hey nerdinthearena,

i too find this area to be fascinating and wish i knew more on the upper end myself. i'm just going to list off a few resources. in my opinion, graduate school will concentrate a lot on progressing your technical knowledge, but will likely not give you a lot of time to hone your intuition (at least in the first few years). so, the more time you spend in undergraduate school doing so, the better.

helpful for intuition and basic understanding

• geometrical vectors by gabriel weinreich - why there's more to vectors and how to draw them. this basically shows visual examples of differential forms and the like.
  
• advanced calculus: a differential forms approach by harold edwards - a fantastic approach to differential forms. it's a treasure, so just read it. :) it is actually well built for skimming and skipping around.
  
• vector calculus, linear algebra, and differential forms: a unified approach - this is a big book, but you could just jump ahead to the later sections covering differential forms for an interesting approach and technical skill building.
  
• div, grad, curl are dead by william burke - pure intuition and similar to geometrical vectors. don't look for anything else here, but definitely read it.
  
• applied differential geometry by william burke - a more complete work than the above and showcases a lot of the connections between geometry and physics.
  
  

  more advanced but still intuitive
  

• an introduction to manifolds - this is at the same level and covers essentially the same material as lee, but in my opinion, it is much better. it is much more concise and to the point with more focused exercises.
  
• differential forms in algebraic topology by raoul bott and loring tu - the sequel to the above, but i haven't read beyond the first chapter. it is well known as a must read classic, but having some topology background will help with this book.
  
• the laplacian on a riemannian manifold: an introduction to analysis on manifolds - an interesting introduction to geometric analysis.
  
• differentiable manifolds: forms, currents, harmonic forms by georges de rham - a classic by one of the greats and also one of the only sources i've found that covers de rham currents (besides research papers that just use them).
  
• riemannian manifolds: an introduction to curvature by john lee - the sequel to lee's smooth manifolds book.
  
  hopefully this helps. if i were to revisit geometric analysis, i would basically use the above books to help bone up my understanding, intuition, and technical skill before moving on. these are also mainly geometry books, so learning analysis (like functional analysis) would be good as well. i mainly have three suggestions there.
  
  

  three general analysis favorites
  

• introductory functional analysis with applications by erwin kreyszig - used everywhere.
  
• calculus of variations by gelfand and fomin
  
• linear differential operators by cornelius lanczos - a geometrical look at the subject. also take a look at the variational principles of mechanics by him as well.

I hope others will chime in here, but I'll answer as well as I can.

Euclidean and Non-Euclidean Geometry

Euclidean and non-Euclidean geometries are interesting and important for various reasons, so I certainly wouldn't say it's a bad idea to study them in depth.

If you want to study these subjects first because you find them interesting and you have plenty of years to spend, then go for it! However, it's not necessary (more on this below).

Multivariable Calculus and Linear Algebra

Before attempting even an elementary treatment of differential geometry, you'll want to have a working knowledge of calculus (single and multivariable) and linear algebra.

Elementary Differential Geometry

You could potentially skip the elementary treatments of differential geometry, but these might be useful for tackling more advanced treatments. Studying elementary differential geometry first is perhaps similar to taking a calculus class (with an emphasis on computation and hopefully on intuition) before taking a class in real analysis (with an emphasis on abstraction and rigorous proofs).

If you do want to work through an elementary treatment, then you have options. One well reviewed book, and the one I learned from as an undergraduate, is Elementary Differential Geometry by Barrett O'Neill.

Note that O'Neill lists calculus and linear algebra as prerequisites, but not Euclidean and Non-Euclidean geometry. Experience with Euclidean geometry is definitely relevant, but if you understand calculus and linear algebra, then you already know enough geometry to get started.

Abstract Algebra, Real Analysis, and Topology

The next step would probably be to study a semester's worth of abstract algebra, a year's worth of real analysis, and optionally, a semester's worth of point-set topology. These are the prerequisites for the introduction to manifolds listed below.

