Prerequisites for Differential Geometry

[hangman1414]

Hello, I was wondering what you guys think is the absolute minimum requirements for learning Differential Geometry properly and also how would you go about learning it once you got to that point, recommended books, websites, etc. I am learning on my own because of some short circuit in my brain that makes me enjoy that sort of thing, lol. Thanks in advance, and nice board you guys have here!

 

[dextercioby]

Hello, I was wondering what you guys think is the absolute minimum requirements for learning Differential Geometry properly and also how would you go about learning it once you got to that point, recommended books, websites, etc. I am learning on my own because of some short circuit in my brain that makes me enjoy that sort of thing, lol. Thanks in advance, and nice board you guys have here!

1).A very solid course on calculus.Starting from "0".From the basic elements of topology,up until triple/multiple integrals and Gauss-Ostrogradsky formula.
2).A course on linear algebra would be useful.But not essential as 1),4)
3).A course on functional analysis including topology and the theory of measure and integration.Again not as important as 1),4)
4).A course (or at least very solid knowledge) of ordinary euclidean 3D&2D geometry.

About books on Differential Geometry in English i cannot give any reference,as i learned the basics for phyiscs from online resources.There are many free online outlines and lecture notes on DiffGeom online,especially at the American Universities/Colleges.I believe you can searsh this forum for many links.

Daniel.

Edit:Check this thread and especially post no.12:
useful link

 

[mathwonk]

ted shifrin has some course notes for diff geom on his website.

http://www.math.uga.edu/~shifrin/

 

[pmb_phy]

1).A very solid course on calculus.Starting from "0".From the basic elements of topology,up until triple/multiple integrals and Gauss-Ostrogradsky formula.

Note that the OP said "absolute minimum" requirements. Topology is not a absolute minimuim requirement to learn differential geometry. In fact I've never studied it myself and I know differential geometry.

A free online tensor calculus/diff geo text is at
http://www.math.odu.edu/~jhh/counter2.html

Pete

 

[dextercioby]

Note that the OP said "absolute minimum" requirements. Topology is not a absolute minimuim requirement to learn differential geometry. In fact I've never studied it myself and I know differential geometry.
A free online tensor calculus/diff geo text is at
http://www.math.odu.edu/~jhh/counter2.html
Pete

What's the basic notion of differential geometry?Is it "manifold"?Does the definition of a manifold assume the topologic structure on a set...??
I would send you to the page 63 of the famous GR course online by Sean M.Carrol.I think you have it on your computer.If not,it has the address:
arXiv:gr-qc/9712019 v1 3 Dec 1997

Daniel.

PS.I hope u're not implying that Diff.Geom. can be thought without the concept of manifold.
That would be the differential geometry of Gauss,wouldn't it??Not the one of Cartan...
What kind of differential geometry do u know...?In whose boat would u be??In the one of Einstein,or in the one of Hermann Weyl and David Hilbert??

 

[dextercioby]

I have the book that u reccomended sinc last year.It's pretty good.Basics,both in GR (calculative examples,brings example of Landau & Lifschitz),pretty thorough in fluid dynamics,has the only online treatment i found on Boltmzann equation treatment of fluids.Pretty weak with em.stuff.Worths reading.

Daniel.

 

[pmb_phy]

What's the basic notion of differential geometry?

Differential geometry is the study of Riemannian manifolds.
Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects.

A precise knowledge of the later is not required for a minimum understanding to start reading Diff geo.

Is it "manifold"?Does the definition of a manifold assume the topologic structure on a set...??

The OP does not need to go out and read a text on topology before he starts reading a text on diff geo.

Keep in mind what the OP asked for. He wanted to know what the absolute minimum requirements are. I didn't say that topology is unrelated to diff geo. This is equivalent to me saying that calculus is required, yet a detailed study of real analysis is not even though the later is the basis for the former.

Pete

 

[dextercioby]

Differential geometry is the study of Riemannian manifolds.

I knew that differential geomatry is a study of MANIFOLDS,not of a subset on which u put a connection and a metric (Riemannian manifolds).Maybe we should ask a mathematician who knows better... :tongue2:

A precise knowledge of the later is not required for a minimum understanding to start reading Diff geo.

Who said a "precise" knowledge??But the definition of manifolds involves a few concepts of topology and differential calculus.

The OP does not need to go out and read a text on topology before he starts reading a text on diff geo.

