**Bachman, David,***A Geometric Approach to Differential Forms*, Birkhauser, 2006, hardcover, 140 pp., ISBN 0817644997.- The goal of this little book is to make the topic of differential forms accessible to students at the sophomore level and above. It contains lots of helpful illustrations, examples, and exercises.
**Bär, Christian,***Elementary Differential Geometry*, Cambridge University Press, 2010, xii + 317 pp. Hardcover, ISBN 9780521896719; paperback, ISBN 9780521721493.- I find this to be a rather sophisticated introduction to the differential geometry of curves and surfaces, though the author says that he avoids the formalism necessary for a deeper study of differential geometry as much as possible. The book starts with a chapter on Euclidean geometry, then studies the local and global geometry of curves and surfaces in three-dimensional space. Along the way, the author discusses the exponential map, parallel transport, Jacobi fields, minimal surfaces, spherical and hyperbolic geometry, cartography, Gauss’ divergence theorem, and the Gauss-Bonnet theorem. Hints for most of the 124 exercises appear at the back of the book.
**Banchoff, Thomas and Lovett, Stephen,***Differential Geometry of Curves and Surfaces*, A K Peters, 2010, hardcover, xvi + 331 pp., ISBN 9781568814568.- This new introduction to differential geometry makes use of Java
applets instead of software to help readers to generate pictures.
All the computer applets are available online at http://www.akpeters.com/DiffGeo.
Students may be disappointed that the text provides no answers to any
exercises. This is the first of a pair of books
that together are intended to bring the reader through classical
differential geometry to the modern formulation of the differential
geometry of manifolds. The companion volume is: Lovett, Stephen,
*Differential Geometry of Manifolds*, A K Peters, 2010, hardcover, xiii + 421 pp., ISBN 9781568814575. That volume is an introduction to differential geometry in higher dimensions, with an emphasis on applications to physics. Neither book directly relies on the other, but knowledge of the content of the first is quite helpful when reading the second. **Chang, Paul,**http://emsh.calarts.edu/~mathart/sw/klein/Klein.html*The Klein Bottle*- Website devoted to the topology of the Klein bottle, with
*Mathematica*code that generates a nice computer picture of one. **Coxeter, H. S. M.,***Introduction to Geometry*, 2e, Wiley, 1969, paperback, 485 pp., ISBN 0471504580.- A sweeping book on geometry by a modern master. Part IV is on differential geometry; part III includes a chapter on hyperbolic geometry.
**do Carmo, Manfredo,***Differential Geometry of Curves and Surfaces*, Prentice Hall, 1976, hardcover, 503 pp., ISBN 0132125897.- Well written, concise, modern, anticipates manifold theory. Lots of powerful global results, interesting problems. Problems more proof-oriented than computational in nature. Best for students who have had advanced calculus.
**Educypedia: Geometry 3D animations and java applets**http://www.educypedia.be/education/mathematicsjavageometry3d.htm- This website has links to various geometric animations. You might want to explore other branches of the Educypedia as well.
**Gauss, Karl Friedrich,***General Investigations of Curved Surfaces*, Dover, paperback, 144 pp., ISBN 048644645X.- Differential geometry of surfaces from the pen of the master, translated into English, with extensive mathematical and historical notes. Originally written in 1825-27.
**Geometry Algorithm Web Sites**http://softsurfer.com/websites.htm- The many websites listed here are not all differential geometry
sites, but they all concern geometry topics, from Euclid’s
*Elements*to topics of current interest. **Goetz, Abraham,***Introduction to Differential Geometry*, Addison-Wesley, 1970, hardcover, 350 pp.- Local differential geometry of curves and surfaces in classical notation. Relies entirely on local coordinate computations.
**Gray, Alfred, Elsa Abbena, and Simon Salamon,***Modern Differential Geometry of Curves and Surfaces with Mathematica*(3e), CRC Press, 2006 (1e, 1993; 2e, 1997), hardcover, 984 pp., ISBN 1584884487.- Large and comprehensive book covering both local and global results. Many illustrations. Uses Mathematica throughout and includes a great deal of useful code. Contains significantly more material than the first two editions. A web site for the book is maintained at http://alpha01.dm.unito.it/personalpages/abbena/gray/. It contains Mathematica notebooks to accompany the text.
