Spivak, A Comprehensive Introduction to Differential Geometry
Volume 4.   3rd edition. x + 390 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:
 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 jth 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.

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.

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" ...
Return to top