1.What is OP??
2.I advise him to do that.Let's see whose advice he follows... I guess it depends on how well he wants to learn...

Keep in mind what the OP asked for. He wanted to know what the absolute minimum requirements are. I didn't say that topology is unrelated to diff geo. This is equivalent to me saying that calculus is required, yet a detailed study of real analysis is not even though the later is the basis for the former.

Okay...absolute minimum requirements would mean reading Gauss' articles on geometry of curves and surfaces. :tongue2:

Daniel.

PS.U know that a hot branch of mathematics is differential topology...

 

[pmb_phy]

Who said a "precise" knowledge??

The OP simple wants to know what he needs to get started learning diff geo.

But the definition of manifolds involves a few concepts of topology and differential calculus.

All one needs to know is what a manifold is and a rigorous definition is not required for an intro to diff geo. If one thinks of a "manifold" simply as something which looks "locally" like a bit of n-dimensional Euclidean space Rⁿ then you're all set.

1.What is OP??

Original Poster

2.I advise him to do that.Let's see whose advice he follows... I guess it depends on how well he wants to learn...

You're missing the point. He wanted to know what he needed to start learning. He didn't imply that he'd stop there. Its good to get a start simply with calculus, linear diff eq, linear algebra and vector analysis. Those are the basics. With that under his belt he can start (not end) learning diff geo.

Pmb

 

[dextercioby]

All one needs to know is what a manifold is and a rigorous definition is not required for an intro to diff geo. If one thinks of a "manifold" simply as something which looks "locally" like a bit of n-dimensional Euclidean space Rⁿ then you're all set.

To make an analogy with a subject dear to me,it's like saying:
Q:"What is the basics/minimum for learning QM??"
A:"If you know that a Hilbert space is a vector space with a scalar product, <<then you're all set>>".

Daniel.

 

[pmb_phy]

To make an analogy with a subject dear to me,it's like saying:
Q:"What is the basics/minimum for learning QM??"
A:"If you know that a Hilbert space is a vector space with a scalar product, <<then you're all set>>".

Daniel.

If I were to use that analogy then the question would be "What minimal math/physics do I need to know to start learning quantum theory". You don't [n]need[/i] to know Hilbert space.

You're giving him those things which are sufficient. He was not asking for that which he needs for a sufficient background. He's asking for the necessary background.

Pete

 

[quetzalcoatl9]

wow

I knew that differential geomatry is a study of MANIFOLDS,not of a subset on which u put a connection and a metric (Riemannian manifolds).Maybe we should ask a mathematician who knows better... :tongue2:

this is the exact kind of response that terrified me when i asked people who supposedly "were in the know". with age i have learned, in the end, that they knew very little.

my answer to the original poster: don't listen to anything anyone on this thread says (including myself). just try things your way.

of course you do not absolutely need to read a book on topology before picking up differential geometry. most differential geometry books will give you a rigorous definition of a differentiable manifold. that is all.

if you want more than that, great, get a topology book and read your heart out.

but there is no sense scaring a beginner about such things.

hangman, what i did was:

1) learned multivariable calculus
2) learned some very basic linear algebra
3) opened a book on differential geometry

as time progressed i learned many other things outside of the scope of that simple list. but if you do that, you will be on your path...

 

[mathwonk]

I would like to point out one small thing that has apparently escaped some people posting advice here. The original question was not "what is the minimum foreknowledge recommended for starting to read differential geometry?" It was, what is the minimum needed to "learn Differential Geometry PROPERLY".

To understand what one is reading "properly", and put it into some context, requires more than simply to start reading. Of course the absolute minimum required to open a book on the subject is nothing but an ability to read the language the book is written in.

 

[Hurkyl]

One interesting approach (that I often take) is this:

Try to read differential geometry, and from that you can learn what other topics you should study.

 

[dextercioby]

One interesting approach (that I often take) is this:
Try to read differential geometry, and from that you can learn what other topics you should study.

In my posts i virtually (wrongly) excluded the idea that he could find a decent book on diff.geom.which has an introductory chapter dealing with the basis of topology.If that kind of books exist,then he should be looking for them.

Daniel.

PS.To the OP:"Good luck!".

