**Spivak, A Comprehensive Introduction to Differential
Geometry**

**Volume 4.** 3rd edition. x + 390 pages.
Clothbound. 1999

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**Outline of Contents:**

7. Higher Dimensions and Codimensions

A. THE GEOMETRY OF
CONSTANT CURVATURE MANIFOLDS

The standard models of the spheres and hyperbolic spaces.

Stereographic
projection and the conformal model of hyperbolic space.

Conformal
maps of Euclidean *n*-space and the isometries of

hyperbolic
space. Totally geodesic submanifolds and geodesic spheres

of hyperbolic
space. Horospheres and equidistant hypersurfaces.

Geodesic
mappings; the projective model of hyperbolic space;

Beltrami's
theorem.

B. CURVES IN A
RIEMANNIAN MANIFOLD

Frenet frames and curvatures. Curves whose *j*th curvature vanishes.

C. THE FUNDAMENTAL
EQUATIONS FOR SUBMANIFOLDS

The normal connection and the Weingarten equations. Second

fundamental
forms and normal fundamental forms; the Codazzi-Mainardi

equations.
The Ricci equations. The fundamental theorem for submanifolds

of Euclidean
space. The fundamental theorem for submanifolds of constant

curvature
manifolds.

D. FIRST CONSEQUENCES

The curvatures of a hypersurface; Theorema Egregium; formula for the

Gaussian
curvature. The mean curvature normal; umbilics; all-umbilic

submanifolds
of Euclidean space. All-umbilic submanifolds of constant

curvature
manifolds. Positive curvature and convexity.

E. FURTHER RESULTS

Flat ruled surfaces in Euclidean space. Flat ruled surfaces in constant

curvature
manifolds. Curves on hypersurfaces.

F. COMPLETE SURFACES
OF CONSTANT CURVATURE

Modifications
of results for surfaces in Euclidean 3-space. Surfaces of|

constant
curvature in the 3-sphere: surfaces with constant curvature 0;

the Hopf map.
Surfaces of constant curvature in hyperbolic 3-space:

Jörgens
theorem; surfaces of constant curvature 0; surfaces of constant

curvature -1;
rotation surfaces of constant curvature between -1 and 0.

G. HYPERSURFACES OF
CONSTANT CURVATURE IN HIGHER DIMENSIONS

Hypersurfaces of constant curvature in dimensions >3. The Ricci tensor;

Einstein
spaces, hypersurfaces which are Einstein spaces. Hypersurfaces

of the same
constant curvature as the ambient manifold.

Addenda. The
Laplacian.

The *
operator and the Laplacian on forms; Hodge's theorem.

When are two
Riemannian manifolds isometric? Better imbedding invariants.

8. The Second Variation

Two-parameter
variations; the second variation formula. Jacobi fields;

conjugate points. Minimizing and
non-minimizing geodesics.

The Hadamard-Cartan Theorem. The
Sturm Comparison Theorem;

Bonnet's Theorem. Generalizations to
higher dimensions; the

Morse-Schoenberg Comparison Theorem;
Meyer's Theorem; the

Rauch Comparison Theorem. Synge's
lemma; Synge's Theorem.

Cut points; Klingenberg's theorem.

9. Variations of Length, Area, and Volume

Variation of are for
normal variations of surfaces in Euclidean 3-space;

minimal surfaces. Isothermal
coordinates on minimal surfaces:

Bernstein's Theorem.
Weierstrass-Enneper representation. Associated

minimal surfaces; Schwarz's Theorem.
Change of orientation;

Henneberg's minimal surface.
Classical calculus of variations in n dimensions.

Variation of volume formula.
Isoperimetric problems. Isothermal coordinates.

Immersed spheres with constant mean
curvature. Imbedded surfaces with

constant mean curvature. The second
variation of volume.

**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 4:__

His unusually extensive chapter on Riemannian submanifolds
goes beyond being a good exposition of readily available material, and performs
a scholarly service. For instance, it pulls together "leftover problems
from classical differential geometry" ...

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