Manifolds

An Introduction to Manifolds by Loring W. Tu will give you the prerequisites to take on graduate-level differential geometry.

Note: the point-set topology is optional, since Tu doesn't assume it; he expects readers to learn it from his appendix, but a course in topology certainly wouldn't hurt.

Differential Geometry

After working through the book by Tu listed above, you'd be ready to tackle Differential Geometry: Connections, Curvature, and Characteristic Classes, also by Loring W. Tu. There may be more you want to learn, but after this second book by Tu, it should be easier to start picking up other books as needed.

Caveat

I myself have a lot left to learn. In case you want to ask me about other subjects, I've studied all the prerequisites (multivariable calculus, linear algebra, abstract algebra, real analysis, and point-set topology) and I've tutored most of that material. I've completed an elementary differential geometry course using O'Neill, another course using Calculus on Manifolds by Spivak, and I've studied some more advanced differential geometry and related topics. However, I haven't worked through Tu's books yet (not much). The plan I've outlined is basically the plan I've set for myself. I hope it helps you too!

Differential geometry track. I'll try to link to where a preview is available. Books are listed in something like an order of perceived difficulty. Check Amazon for reviews.

Calculus

Thompson, Calculus Made Easy. Probably a good first text, well suited for self-study but doesn't cover as much as the next two and the problems are generally much simpler. Legally available for free online.

Stewart, Calculus. Really common in college courses, a great book overall. I should also note that there is a "Stewart lite" called Calculus: Early Transcendentals, but you're better off with regular Stewart. Huh, it looks like there's a new series called Calculus: Concepts and Contexts which may be a good substitute for regular Stewart. Dunno.

Spivak, Calculus. More difficult, probably better than Stewart in some sense.

Linear Algebra

Poole, Linear Algebra. I haven't read this one but it has great reviews so I might as well include it.

Strang, Introduction to Linear Algebra. I think the Amazon reviews summarize how I feel about this book. Good for self-study.

Differential Geometry

Pressley, Elementary Differential Geometry. Great text covering curves and surfaces. Used this one in my undergrad course.

Do Carmo, Differential Geometry of Curves and Surfaces. Probably better left for a second course, but this one is the standard (for good reason).

Lee, Riemannian Manifolds: An Introduction to Curvature. After you've got a grasp on two and three dimensions, take a look at this. A great text on differential geometry on manifolds of arbitrary dimension.

------

Start with calculus, studying all the single-variable stuff. After that, you can either switch to linera algebra before doing multivariable calculus or do multivariable calculus before doing linear algebra. I'd probably stick with calculus. Pay attention to what you learn about vectors along the way. When you're ready, jump into differential geometry.

Hopefully someone can give you a good track for the other geometric subjects.

Have you had a look at Carroll's general relativity notes? Chapters 2 and 3 are predominantly about developing the mathematics behind GR, and are very good introductions to this. I have a copy of Carroll's book and I can promise you that those chapters are almost unchanged in the book as compared to the lecture notes. This is my main suggestion really, as the notes are freely available, written by an absolute expert and a joy to read. I can't recommend them (and the book really) enough.

Most undergraduate books on general relativity start with a "physics first" type approach, where the underlying material about manifolds and curvature is developed as it is needed. The only problem with this is that it makes seeing the underlying picture for how the material works more difficult. I wouldn't neccessarily say avoid these sort of books (my favourite two of this kind would be Cheng's book and Hartle's.) but be aware that they are probably not what you are looking for if you want a consistent description of the mathematics.

I would also say avoid the harder end of the scale (Wald) till you've at least done your course. Wald is a tough book, and certainly not aimed at people seeing the material for the first time.

Another useful idea would be looking for lecture notes from other universities. As an example, there are some useful notes here from cambridge university. Generally I find doing searches like "general relativity site:.ac.uk filetype:pdf" in google is a good way to get started searching for decent lecture notes from other universities.