 

[mathwonk]

some people have apparently learned differential geometry without knowing topology. I will add that from the opposite perspective I have studied topology and still do not feel I understand differential topology. It is hopeless to be fully prepared to understand anything. Learning is a process of going up and back and also around and around with a topic as it gradually becoems more clear. What one is satisifed as "understanding a subject" also evolves. With me I feel I understand a subject if I can see how every important idea was thought of and how every proof works, and have become independent of written expositions of the subject, since I see so clearly the workings that I never again need to refer to them. In particular I can give every proof, and even contrive new proofs, of every idea. I also wish to be able to explain everything, motivate everything, and illustrate everything in simple examples. Also one should be able to apply it and teach it. It takes me a long time to reach this level, and in fact I have not yet reached it with any subject, even calculus of one variable, although I have been studying and teaching it for over 40 years. indeen every year i learn something new about calculus as i teach it and try to explain it again. this year for instance I learned the proper hypotheses for the mean value theorem. i.e. how much should assume to have a constant function? continuity and derivative zero a.e. is not enough, but lipschitz continuity and derivative zero a.e. is enough. So lipschtiz continuity is a crucial property that an indefinite integral has, stronger than continuity. i.e. the indefinite integral of a riemann integrable function is a lipschitz continuous function that has a derivative almost everywhere. moreover within the realm of lipschitz continuous functions it is determined up to a constant by that derivative a.e.

I had read this theory in real variable courses in reference to lebesgue integration (the relevant concept there is "absolute continuity") but never understood it until this year. I have long referred to the radon nikodym theorem ignorantly as a typical example of a useless technical theorem in real analysis, and have argued for its omission from the phd syllabus. now after thoughtfully teaching one variable inhtegral calculus yet again, I finally understand its importance as characterizing which functions are integrals of other functions. so anytime anyone says you do not need to know such and such, consider who is saying this. maybe they themselves also do not understand the topic. at least this has often been true of me and my advice of this nature. all it means is I myself do not understand why something is needed or useful.

to make a long story longer, you will never regret any amount of topology you learn, as a backdrop to geometry or anything else. topology is the most basic aspect of geometry. but if your main interest is differentisal geometry, you may just plunge in learning it. however remember that your understanding will deepen as you learn more auxiliary subjects like topology and even algebra. Indeed tensors cannot be properly understood without some multilinear algebra.

 

[hangman1414]

Hello again, I thank all of you for your thoughtful replies. I think Mathwonk hit the nail on the head with his interpretation of my original post. By properly I meant I do not wish to take many shortcuts that may hinder my understanding of differential geometry, but I also do not want to learn anything that is not neccesary or related to the subject. A good understanding of differential geometry is my ultimate goal at this point, and if that takes learning a lot of linear algebra (which I have a decent understanding of right now) then that is fine, if that's what it takes to realize my goal.
Also, the other reason I posted was I had tried a number of years ago to read a differential geometry book and was lost almost from the beginning, I didn't want that to happen again.
Thank you for your time, and take care!

 

[mathwonk]

I will not necessarily claim that one must know differential manifolds to study differential geometry but it is entirely possible that it is necessary in order to understand the subject properly. michael spivak's treament spends the first volume on a detailed introduction to differential manifolds and then in the second volume, explains in equal detail the fundamental controbutions to differential geometry of Gauss and Riemann, including translations of their original papers. Of course the theory in spivak's volume one did not exist when gauss and riemann did their work, so it cannot have been necessary to it, but it is very helpful to us in understadning what they did. here are spivaks own remarks from his preface:

"For many years I have wanted to write the Great American Differential Geometry book. Today a dilemma confronts anyone intent on penetrating the mysteries of differential geometry. On the one hand, one can consult numerous classical treatments of the subject in an attempt to form some idea how the concepts within it developed. Unfortunately, a modern mathematical education tends to make classical mathematical works inaccessible, particularly those in differential geometry. On the other hand, one can now find texts as modern in spirit, and as clean in exposition, as Bourbaki's Algebra. But a thorough study of these books usually leaves one unprepared to consult classical works, and entirely ignorant of the relationship between elegant modern constructions and their classical counterparts. ... no one denies that modern definitions are clear, elegant, and precise; it's just that it's impossible to comprehend how anyone ever thought of them. And even after one does master a modern treatment of differential geometry, other modern treatments often appear simply to be about totally different subjects.

There are two main premises on which these notes are based. The first premise is that it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely in order to rigorize the concepts of classical differential geometry.

The second premise for these notes is that in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary; there is no point in introducing the curvature tensor without explaining how it was invented and what it has to do with curvature. The second volume of these notes gives a detailed exposition of the fundamental papers of Gauss and Riemann."

i myself suggest getting both books and reading volume two for the exciting geometry and referring back to volume one as one finds oneself wanting to understand volume two better. i.e. use vol 2 as motiovation to plow through vol 1.