**Henderson, David W.,***Differential Geometry: A Geometric Introduction*, Prentice Hall, 1998, hardcover, 250 pp., ISBN 0135699630.- Very unusual book. Presents elementary differential geometry via the discovery approach. Very intuitive, nice pictures, few details. Includes quite a bit of computer code in Maple (accessible on the Web from http://www.math.cornell.edu/~henderson/books/dg/scripts/) and a bibliography covering a broad selection of mathematical topics.
**Hicks, Noel,**See Ch. 2, 3.*Notes on Differential Geometry*, Van Nostrand, 1965, paperback, 183 pp.- Good for sophisticated students. Much more advanced than other
books on this list. Covers huge amount of material (including manifold
theory) very efficiently. Treats hypersurfaces in
**R**^{n+1}. **Hilbert, David and S. Cohn-Vossen,**Original German edition, ~1932. See Ch. 4.*Geometry and the Imagination*, Chelsea, 1952, hardcover, 357 pp.- Not a text. Imprecise and sometimes incorrect, no problems - just enjoyable reading. Extremely intuitive, full of insights and heuristic arguments, excellent illustrations.
**Hsiung, Chuan-Chih,***A First Course in Differential Geometry*, Wiley-Interscience, 1981, hardcover, 343 pp., ISBN 0471079537.- Elegant modern treatment with lots of global results, but probably less readable than do Carmo. The long chapter on preliminaries makes the book self-contained and enables the author to streamline proofs later on. Aimed at advanced undergraduates and beginning graduate students.
**Klingenberg, Wilhelm,**(original German edition, 1972), hardcover.*A Course in Differential Geometry*, Springer-Verlag, 1978, hardcover, 178 pp., ISBN 0387902554- Terse, elegant exposition aimed at beginning graduate students. Includes global theorems. Unusually interesting exercises with extensive references to recent (as of 1978) articles.
**Kühnel, Wolfgang,***Differential Geometry: Curves - Surfaces - Manifolds*, 2e, AMS, 2006, paperback, 392 pp., ISBN 0-8218-3988-8.- Local and global theory of curves and surfaces, including curves and
surfaces in Minkowski space, surfaces of revolution, ruled surfaces,
minimal surfaces, hypersurfaces in
**R**^{n+1}, and the Gauss-Bonnet theorem. The second half of the book covers Riemannian manifolds, spaces of constant curvature, and Einstein spaces. **Laugwitz, Detlef,**(original German edition, 1960).*Differential and Riemannian Geometry*, Academic Press, 1965, hardcover, 238 pp.- Elegant and efficient presentation of modern material in classical notation. Very well written, with historical notes and references. Includes global theorems.
**Lipschutz, Martin M.,***Theory and Problems of Differential Geometry*(Schaum Outline), McGraw-Hill, 1969, paperback, 269 pp., ISBN 070379858.- Good source of practice problems. Organized around examples and problems, so can’t easily be read as a book. Classical notation. Few global results.
**The MacTutor History of Mathematics Archive**http://www-gap.dcs.st-and.ac.uk/~history/- Includes biographies of mathematicians, mathematicians of the day, birthplace maps, timelines of mathematicians, a chronology of important dates in the history of mathematics, an index of historical topics, and an index of famous plane curves. At the bottom of the index of famous curves, there is a link to an index of curves for which there are Java applets allowing you to experiment with changing the curve parameters and viewing the results.
**Math Archives: Differential Geometry**http://archives.math.utk.edu/topics/differentialGeom.html- University of Tennessee Knoxville website featuring links to many differential geometry sites.
**The Math Forum at Drexel**http://mathforum.org/library/topics/differential_g/- Lists dozens of useful websites.
**The Mathematical Atlas, 53: Differential geometry**http://www.math.niu.edu/~rusin/known-math/index/53-XX.html- Includes lots of useful links to texts, tutorials, software, tables, web sites, and discussions of differential geometry topics.
