A First Course in Geometric Topology and Differential Geometry

After a brief overview of the elementary topology of subsets of Euclidean space in chapter 1, topological surfaces are discussed in chapter 2. Surfaces are built up from arcs, disks, and one-spheres. Unfortunately, the proofs of the theorem of invariance of domain and the Schonflies Theorem are not included, but references are given. Gluing techniques though are effectively discussed, and the author does not hesitate to use diagrams to explain the relevant concepts. The more popular constructions in surface topology, namely the Mobius strip and the Klein bottle are given as examples of the cutting and pasting techniques. The amusing fact that the Klein bottle can be obtained from gluing two Mobius strips along their boundaries is proven.

The theory of simplicial surfaces is discussed in the next chapter. Simplicial surfaces are much easier to deal with for beginning students of topology. Simplicial complexes are introduced first, and the author then studies which simplicial complexes have underlying spaces that are topological surfaces. He proves that this is the case when each one-dimensional simplex of the complex is the face of precisely two two-dimensional simplices, and the underlying space of each link of each zero-dimensional simplex of the complex is a one-dimensional sphere. Unfortunately, the author does not prove that any compact topological surface in n-dimensional Euclidean space can be triangulated. The Euler characteristic is defined first for 2-complexes and it is shown that it is the same for two simplicial surfaces that triangulate a compact topological surface. The author does prove in detail the classification of compact connected surfaces. Interestingly, the author also proves a simplicial analogue of the Gauss-Bonnet theorem, and gives a proof of the Brouwer fixed point theorem.

The author turns to smooth surfaces in the next few chapters, wherein curves are defined along with the relevant differential-geometric notions such as curvature and torsion. The fundamental theorem of curves is proven. The reader is first introduced to the concept of what in more advanced treatments is called a differentiable manifold, and several concrete examples are given of smooth surfaces. The differential geometry of smooth surfaces is outlined, with the first fundamental form and directional derivatives discussed in great detail. The reader should be familiar with the inverse function theorem to appreciate the discussion of regular values.

Even more interesting differential geometry is discussed in chapter 6, which covers the curvature of smooth surfaces. The important Gauss map is defined, along with the Weingarten map and the second fundamental form. This allows an intrinsic notion of curvature, but the author does perform explicit computations of curvature using various choices of coordinates. The proof that Gaussian curvature is intrinsic (Theorema Egregium) is proven, along with the fundamental theorem of surfaces. Geodesics, so important in physical applications, are discussed in the next chapter. The reader gets a first look at the "Christoffel symbols", even though they are not designated as such in the book.

The book ends with a thorough treatment of the Gauss-Bonnet theorem for smooth surfaces. The smooth case is much more difficult to prove than the simplicial case, as the reader will find out when studying this chapter. The author also gives a very brief introduction to non-Euclidean geometry.


a remark on omissions   February 18, 2005
twit
5 out of 11 found this review helpful

I have not read the book, only the reviews. In one excellent review here it is remarked that it is "unfortunate" that the author does not prove the Schoenflies theorem and the triangulability of surfaces.

later this same reviewer observes that the proof of the smooth Gauss Bonnet theorem in the book seems relatively hard. I merely wish to point out that the author has made choices in the reader's interest both by what he includes and what he omits.

The two theorems named above which are not proved, could well take another entire book to prove. They are far harder than the smooth Gauss Bonnet theorem.

I have seen entire books devoted to proving triangulability, and Schoenflies theorem was the subject of weeks of tedious work in a topology course I took as a student. I still dislike even hearing of this result. So if these omissions are the reviewer's only criticisms of the book, they should rightly be considered pluses.

Hence I also give the book at least 4 stars, by logical deduction.


Just a mediocre book of lesser extent   May 8, 1998
1 out of 34 found this review helpful

First, the title reads fine. But there's a catch. This kind of title sounds like it covers all. The truth is. It ain't true. Second. The author's attitude. I'd rather say the author is talking to himself.