 

[mathwonk]

on reflection, since volumes 1 and 2 of spivaks differential geometry comprise over 1000 pages I guess, i suggest that the minimum background needed to understand differential geometry properly is a good grasp of several variables calculus, as explained wonderfully and succintly in spivaks calculus onmanifolds, about 140 pages.

 

[Perion]

There's like five volumes in the total set of Spivaks A Comprehensive Introduction to Differential Geometry. See the five volume links at http://www.mathpop.com/bookhtms/dg.htm for volume contents and reviews. How would you like the totality of that information swimming about in your head

BTW - anyone thinking about Spivak's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus as some sort of introduction should read the reviews (especially Kishan Yerubandi's) at .

Perion

 

[mathwonk]

After reading the suggested reviews, Kishan Yerubandi's is almost the only one that does not rave about spivak's wonderful book. I myself am praising it from personal experience reading it (30 years ago), not reading reviews. yes it was too hard for my weak students, but ideal for my ambitious ones. a good student should never listen seriously to the fears of a poor student.

some people are scared by any form of exposition. "calculus on manifolds" is a short book for people who prefer that. of course the material is dense. the diff geom series is long, for people who like all the details lavishly explained. volume one of the diff geom series retreats the material of calc on manifolds in much greater detail.

i recommend the short one. it is lighter to carry around, and seems more manageable. if you finish those 140 pages, or even the first 108, you will have climbed out of the pit of the mathematically challenged, that so many viewers of this site are in, and into the ranks of the knowledgeable users of modern mathematics. knowing or not knowing about chains and forms and manifolds, essentially separates the sheep from the goats on this site at the upper levels.

people here who are struggling to learn tensors in physics would probably be better off going back and learning calculus the way spivak presents it, and then re-approaching tensors. in fact they will already know what tensors are after chapter 4 of spivak's little book.

 

[Perion]

Concerning Spivak's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, I now regret mentioning that single, negative, review at Amazon.com. After reading several other (non-Amazon) reviews last night and spending time studying the book's first chapter, table of contents, and index (previewed at Amazon), I'm convinced that your praise of the book is justified. [In fact, I decided to order it for myself which I'm sure will give my wife great cause for celebration... NOT!]

BTW - I'm new to these forums but have gained a great deal from many of the detailed posts that can be found here. Among others, I printed out your "primer on linear algebra" post the other day and found it to be an interesting overview. I also spent more time than I probably should have studying the GREAT discussion in the topic What is a tensor?. The discussions in these forums are gold mines!

You seem to post in unusual detail and pertenence. I appreciate you (and many others) taking the time to offer us hungry travelers meat for our sojourn. Not being able to attend a formal school, at least for now, I am very thankful for finding some forums where there are serious, knowlegable, and dedicated folks that can offer guidence to people like me who struggle in our persuits, on our own, with whatever time and resources we can corral (in my case, while juggling a 70 hr. work week and family responsibilities ). Keep up the good work.

Best wishes,
Perion

 

[mathwonk]

that is the nicest most generous response anyone could hope for. thank you. i am a math teacher who enjoys the chance to explain topics i love and enjoy here. i post mostly during holidays, when my teaching leaves me free. my weakness is a tendency to impatience and dogmatism. there are many people posting here who are great role models in knowledge, objectivity, energy and patience (hurkyl, matt grime,...). but it is hard to fully emulate them. the linear algebra post was just off the top of my head one night so not so well organized nor edited. i prefer posts with mathematical content. so when bragging on some book other, i sometimes just try to write a primer of the topic. best wishes.

 

[majutsu]

This is a really awesome thread. Thanks for all who contributed. I learned a lot. Mathwonk, you especially deserve kudos for the post discussing calculus and the PhD syllabus, as that personal tidbit really brought the concept of "deepened understanding" to life. Thanks a lot.

 

[gvk]

If someone does not know anything exept calculus, I would recommend the excellent book of Miishchenko A., Fomenko A.
A course Differential Geometry and Topology, Mir Publisher 1988

This is the only book on differential geomentry which has more than 350 illustrations and many simple examples.

The book is cheap, but difficult to find. However, it is available in many libraries.

