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

**Volume 2. ** 3rd edition. xiv + 361 pages.
Clothbound. 1999

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**Outline of Contents:
** 1. Curves in the Plane and in Space

Curvature of plane curves. Convex curves. Curvature and torsion of

space curves. The Serret-Frenet formulas. The natural from on a Lie group.

Classification of plane curves under the group of special affine motions.

Classification of curves in Euclidean n-space.

2. What they knew about Surfaces before Gauss

Euler's Theorem. Meusnier's Theorem.

3. The Curvature of Surfaces in Space

A. HOW TO READ GAUSS

B. GAUSS' THEORY OF SURFACES

The Gauss map. Gaussian curvature. The Weingarten map; the first and

second fundamental forms. The Theorema Egregium. Geodesics

on a surface. The metric in geodesic polar coordinates. The integral

of the curvature over a geodesic triangle.

Addendum. The formula of Bertrand and Puiseux; Diquet's formula.

4. The Curvature of Higher Dimensional Manifolds

A. AN INAUGURAL LECTURE

"On the Hypotheses which lie at the Foundations of Geometry"

B. WHAT DID RIEMANN SAY?

The form of the metric in Riemannian normal coordinates.

C. A PRIZE ESSAY

D. THE BIRTH OF THE RIEMANN CURVATURE TENSOR

Necessary conditions for a metric to be flat. The Riemann curvature

tensor. Sectional curvature. The Test Case; first version.

Addendum. Finsler metrics.

5. The Absolute Differential Calculus (The Ricci Calculus)

Covariant derivatives. Ricci's Lemma. Ricci's identities. The curvature tensor.

The Test Case; second version. Classical connections. The torsion tensor.

Geodesics. Bianchi's identities.

6. The Dell Operator

Kozul connections. Covariant derivatives. Parallel translation.

The torsion tensor. The Levi-Civita connection. The curvature tensor.

The Test Case; third version. Bianchi's identities. Geodesics.

The First Variation Formula.

Addenda. Connections with the same geodesics. Riemann's

invariant definition of the curvature tensor.

7. The Repère Mobile (The Moving Frame)

Moving frames. The structural equations of Euclidean space.

The structural equations of a Riemannian manifold. The Test Case;

fourth version. Adapted frames. The structural equations in polar

coordinates. The Test Case; fifth version. The Test Case; sixth version.

"The curvature determines the metric". The 2-dimensional case.

Cartan connections. Covariant derivatives and the torsion and curvature

tensors. Bianchi's identities.

Addenda. Manifolds of constant curvature: Schur's Theorem;

The form of the metric in normal coordinates. Conformally equivalent manifolds.

É. Cartan's treatment of normal coordinates.

8. Connections in Principal Bundles

Principal bundles. Lie groups acting on manifolds. A new definition of

Cartan connections. Ehresmann connections. Lifts. Parallel translation

and covariant derivatives. The covariant differential and the curvature

form. The dual form and the torsion form. The structural equations.

The torsion and curvature tensors. The Test Case; seventh version.

Bianchi's identities.

Addenda. The tangent bundle of

Connections in vector bundles. Flat connections.

**Arthur D. Kramer, American Mathematical Monthly, April 1973**

Volume II with its quasi-historical approach is truly a gem. ... I have only praise for this work, which should be digested by all students of differential geometry ...

**N. J. Hicks, Mathematical Reviews, volume 42, #6726**

The first five chapters of the book are quite unique and provide such valuable background material that I feel many people will be grateful to the author, as I am, for this fine piece of work.

**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 2:

These works [of Gauss and Riemann] are of tremendous
historical interest, and Spivak must be thanked for his illuminating exposition
of them.

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