**Mathtools.net**http://www.mathtools.net/Java/Mathematics/index.html- Links to various mathematical Java applets.
**McCleary, John,***Geometry from a Differentiable Viewpoint*, Cambridge University Press, 1994, paperback, 308 pp., ISBN 0521424801.- Well-written book with historical outlook. Interesting section on map projections. Includes translation of Riemann’s thesis, which laid the groundwork for modern manifold theory. CONTENTS: Part A: synthetic (axiomatic) geometry (Euclidean and non-Euclidean). Part B: curves in the plane and in space, surfaces, map projections, curvature, goedesics, Gauss-Bonnet theorem, and constant curvature surfaces. Part C: abstract surfaces, models of non-Euclidean geometry, introduction to manifolds. Appendix: Riemann’s Habilitationsvortrag.
**Millman, R. S. and G. D. Parker,***Elements of Differential Geometry*, Prentice-Hall, 1977, hardcover, 265 pp., ISBN 0132641437.- Readable modern treatment that relies heavily on local coordinate computations. Good source for global theorems. Shorter and requires less background than do Carmo.
**Montiel, Sebastián and Ros, Antonio,***Curves and Surfaces*, AMS, 2005, 390 pp., ISBN 0821838156.- Local and global geometry of curves and surfaces, with chapters on
separation and orientability, integration on surfaces, global extrinsic
geometry, intrinsic geometry of surfaces (including rigidity of ovaloids),
the Gauss-Bonnet theorem, and the global geometry of curves
**O’Neill, Barrett,***Elementary Differential Geometry*(revised 2e), Academic Press, 2006 (1e, 1966; 2e, 1997), hardcover, 503 pp., ISBN 0120887355.- Modern, assumes little background, but has considerable depth and anticipates manifold theory. Excellent problems. Uses differential forms and the method of moving frames as primary tools. This adds depth and computational power, but also lengthens the book. Uses invariant index-free notation throughout. Second edition adds a couple of global results, plus computer exercises, brief tutorials on Maple and Mathematica, and useful chunks of code in Maple and Mathematica. See Prof. O’Neill’s web site at http://www.math.ucla.edu/~bon/ for errata and other useful materials.
**Oprea, John,***Differential Geometry and Its Applications*(2e), Mathematical Association of America, 2007 (originally published by Prentice Hall: 1e, 1997; 2e, 2004), hardcover, ISBN 0883857480.- Differential geometry with an emphasis on applications involving the calculus of variations. No global theory of curves. Good, interesting problems. Uses Maple throughout to help with calculations and visualization. Useful chunks of Maple code are provided. See the web site for the book at http://academic.csuohio.edu/oprea_j/dgbook/dgbook4.html for errata and Maple files.
**Polthier, Konrad,**http://plus.maths.org/issue26/features/mathart/index-gifd.html*Inside the Klein bottle*- Website with lots of information and wonderful pictures, some animated, of Klein bottles.
**Pressley, Andrew,***Elementary Differential Geometry*(Second Edition, Springer, 2010, paperback, xi + 473 pp., ISBN 9781848828902.- Presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to a minimum. Attempts to use the most direct and straghtforward approach to each topic. Hints or solutions to all the exercises appear at the back of the text. The second edition features a new section on spherical geometry and a whole new chapter on hyperbolic geometry.
**Spivak, Michael,***A Comprehensive Introduction to Differential Geometry*(2e), Volumes 2 and 3, Publish or Perish, 1979.- Part of a huge 5-volume set. See Vol. 3 especially. Very chatty and intuitive. Includes lots of material hard to find elsewhere. Modern - uses manifold point of view. Lots of pictures. For beginning graduate students. Vol. 2 has fascinating historical sections.
**Stoker, J. J.,***Differential Geometry*, Wiley-Interscience, 1969, hardcover. Reprint edition, 1989, paperback, ISBN 0471504033.- Well written. Extensive topical coverage. Considers every possible point of view for comparison purposes. Lots of global theorems, chapter on general relativity. Rather long, notation classical. Rather chatty, well motivated. Not many problems. They deal more with concepts than computations.