If you're willing to dive in a bit more to the mathematics, the riemannian geometry book by DoCarmo is supposed to be excellent, although I've only seen his differential geometry book (which was very good). As a word of warning, this book might assume knowledge of differential geometry from his earlier book. The book you linked by Bishop also looks fine, and there is also the book by Schutz which is supposed to be great and this book by Sternberg which looks pretty good, although quite tough.

Finally, if you would like I have a dropbox folder of collected together material for GR which I could share with you. It's not much, but I've got some decent stuff collected together which could be very helpful. As a qualifier, I had to teach myself GR for my undergrad project, so I know how it feels being on your own with it. Good luck!

I could be interested in reading that paper, however I might need a discussion on the Atiyah-Singer Index Theorem first - It's something I haven't really had to use, but something I'd like to know.

My own personal interests lie in manifolds with special holonomy, and I'd be particularly interested in discussing G2 manifolds, if anyone else is.

Another, more basic, option would be Frenkel's 'Geometry of Physics' book, which has a lot of nice physics formulated in the language of differential geometry. This may be a good option for people with physics backgrounds with little formal DG training, as it does all of DG from scratch while being sure to tie all the math to physics (E&M, Lagrangian/Hamiltonian Mechanics, Relativity, Yang-Mills Theory etc.) Check it out here: https://www.amazon.com/Geometry-Physics-Introduction-Theodore-Frankel/dp/1107602602

I heard good things about it, but honestly as an applied mathematician I found its table of contents too lackluster. Its coverage appears to be in a weird spot between "for physicists" and "for mathematicians" and I don't know who its target audience is. I think the standard recommendation for classical mechanics from the physics side is Goldstein, which is a perfectly good book with plenty of math!

For an actual mathematicians' take on classical mechanics, you'll have to wait until you take more advanced math, namely real analysis and differential geometry. Common references are Spivak and Tu. When you have that background, I think Arnold has the best mathematical treatment of classical mechanics.

Great source! That's something like what I was looking for. I've found the seminal work on the subject (Amari), but I was hoping for something succinct and possibly easier to understand.

Differential geometry is a fascinating field; definitely look into it. I suggest you check out this book, which we're using for our class.

When I was in your position I learned some representation theory of finite groups, from this book. It was at the perfect level for somebody who only has one semester's background in group theory. It'll gently introduce you to some things that you'll constantly need when you get further into algebra, like tensor products. Also, it's a topic which doesn't get covered at all in most undergrad abstract algebra courses, so it's a good thing to learn by yourself.

On the other hand, if you liked topology more than you liked group theory, you'd probably like Tu's Introduction to Manifolds.

Just want to add to your post with an example. Consider the expression Δu = 0. The Laplacian operator Δ can be interpreted as a measure of how much the value of a field differs from the
average value taken over the surrounding points. This geometric statement remains the
same regardless of how we express positions in space; it’s coordinate-free. Now if we choose,
say, Cartesian coordinates, the geometric statement manifests as uₓₓ + uᵧᵧ = 0. Suppose
we wanted to choose polar coordinates instead because they are more convenient for the
given problem (eg. the function u is radial). How do we turn the geometric meaning of ∆
into an expression with respect to polar coordinates? Do we get the analagous expression
uᵣᵣ +uᵩᵩ = 0? What we really want is an expression that uses coordinates, but doesn’t pick a specific one. A type of
expression that has the generality of ∆u as well as the practicality of uₓₓ + uᵧᵧ. Tensor calculus gives us our happy medium.

Also, as far as the history goes, I like to think that Einstein's crowning achievement was his discovery of the summation convention for matching upper and lower indices. I can't imagine working with tensors without it.

Also, there is an excellent lecture series on youtube. There is an accompanying book (written by the lecturer himself), but the videos are so good that you don't even need it.

