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7 of 7 people found the following review helpful:
4.0 out of 5 stars
A good start, May 27, 2001
This book could be considered as the second semester of an advanced calculus course and serves as an excellent introduction to differential geometry. The approach is rigorous, but the author does employ a great deal of illustrations to explain the relevant concepts. The first five chapters cover vector fields on curves and surfaces. The many concrete examples given by the author illustrate effectively the normal and tangent vector fields. The Gauss map is then appropriately introduced in Chapter 6 and shown to be onto for compact, connnected, oriented n-dimensional surfaces in n+1-dimensional Euclidean space. This is followed by a discussion of geodesics and parallel transport in the next two chapters. The important concept of holonomy is introduced in the exercises along with the Fermi derivative. These ideas are extremely important in physical applications and must be understood in depth if the reader is to go into areas such as general relativity and high energy physics. The next chapter considers the local behavior of curvature on an n-surface via the Weingarten map. The important concept of the covariant derivative is introduced. The concept of a geodesic spray, so important in the theory of differential equations, is introduced in the exercises. The curvature of plane curves is treated in Chapter 10 with the circle of curvature introduced. The Frenet formulas, which relate the tangent and normal vectors to the curvature and torsion, are discussed in the exercises. The curvature of surfaces is discussed later in Chapter 12 with the first and second fundamental form introduced, along with the very important Gauss-Kronecker curvature. And in this chapter the author introduces the idea of local and global properties of an n-surface. Although not rigorous, the discussion is helpful for students first introduced to these concepts. After a nice overview of convex surfaces, the parametrization of surfaces is discussed in the next two chapters, where the inverse function theorem for n-surfaces is proved. This is followed by a consideration of focal points with Jacobi fields discussed in the exercises. More measure-theoretic concepts are discussed in the next chapter on surface area and volume. Partitions of unity are brought in so as to define the integral of an n-form over a compact oreinted n-surface. Exterior products of forms are introduced in the exercises. Soap bubble enthusiasts will appreciate the discussion on minimial surfaces in Chapter 18. Although very short, the author's treatment does bring out the important ideas. Minimal surfaces have taken on particular important in the new membrane theories in high energy physics recently. This is followed by a detailed treatment of the exponential map in Chapter 19. Once again, techniques with a variational calculus flavor are used to characterize geodesics as shortest paths. After a discussion of surfaces with boundary in Chapter 20 the Gauss-Bonnet theorem is proved in Chapter 21 using Stoke's theorem. The discussion of this important result is crystal clear and should prepare the reader for more advanced statements of it in the general context of differentiable manifolds. This is followed by a brief discussion of rigid motions and isometries in the next two chapters. The book ends with ta discussion of Riemannian geometry, a topic of upmost importance in physics and discussed here with care. A very good book and one that will be useful to beginning students of differential geometry, and also physics students going into the areas of gravitational physics or high energy physics.
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by Manfredo Do Carmo
by Timothy Gowers
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