**Struik, Dirk J.,***Lectures on Classical Differential Geometry*(2e), originally published by Addison-Wesley, 1961 (1e, 1950). Dover edition (first published by Dover in 1988), paperback, 240 pp., ISBN 0486656098.- Lots of interesting examples, problems, historical notes, and hard-to-find references (refers to original foreign language sources). Beautiful illustrations. Old-fashioned, hard to read in places.
**Thorpe, John A.,***Elementary Topics in Differential Geometry*, Springer-Verlag, 1979, hardcover, 253 pp., ISBN 0387903577.- An unusual book. The first half deals from the outset with
orientable hypersurfaces in
**R**^{n+1}, described as solution sets of equations. The second half uses parametrized surfaces. Little discussion of curves or phenomena specific to**R**^{3}. Mainly concerned with concepts that generalize to manifolds. Uses invariant, index-free notation. For advanced undergraduates. **VIDIGEO (Visual Interactive DIfferential GEOmetry)**http://www2.mat.dtu.dk/info/mathematics/VIDIGEO/- A website whose goal is to give students a chance to see and experience the connection between formal mathematical descriptions and their visual interpretations.
**Visual Geometry Pages**http://www.vismath.de/vgp/- Mathematical visualization of problems from differential geometry.
**Willmore, T. J.,***Differential Geometry*, Oxford University Press, 1959, hardcover, 317 pp.- Extensive topical coverage, including many global theorems. Main drawbacks are dry style and classical notation.
**Wilson, P. M. H.,***Curved Spaces: From Classical Geometries to Elementary Differential Geometry*, Cambridge University Press, 2008, 198 pp., hardcover, ISBN 9780521886291; paperback, ISBN 9780521713900.- As the title implies, this book covers both classical geometries and differential geometry. Its chapter titles are: Euclidean geometry, Spherical geometry, Triangulations and Euler numbers, Riemannian metrics, Hyperbolic geometry, Smooth embedded surfaces, Geodesics, and Abstract surfaces and Gauss-Bonnet.
**Wolfram Demonstrations Project - Differential Geometry**http://demonstrations.wolfram.com/- Go to this site, click on
**Mathematics**, then**Geometry**. From the drop-down list, choose**Differential Geometry**. Also look at**Curves**and**Hyperbolic Geometry**. Each topic is illustrated with*Mathematica*demonstrations. These can be previewed online, but they work better if you first download live versions to your computer. **Wolfram Research Inc.’s MathWorld.com, differential geometry section**http://mathworld.wolfram.com/topics/DifferentialGeometry.html- Xah Lee calls mathworld.wolfram.com the best mathematics resource on the web. There is a huge amount of information here. The first link takes you to the page that leads to the material on differential geometry.
**Xah Lee’s Curve Family Index**http://xahlee.org/SpecialPlaneCurves_dir/Intro_dir/familyIndex.html- This site contains a wealth of information about plane curves. There’s also a fine list of related websites at http://www.xahlee.org/SpecialPlaneCurves_dir/Intro_dir/relatedHyperLinks.html
**Xah Lee’s Visual Dictionary of Plane Curves**http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html- This site goes hand in hand with the previous one.
*My last two entries are lists of books on differential geometry:***Mathematical Association of America (MAA) Basic Library List of Geometry Books**http://mathdl.maa.org/mathDL/19/?pa=content&sa=viewDocument&nodeId=3234- This is a 1991 list of recommended books on various geometry topics. Differential geometry appears near the end of the geometry list. The home page for the Basic Library List is here. Links at the bottom of the page take you to an explanation of the ratings, changes to the list made in July 2010, and to an index for all the mathematical subjects in the Basic Library List.
**isbn.nu**http://isbn.nu/- This is a website devoted to searching for books and comparing prices. You can search by words in title, author, subject, or ISBN. I suggest entering the words “differential geometry” either as Words in Title or as Subject. After you press Search, you can arrange the results alphabetically, or chronologically by publication date.