at first that I would find differential geometry too analytic, but it's just so fun you don't mind. The only parts of the book I didn't love were the sections explicitly about solving DEs; I just don't care about that. But it's not at all surprising to me that this has become the standard book on the subject; I couldn't recommend it more. Frankly it's kinda spoiled me for other books which now feel so poorly written by comparison.
Read
October 13, 2020There is a lot of detail and the exercises require local computations. This may feel tedious but can be well worth the effort. Even if one does not end up working in Riemannian geometry, it is good psychological training: do not fear computations. However, I do agree that it can be slow going to get through this book.
March 18, 2023
czytanie tej ksiazki oddaje
May 27, 2025
Oh all smooth manifolds are paracompact? Why don’t you para come pack this bussy.
August 30, 2011
Should be pithier.
Displaying 1 - 11 of 11 reviews