Differential geometry

As can be seen from the title of this book, our treatment of mechanical control systems is geometric, more precisely, differential geometric. In this chapter we review the essential differential geometric tools used in the book. Since a thorough treatment of all parts of the subject necessary for our objectives would be even more lengthy than what we currently have, we essentially present a list of definitions and facts that follow from these definitions. The treatment we give, while comparatively brief, is not as fast-paced as was the case in Chapter 2. In this chapter we spend more time on examples that illustrate the main ideas. Nevertheless, a reader who knows no differential geometry can expect to invest some time in learning the subject, and might find the review in this chapter too hasty. In this case, additional references (see below) will be helpful in providing context and further examples. Readers having a passing acquaintance with differential geometry may wish to quickly scan this chapter to see if there are any major topics that are unfamiliar to them.