**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.

**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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