Contents
Chapter 1. INTRODUCTION TO DIFFERENTIAL GEOMETRY ... 12
1.1. Curvilinear Coordinate Systems. Simple Examples ... 12
1.1.1. Background...... 12
1.1.2. Cartesian and Curvilinear Coordinates... 15
1.1.3. Simple Examples of Curvilinear Coordinate Systems 21
1.2. The Length of Curve in a Curvilinear Coordinate System ... 25
1.2.1. The Length of Curve in a Euclidean Coordinate System 25
1.2.2. The Length of Curve in a Curvilinear Coordinate System 28
1.2.3. The Concept of the Riemannian Metric in a Euclidean Domain ...... 32
1.2.4. Indefinite Metrics..... 35
1.3. Geometry on a Sphere and on a Plane... 3&
1.4. Pseudosphere and Lobachevskian Geometry... 46
Chapter 2. GENERAL TOPOLOGY.... 67
2.1. Definition and Basic Properties of Metric and Topological Spaces..... 67
2.1.1. Metric Spaces...... 67
2.1.2. Topological Spaces..... 70
2.1.3. Continuous Mappings.... 72
2.2. Connectedness. Separation Axioms ... 76
2.2.1. Connectedness...... 77
2.2.2. Separation Axioms..... 79
2.3. Compact Spaces..... 82
2.3.1. Definition ..... 82
2.3.2. The Properties of Compact Spaces... 83
2.3.3. Metric Compact Spaces... 84
2.3.4. Operations Over Compact Spaces... 86
2.4. Functional Separability. Partition of Unity... 87
2.4.1. Functional Separability... 87
2.4.2. Partition of Unity..... 90
Chapter 3. SMOOTH MANIFOLDS (GENERAL THEORY)... 92
Introduction ...... 92
3.1. The Concept of a Manifold... 93
3.1.1. Fundamental Definitions... 93
3.1.2. Functions of Coordinate Transformation. Definition of a Smooth Manifold...... 98
3.1.3. Smooth Manifolds. D isomorphism... 104
3.2. Definition of Manifolds by Equations... 108
3.3. Tangent Vectors. Tangent Space.... 113
3.3.1. Simple Examples ..... 113
3.3.2. General Definition of a Tangent Vector... 117
3.3.3. Tangent Space TPo(M)... 118
3.3.4. Sheaf of Tangent "Curves... 120
3.3.5. Directional Derivative of a Function... 122
3.3.6. Tangent Bundles...... 127
3.4. Submanifolds...... 129
3.4.1. Differential of a Smooth Manifold... 130
3.4.2. Differential and Local Properties of Mappings... 135
3.4.3. Sard's Theorem...... 138
3.4.4. Embedding of Manifolds in a Euclidean Space... 141
Chapter 4. SMOOTH MANIFOLDS (EXAMPLES)... 147
4.1. The Theory of Curves on a Plane and in a Three-Dimensional Space ..... 147
4.1.1. The Theory of Curves on a Plane. Frenet Formulas . . 147
4.1.2. The Theory of Spatial Curves. Frenet Formulas ... 154
4.2. Surfaces. First and Second Fundamental Forms... 160
4.2.1. The First Fundamental Form.... 160
4.2.2. The Second Fundamental Form... 163
4.2.3. An Elementary Theory of Smooth Curves on a Hyper-surface ...... 169
4.2.4. The Gaussian and Mean Curvatures of a Two-Dimensional Surface...... 178
4.3. Transformation Groups..... 200
4.3.1. Simple Examples of Transformation Groups ... 200
4.3.2. Matrix Transformation Groups... 215
4.4. Dynamical Systems...... 229
4.5. Classification of Two-Dimensional Surfaces... 246
4.5.1. Manifolds with Boundary... 247
4.5.2. Orientable Manifolds.... 249
4.6. Riemannian Surfaces of Algebraic Functions... 270
Chapter 5. TENSOR ANALYSIS AND RIEMANNIAN GEOMETRY 294
5.1. The General Concept of a Tensor Field on a Manifold ... 294
5.2. Simple Tensor Fields ..... 300
5.2.1. Examples ..... 300
5.2.2. Algebraic Operations over Tensors... 305
5.2.3. Skew-Symmetric Tensors... 309
5.3. Connection and Covariant Differentiation... 322
5.3.1. The Definition and Properties of Affme Connection . . . 322
5.3.2. Riemannian Connections..... . 332
5.4. Parallel Displacement. Geodesies.... 336
5.4.1. Preliminaries ...... 336
5.4.2. The Equation of Parallel Displacement... 339
' 5.4.3. Geodesies...... 341
5.5. Curvature Tensor ...... 357
5.5.1. Preliminaries ...... 357
5.5.2. Coordinate Definition of the Curvature Tensor... 360
5.5.3. Invariant Definition of the Curvature Tensor... 361
5.5.4. Algebraic Properties of the Riemann Curvature Tensor 362
5.5.5. Certain Applications of the Riemann Curvature Tensor 366
Chapter 6. HOMOLOGY THEORY.... 371
6.1. Calculus of Exterior Differential Forms. Cohomology Groups 372
6.1.1. Differentiation of Exterior Differential Forms... 372
6.1.2. Cohomology Groups of a Smooth Manifold (the de Rham Cohomology Groups).... 378
6.1.3. Homotopic Properties of Cohomology Groups... 381
6.2. Integration of Exterior Forms..... 387
6.2.1. The Integral of a Differential Form over a Manifold . . . 388
6.2.2. Stokes' Theorem...... 393
6.3. The Degree of Mapping and Its Applications... 399
6.3.1. Example ...... 399
6.3.2. The Degree of Mapping... 400
6.3.3. The Fundamental Theorem of Algebra... 402
6.3.4. Integration of Forms.... 403
6.3.5. Gaussian Mapping of a Hypersurface... 404
Chapter 7. SIMPLE VARIATIONAL PROBLEMS IN RIEMANNIAN
GEOMETRY ...... 407
7.1. Functional. Extremal Functions. Euler's Equations ... 407
7.2. Extremality of Geodesies.... 416
7.3. Minimal Surfaces..... 424
7.4. Calculus of Variations and Symplectic Geometry... 430
Name Index ...... 451
Subject Index...... 452

