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A Comprehensive Introduction to Differential Geometry, Vol. III

Michael Spivak
Publish or Perish, Inc.
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

  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.