Urs Schreiber is one of the founders of nLab, which is a great wiki on mathematics and physics from the (higher) categorical point of view.

Also, Cartan geometry is pretty awesome. If you're interested in it, there's a great book by R. W. Sharpe on this topic.

For differential geometry a great book is: [Analysis and Algebra on Differentiable Manifolds] (https://www.amazon.com/Analysis-Algebra-Differentiable-Manifolds-Mathematics/dp/9400793308/ref=sr_1_6?ie=UTF8&qid=1484591942&sr=8-6&keywords=problems+differential+geometry+and+manifolds+springer). It maybe doesn't have tremendous creativity required to solve the problems, but it'll give you lots of good practice.

For linear algebra and abstract, if you're not satisfied using Dummit and Foote (with easily accessible solutions online) or Lang Algebra (with harder to find solutions), Lang has some great exercises by the way, then I recommend the [Berkeley Problems in Mathematics book] (https://www.amazon.com/Berkeley-Problems-Mathematics-Problem-Books/dp/0387008926/ref=sr_1_sc_2?ie=UTF8&qid=1484591875&sr=8-2-spell&keywords=problems+in+abstracct+algebra).

Both of these books have complete solutions for problems and should be very useful for you.

An Introduction to Manifolds by Tu is a very approachable book that will get you up to Stokes. Might as well get the full version of Stokes on manifolds not just in analysis. From here you can go on to books by Ramanan, Michor, or Sharpe.

A Guide to Distribution Theory and Fourier Transforms by Strichartz was my introduction to Fourier analysis in undergrad. Probably helps to have some prior Fourier experience in a complex analysis or PDE course.

Bartle's Elements of Integration and Legesgue Measure is great for measure theory. Pretty short too.

Intro to Functional Analysis by Kreysig is an amazing introduction to functional analysis. Don't know why you'd learn it from any other book. Afterwards you can go on to functional books by Brezis, Lax, or Helemskii.

Many people here have suggested Munkres, an excellent resource in introductory Topology (especially for self-teaching).

I would also like to suggest J. M. Lee's Introduction to Topological Manifolds. It's not nearly as pricey as Munkres, but maintains the same solid and clear exposition. I also enjoy how Lee emphasizes manifolds throughout the text as a type of motivation for introductory topology.

I've become greatly interested in geometric concepts in physics. I would like some opinions on these text for self study. If there are better options, please share.

For a differential geometry approach for Classical Mechanics:
Saletan?

For a General self study or reference book:
Frankel or Nakahara?

For applications in differential geometry:
Fecko or Burke?



Also, what are good texts for Geometric Electrodynamics that includes spin geometry?

Geometry, Topology, and Physics isn't a complete overview of math (as suggested by the title, it focuses on, well, geometry and topology), but if you're interested in learning about those specific subfields and their application to physics, I'd definitely recommend it.

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
  

It's a book on the math of GR (Differential Geometry by Erwin Kreyszig). Pretty great book and it's like 12 bucks on amazon.

Some of my physics major friends liked Nakahara. If you want to instead just do Riemannian geometry computations like a physicist you can try a general relativity book like Wald or Carroll.

I'm not too sure about it personally, but several friends have taught from Nakahara, and have a lot of good things to say about it. It's graduate level.

Want a challenge read Tu's introduction to Manifolds (https://www.amazon.com/gp/aw/d/B014P9I1MU/ref=cm_cr_arp_mb_bdcrb_top?ie=UTF8). It'll stretch you no doubt, but is great motivation for both linear algebra and calculus.

This is one is the best textbook for self-study I've find: Elementary Differential Geometry - A.N. Pressley.
Every self-study book should be like this one, well written and with answers to every exercises.

Learn math first. Physics is essentially applied math with experiments. Start with Calculus then Linear Algebra then Real Analysis then Complex Analysis then Ordinary Differential Equations then Partial Differential Equations then Functional Analysis. Also, if you want to pursue high energy physics and/or cosmology, Differential Geometry is also essential. Make sure you do (almost) all the exercises in every chapter. Don't just skim and memorize.

