Tubes
The concept of a tube of radius \(r\) is elementary and clear in Euclidean space of dimensions 2 or 3, where it is defined as the set of points \(x\) whose distance to a given submanifold \(P\) of dimension 0, 1 or 2 is \(\leq r\). Since mainly metric notions are involved in this definition, it is easily generalized to tubes \(T^M(P,r)\) of radius \(r\) around a submanifold \(P\) of an arbitrary Riemannian manifold \(M\). The main interest is taken by the volume \(V^M_P(r)\) of \(T^M(P,r)\). For the case of a \(q\)-dimensional submanifold \(P\subset{\mathbb R}^n\), \textit{H. Weyl} [Am. J. Math. 61, 461--472 (1939; Zbl 0021.35503)] proved a formula, representing \(V^{{\mathbb R}^n}_P(r)\) as the polynomial \[ V^{{\mathbb R}^n}_P(r)=\sum_{c} \frac{\Omega_m\,r^{m+c} k_c(P)}{(m+2)(m+4)\ldots(m+c)}\quad (m=n-q, c\text{ is even and }0\leq c\leq q) \] in \(r\) with coefficients depending only on certain integral invariants \(k_{2c}(P)\) of the Riemannian metric of \(P\). The proof of this formula occupies chapters 1.--4., including motivations, preliminary and technical considerations, a quick introduction to Riemannian geometry and the main tool: Fermi coordinates, generalizing normal coordinates around a point of \(M\) to the neighborhood of a submanifold \(P\subset M\). In chapter 5, the generalized Gauss-Bonnet theorem for hypersurfaces is proved with the help of this formula. For arbitrary manifolds \(M\) and their submanifolds a formula similar to that of H.\,Weyl will not exist. Imposing additional structures, for instance that of being a Hermitean manifold, or homogeneous, especially of constant sectional curvature, leads to similar results. Chapters 6 and 7 of the book are devoted to the study of submanifolds of (real and complex) manifolds of constant sectional curvature (resp. constant holomorphic sectional curvature) and formulae for the volume of tubes of their submanifolds. Chapters 8--11 are concerned with comparison theorems and power series expansions of \(V^{{\mathbb R}^n}_P(r)\), Steiner's formula and mean-value theorems in relation with the Laplacian. Each chapter ends with a number of problems. Appendix~A is concerned with moments and formulae for the volume of geodesic balls in spaces of constant curvature. Appendix~B describes routines for the computer software ``Mathematica'' to compute the tangent, normal and binormal vector of a curve \(\alpha(t)\) in \({\mathbb R}^3\) and to produce plots of tubes around \(\alpha\). This second edition of Alfred Gray's monograph comes with newly added historical notes and figures in ``Mathematica''. The only changes are the addition of section 8.9 on comparison theorems for Riemannian manifolds, the inclusion of some new results on harmonic spaces, the structure of Hopf hypersurfaces in complex projective spaces and the conjecture about the volume of geodesic balls. Basic references associated with these additions have also been inserted. For the review of the first edition see Zbl 0692.53001.
scientific article; zbMATH DE number 2019760