Tubes
type: Item
instance of: scholarly article
title: Tubes
zbMATH Open document ID: 1048.53040
author: Alfred Gray
published in: Progress in Mathematics
publication date: 2003-12-18T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 2019760
zbMATH Keywords: volume of a tube
zbMATH Keywords: Weyl's tube formula
zbMATH Keywords: Fermi coordinates
zbMATH Keywords: Fermi vector fields
zbMATH Keywords: comparison theorems
zbMATH Keywords: Bishop-Günther inequalities
zbMATH Keywords: Gauss-Bonnet theorem
zbMATH Keywords: complex projective space
zbMATH Keywords: Chern forms
zbMATH Keywords: Steiner's formula
zbMATH Keywords: parallel hypersurfaces
zbMATH Keywords: geodesic balls
reviewed by: Hubert Gollek
describes a project that uses: Mathematica
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2019760