This is a lot of math to learn, but if you are determined enough you can probably master Calculus to Real Analysis, and that will give you a big head start and a deeper understanding of university-level physics.

It's hard to give an objective answer, because any sufficiently advanced book will be bound to not appeal to everyone.

You probably want Daddy Rudin for real analysis and Dummit & Foote for general abstract algebra.

Mac Lane for category theory, of course.

I think people would agree on Hartshorne as the algebraic geometry reference.

Spanier used to be the definitive algebraic topology reference. It's hard to actually use it as a reference because of the density and generality with which it's written.

Spivak for differential geometry.

Rotman is the group theory book for people who like group theory.

As a physics person, I must have a copy of Fulton & Harris.

Agnostic here. I'm afraid it is not so easy to rule out the presence of brilliance and religion in a scientist or mathematician. Here is a list of living scientists who are christians (it is only a part of a much larger list going back several centuries).

Here are some examples with whose work I am more or less directly familiar.

John Polkinghorne was a student of Paul Dirac, and he has written a couple of books that are very lucid introductions to Quantum Mechanics.

Christopher Isham has written books on

• Quantum Theory,
  
• Modern Differential Geometry for Physicists and
  
• Lectures On Groups And Vector Spaces For Physicists.
  
  I have only actually read the first one on Quantum Theory.
  
  Edward Nelson is a mathematical physicist who has also contributed to quantum theory.
  
  Don Knuth is a mathematician and computer scientist who created the computer typesetting system, TEX, as well as a number of important theoretical developments, (e.g. the Robinson–Schensted–Knuth correspondence, and the Knuth–Bendix completion algorithm)
  
  Maddening, ain't it? But that's just how it is.

For elementary differential geometry, just calculus and linear algebra should be sufficient. You can use a book like this for that purpose. For more advanced differential geometry, you will need to know topology and analysis and maybe some algebra as well.

The mathematics necessary for theoretical physics varies based upon what type of theoretical physics you want to work in.

I assume you are a rising senior?

Long term the best book I've seen for an overview of what you want is Geometry, Topology and Physics by Nakahara link

Although this book is really suited for graduate students with extensive mathematical background. But think of this book as a goal!

For you you'll want to read up a good deal on Abstract Algebra. That paves the way for understanding Lie Algebras, Topology etc. And you'll also want to do some analysis at the same time.

I am not a mathematical physicist but representation theory has a lot of applications in physics so I know a good bit of literature if you have more specific questions about books for self-studying some of the courses that people have listed below. (Understand though that you will have to retake them once you get to college).

The content of the upper level math courses tend to vary depending on the professor and what they feel like teaching on any given year. I took fundamentals of Geometry with prof. Piper a few years ago. We covered most everything in this book (you can read through the index to get a good idea of what the course contained)

http://www.amazon.com/Elementary-Differential-Geometry-Undergraduate-Mathematics/dp/184882890X/ref=sr_1_2?ie=UTF8&qid=1320607881&sr=8-2

We also did a bit with the more computational side of things, representing geometric transformations with quaternions or matrices, did Maple projects, etc.

I've always preferred the 1st edition of Barrett O'Neill's book to doCarmo's. Struik's book is another good one at that level.

Let me point you to my friend Nakahara.

Modern Differential Geometry for Physicists seems the most legit. It covers everything important (though it could have more cohomology) and is written well. It also navigates the line between physics and math, which is what you seem to be doing.

I'll second Spivak's two calculus texts. Apostol and Courant are good alternatives if you have some reservations about Spivak.

I'd go with Do Carmo's Differential Geometry of Curves and Surfaces instead of Spivak's five volume sequence.

Maybe this book?

