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In this chapter we shall be concerned with the geometry of tubes about a submanifold P of a general Riemannian manifold M (and not specifically tubes in Euclidean space). In we define and discuss normal and Fermi coordinates. is devoted to a quick review of the curvature tensor of a Riemannian manifold and its various contractions. Instead of working directly with Fermi coordinates, it is usually easier to use certain vector fields, which we call Fermi fields and define in There is a close relation between Fermi fields and the more familiar Jacobi fields (see Corollaries 2.9 and 2.10). In Chapter 3 we shall derive three fundamental equations to describe the geometry of tubes using Fermi fields. Since Fermi coordinates are a generalization of normal coordinates, it is not surprising that there is a tube generalization of the well-known Gauss Lemma; this we prove in |
