Spivak, A Comprehensive Introduction to Differential Geometry
Volume 3.   3rd edition. xii + 314 pages.  Clothbound.  1999

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Outline of Contents:
 1. The Equations for Hypersurfaces
        Covariant differentiation in a submanifold of a Riemannian manifold.
        The second fundamental form, the Gauss formulas, and Gauss' equation;
        Synge's inequality. The Weingarten equations and the Codazzi-Mainardi
        equations for hypersurfaces. The classical tensor analysis description.
        The moving frame description. 
        Addendum. Auto-parallel and totally geodesic submanifolds.

 2. Elements of the Theory of Surfaces in Euclidean 3-Space
        The first and second fundamental forms. Classification of points on a surface;
        the osculating paraboloid and the Dubin indicatrix. Principal directions
        and curvatures, asymptotic directions, flat points and umbilics; all-umbilic
        surfaces. The classical Gauss formulas, Weingarten equations, Gauss
        equation, and Codazzi-Mainardi equations. Fundamental theorem of
        surface theory. The third fundamental form. Convex surfaces; Hadamard's
        theorem. The fundamental equations via moving frames. Review of
        Lie groups. Applications of Lie groups to surface theory; the fundamental
        equations and the structural equations of SO(3). Affine surface theory;
        the osculating paraboloids and the affine invariant conformal structure.
        The special affine first fundamental form. Quadratic and cubic forms;
        apolarity. The affine normal direction; the special affine normal. 
        The special affine Gauss formulas and special affine second fundamental
        form. The Pick invariant; surfaces with Pick invariant 0. The special
        affine Weingarten formulas. The special affine Codazzi-Mainardi equations;
        the fundamental theorem of special affine surface theory.
 3. A Compendium of Surfaces
        Basic calculations. The classical flat surfaces. Ruled surfaces. Quadric
        surfaces. Surfaces of revolution; rotation surfaces of constant curvature.
        Minimal surfaces. 
        Addendum. Envelopes of 1-parameter families of planes.

 4. Curves on Surfaces
        Normal and geodesic curvature. The Darboux frame; geodesic torsion.
        Laguerre's theorem. General properties of lines of curvature, asymptotic
        curves, and geodesics. The Beltrami-Enneper theorem. Lines of curvature
        and Dupin's theorem. Conformal maps of Euclidean 3-space; Liouville's
        theorem. Geodesics and Clairaut's theorem. Special parameter curves. 
        Singularities of line fields.

 5. Complete Surfaces of Constant Curvature
        Hilbert's lemma; complete surfaces of constant curvature K>0. 
        Analysis of flat surfaces; the classical classification of developable surfaces.
        Complete flat surfaces. Complete surfaces of constant curvature K<0.

 6. The Gauss-Bonnet Theorem and Related Topics
        The connection form for an orthonormal moving frame on a surface;
        the change in angle under parallel translation. The integral of K dA
        over a polygonal region. The Gauss-Bonnet theorem; consequences.
        Total absolute curvature of surfaces. Surfaces of minimal total absolute
        curvature. Total curvature of curves; Fenchel's theorem, and the
        Fary-Milnor theorem.
        Addenda.  Compact surfaces with constant negative curvature.
        The degree of the normal map.

From reviews:

N. J. Hicks, Mathematical Reviews, volume 52, #15245b

For the 5 volume set:

The author has pulled together the main body of "classical differential geometry" that forms the background and origins of the state of the theory today. He has presented this material in an uncompromisingly clear, fresh, and readable fashion, trying always to present the intuition behind the ideas when possible.  There are many excellent illustrations, and there is an extensive bibliography of books and articles ... The author has a style that contacts the joy of doing mathematics and an admirable attitude when faced with the occasional gross but necessary computation, i.e., he does it. ... There is no doubt that these books will contribute strongly to the further development of differential geometry.

Stephanie Alexander, Bulletin of the AMS, volume 84, number 1, January 1978

For the 5 volume set:                                                                                                                         

The Comprehensive introduction is probably best suited for leisurely and enjoyable background reference by almost anyone interested in differential geometry. Great care has been taken to make it accessible to beginners, but even the most seasoned reader will find stimulating reading here ... The appeal of the book is due first of all to its choice of material, which is guided by the liveliest geometric curiosity. In addition, Spivak has a clear, natural and well-motivated style of exposition; in many places, his book unfolds like a novel.

The Comprehensive introduction will be widely read and enjoyed, and will surely become a standard reference for graduate courses in differential geometry. Spivak is greatly to be thanked for this spontaneous, exuberant and beautifully geometrical book.

Specifically for Volume 3:  

An excellent selection of fundamental theorems on surfaces is the main subject of this third volume. Some modern work is included ... In addition, there is a systematic and well illustrated compendium of examples.
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