Or a standard Riemannian geometry textbook like do Carmo might suit your needs.

There are two classes you might have slept through with that name. The "classical" differential geometry of curves and surfaces (I think the standard is do Carmo), or a class on Riemannian geometry (I can recommend Lee).

Manifolds that are Euclidean locally are called Riemann manifolds, but in general, not all manifolds have that property.

My only experience with manifolds is from shape analysis, so I used a Riemann manifold to measure differences in 2-d closed curves by geodesics. I still don't 'get' them, but you might want to check out the book by Do Carmo on Differential Geometry

http://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897

From my limited understanding, a Riemann manifold is a kind of generic space to compare curves in other spaces that might not normally be comparable because of curvature. Like if you want to compare a line in Euclidean coordinates to a 'line' in spherical coordinates, you'd transform each curve using the xyz or R,theta, phi, plop them on a manifold and calculate the difference using an inner product on the tangent space.

Don't know your background, but I'd look at Pollack's https://www.amazon.com/Differential-Topology-AMS-Chelsea-Publishing/dp/0821851934 and, of course, Spivak's https://www.amazon.com/Comprehensive-Introduction-Differential-Geometry-Vol/dp/0914098705

MIT has lectures on OCW, as well.

I like Lee's Introduction to Topological Manifolds.

Νι διαβάζω αυτόν τον καιρό, αυτό http://www.amazon.com/Comprehensive-Introduction-Differential-Geometry-Edition/dp/0914098705, μόνο που δεν το αγόρασα το βρήκα σε djvu. Δε με λες βούρλο, αυτό εσύ το βγάζεις;

Πέρα απ την πλακά, προτιμώ κάποιος να μην έχει ανοίξει βιβλίο στην ζωή του παρά να έχει attitude "your opinions are not worth discussing". Αυτό είναι παταγωδώς γελοίο, και μιας και έχουμε στο θέμα αυτούς τους μαλέες είναι κόντρα στην φιλοσοφία των Α Ελλήνων.

Είναι anti intellectualism και πνευματικός μεσαίωνας να λες διάβασε βιβλία η γνώμη σου δεν αξίζει σχολίου. Ντροπή σου χοντροκέφαλε ps: μας ζάλισες τα αρχίδια με την επαρχία.

Before it was re-published by Dover, Differential Geometry of Curves and Surfaces was green too; now it's blue, and the only green book by do Carmo still in publication is Riemannian Geometry.

It is algebra. But Lie algebras and the jacobi identity are standard subjects in any book on differential geometry.

Try Tu.

Actually there are some books which aren't reprints which are published by Dover. See: Curvature in Mathematics and Physics - Sternberg

Do Carmo's Differential Geometry of Curves and Surfaces

Gallian's Contemporary Abstract Algebra

Axiomatic "non-euclidean geometry" is something that was studied hundreds of years ago but isn't really an area of math that one studies or learns about anymore. Today, "non-euclidean geometry" (like the geometry of spheres or the hyperbolic plane) is part of differential geometry. There are many undergraduate-level books on manifolds and differential geometry, but I've never really looked at these. One you could try is Elementary Differential Geometry by Pressley.

TL;DR Start here

_

I think the classic introduction to the topic is Do Carmo's Riemannian Geometry. One that my colleagues use a lot (and is always taken out of the library, grrr) is Jurgen Jost's Riemannian Geomery and Geometric Analysis this second book is more recent and put out by springer.

There's another set of books that, from what I understand, approaches much more the algebraic aspects of this topic, but I have no experience with it. But I've read a lot of people in that area think it's the bee's knees. This is the 4 volume work by Spivak, A Comprehensive Introduction to Differential Geometry

the only thing that comes to mind is Frankel's geometry of physics

http://www.amazon.com/The-Geometry-Physics-An-Introduction/dp/1107602602

it's not really a math book as such (not the most rigorous proofs, and few at that) and it has way more.

i'm no expert though.