Spivak, A Comprehensive Introduction to Differential Geometry
Volume 1.   3rd edition.    xviii + 491 pages.  Clothbound.  1999

Prerequisites: Basic knowledge of metric spaces and of advanced calculus
using linear algebra (essentially the material in Spivak, Calculus on Manifolds).

Click on cover picture for more detailed image

Outline of Contents:
 1. Manifolds
        Elementary properties of manifolds. Examples of manifolds.
 2. Differential Structures
        Smooth structures. Smooth functions. Partial derivatives. Critical points.
        Immersion theorems. Partitions of unity.
 3. The Tangent Bundle
        The tangent space of Euclidean space. The tangents space of an imbedded
        manifold. Vector bundles. The tangent bundle of a manifold. Equivalence
        classes of curves, and derivations. Vector fields. Orientation.
        Addendum: Equivalence of tangent bundles.
 4. Tensors
        The dual bundle. The differential of a function. Classical versus modern
        terminology. Multilinear functions. Covariant and contravariant tensors.
        Mixed tensors, and contraction.
 5. Vector Field and Differential Equations
        Integral curves. Existence and uniqueness theorems. The local flow.
        One-parameter groups of diffeomorphisms. Lie derivatives. Brackets.
        Addenda: Differential equations. Parameter curves in two dimensions.
 6. Integral manifolds
        Prologue; classical integrability theorems. Local Theory; Frobenius
        integrability theorem. Global Theory.
 7. Differential Forms
        Alternating functions. The wedge product. Forms. Differential of a form.
        Frobenius integrability theorem (second version). Closed and exact forms.
        The Poincaré Lemma.
 8. Integration
        Classical line and surface integrals. Integrals over singular cubes.
        The boundary of a chain. Stokes' Theorem. Integrals over manifolds.
        Volume elements. Stokes' Theorem. de Rham cohomology.
 9. Riemannian Metrics
        Inner products. Riemannian metrics. Length of curves. The calculus of
        variations. The First Variation Formula and geodesics. The exponential
        map. Geodesic completeness. 
        Addendum: Tubular neighborhoods.
10. Lie Groups
        Lie groups. Left invariant vector fields. Lie algebra. Subgroups and
        subalgebras. Homomorphisms. One-parameter subgroups.
        The exponential map. Closed subgroups. Left invariant forms.
        Bi-invariant metrics. The equations of structure.
11. Excursion in the Realm of Algebraic Topology
        Complexes and exact sequences. The Mayer-Vietoris sequence.
        Triangulations. The Euler characteristic. Mayer-Vietoris sequence
        for compact supports. The exact sequence of a pair. Poincaré Duality.
        The Thom class. Index of a vector field.  Poincaré-Hopf  Theorem.

From the Preface:

For many years I have wanted to write the Great American Differential Geometry book. Today a dilemma confronts any one intent on penetrating the mysteries of differential geometry. On the one hand, one can consult numerous classical treatments of the subject in an attempt to form some idea how the concepts within it developed. Unfortunately, a modern mathematical education tends to make classical mathematical works inaccessible, particularly those in differential geometry. On the other hand, one can now find texts as modern in spirit, and as clean in exposition, as Bourbaki's Algebra. But a thorough study of these books usually leaves one unprepared to consult classical works, and entirely ignorant of the relationship between elegant modern constructions and their classical counterparts. ... no one denies that modern definitions are clear, elegant, and precise; it's just that it's impossible to comprehend how any one ever thought of them. And even after one does master a modern treatment of differential geometry, other modern treatments often appear simply to be about totally different subjects.

There are two main premises on which these notes are based. The first premise is that it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely in order to rigorize the concepts of classical differential geometry.

The second premise for these notes is that in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary; there is no point in introducing the curvature tensor without explaining how it was invented and what it has to do with curvature. The second volume of these notes gives a detailed exposition of the fundamental papers of Gauss and Riemann. 
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From reviews:

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.

Specifically for Volume 1:

N. J. Hicks, Mathematical Reviews, volume 42, #2369.

This fine book is an education in its area. ... The author spends a good deal of effort in careful motivation of crucial concepts ... His style is a combination of the naïve and the sophisticated that is quite refreshing.  Somehow an impression of honesty and complete integrity underlies his writing at all times, even in his humor.

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 1:                                            

The account is distinguished by its elementary prerequisites ... and by its careful attention to motivation. It is also a lively account, full of examples, excellent ... drawings which function as part of the text ...

A particularly good feature of this volume is its treatment of algebraic topology from the differentiable viewpoint.
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