 

[ForMyThunder]

I don't think there are any prerequisites for any math courses. Just that you understand it. For example, when I was 11, I taught myself Calculus from a book I got at the library without learning Algebra first.

 

[NoMoreExams]

I find that somehow hard to believe, you had to at least know some algebra to do most of the problems. If you were told to integrate a function that required you to know how to use partial fractions did you just derive that on the spot? Also, did you just know how to do the problems or did you understand the theory behind it?

 

[ForMyThunder]

I was learning it during the summer and when I saw an explanation that I didn't understand a step in, I studied it for a while and after a while, it started to make sense. I kinda learned algebra along the way. It's not like I understood everything about Calculus from the first book I read. I probably went through about three or four during the summer.

 

[wofsy]

if you know multivariate calculus you can learn Differential Geometry.

in modern differential geometry you need to know calculus on manifolds. The hands down best book ever is Spivak's Calculus on Manifolds.

After that the book you use depends on your goal and on how you like to learn.

If you like to start simple and have concrete examples that you can visualize, then classical differential geometry of surfaces is the way. Elementary Differential Geometry by Barrett O'Niell is a great book. You will love it. I would recommend this book no matter what else you want to know.

If you want to learn basic differential geometry for mathematicians I recommend Milnor's book Morse Theory. This books is hard and formidable but is a standard text for first to second year graduate students and the applications of geometry in there are breathtaking.

A more complete mathematician's book is Hick's Notes on Differential Geometry. It has lots of stuff but is not at all pedagogical. I hate this book but it is also a standard and it has many instructive exercises.

If you just want to learn Differential Geometry for General Relativity then you should read a Physics Book because a co-ordinate laden tensor notation approach is what you need. Many mathematical books downplay this tensor notation.

Not all differential Geometry uses a Riemannian metric. One can talk about parallel translation and connections independently of a metric. In Riemannian geometry the connection is chosen to be compatible with the metric. This means that parallel translation preserves lengths and angles between tangent vectors. But there are connections that are not compatible with any metric. This branch of Differential Geometry is called Gauge Theory
and its applications are advanced. An introduction to this that is off the charts wonderful is the Appendix in Milnor's Characteristic Classes.

 

[wofsy]

"Differential geometry is the study of Riemannian manifolds. "

Differential geometry is not the study of Riemannian manifolds. It is the study of vector bundles with a connection. A Riemannian metric though usually used is not required and there is a rich theory of connections that does not require a metric. In Riemannian geometry (as opposed to differential geometry) the connection must be compatible with the metric i.e. parallel translation must be an isometry of tangent spaces. But there are many examples of connections on vector bundles that are not compatible with any metric.