Spivak, A Comprehensive Introduction to Differential
Volume 3. 3rd edition. xii + 314 pages. Clothbound. 1999
|Click on cover picture for more detailed image||Return to Set listing|
|Click here for quotations from reviews||Return to book list||Return to home page|
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.
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
Addenda. Compact surfaces with constant negative curvature.
The degree of the normal map.
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.
Return to top