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2004 | Buch | 2. Auflage

An Introduction to Weyl’s Tube Formula Kapitel lesen Erstes Kapitel lesen

Tubes

verfasst von: Alfred Gray

Verlag: Birkhäuser Basel

Buchreihe : Progress in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

In July 1998, I received an e-mail from Alfred Gray, telling me: " . . . I am in Bilbao and working on the second edition of Tubes . . . Tentatively, the new features of the book are: 1. Footnotes containing biographical information and portraits 2. A new chapter on mean-value theorems 3. A new appendix on plotting tubes " That September he spent a week in Valencia, participating in a workshop on Differential Geometry and its Applications. Here he gave me a copy of the last version of Tubes. It could be considered a final version. There was only one point that we thought needed to be considered again, namely the possible completion of the material in Section 8. 8 on comparison theorems of surfaces with the now well-known results in arbitrary dimensions. But only one month later the sad and shocking news arrived from Bilbao: Alfred had passed away. I was subsequently charged with the task of preparing the final revision of the book for the publishers, although some special circumstances prevented me from finishing the task earlier. The book appears essentially as Alred Gray left it in September 1998. The only changes I carried out were the addition of Section 8. 9 (representing the discus­ sion we had), 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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. An Introduction to Weyl’s Tube Formula
Abstract
The subject of this book is the computation and estimation of the volume of a tube about a submanifold of a Riemannian manifold. We explain some elementary aspects of the formula in Before beginning the general theory, we carry out in the computations in those low dimensional cases where the tubes can be easily visualized.
Alfred Gray
Chapter 2. Fermi Coordinates and Fermi Fields
Abstract
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
Alfred Gray
Chapter 3. The Riccati Equation for the Second Fundamental Forms
Abstract
Our goal in this chapter is to study the geometry of a Riemannian manifold M in the neighborhood of a topologically embedded submanifoldP. The principal tool that we shall use is the fact that the second fundamental forms of the tubular hypersurfaces about P satisfy a Riccati differential equation.
Alfred Gray
Chapter 4. The Proof of Weyl’s Tube Formula
Abstract
In Theorem 3.15 we wrote down a formula for the volume \( V_P^{\mathbb{R}^n } (r) \) of a tube about a submanifold P of Euclidean space \( \mathbb{R}^n. \) Although this formula has a great deal -of interest, it is not our principal concern, because the integrand is a function of the second fundamental form T of P. As Weyl says in [Weyll]:
So far we have hardly done more than what could have been accomplished by any student in a course of calculus.
Alfred Gray
Chapter 5. The Generalized Gauss-Bonnet Theorem
Abstract
In this chapter we shall prove the Generalized Gauss-Bonnet Theorem using tubes. The principal ingredients are: (1) H. Hopf’s generalization [Hopfl], [Hopf2] of the Gauss-Bonnet Theorem for hypersurfaces in ℝn, (2) the Nash Embedding Theorem [Nash], (3) Weyl’s Tube Formula [Weyll], and (4) some elementary calculations with volumes of tubes and Euler characteristics. This proof using tubes is neither the most direct nor the most elegant; Chern’s proof [Chernl] excels in both of these respects (see the end of this chapter). However, the proof via steps (1)—(4) has many interesting features; it is due to Allendoerfer [All] and Fenchel [F12]. Furthermore, some of the techniques of the present chapter will be useful when we discuss complex versions of the Weyl Tube Formula in Chapters 6 and 7.
Alfred Gray
Chapter 6. Chern Forms and Chern Numbers
Abstract
We interrupt our study of tubes in order to present some basic information about complex manifolds that we shall need in Chapter 7 when we prove the Complex and Projective Weyl Tube Formulas. In Section 6.1 we start by recalling some basic facts about Kähler manifolds. Then we define the Chern forms of the tangent bundle of a Kähler manifold in terms of curvature and discuss the basic properties of these forms. Section 6.2 is devoted to Kähler manifolds with constant holomorphic sectional curvature. In Section 6.3 we recall some facts about locally symmetric spaces that we shall eventually need. More can be said about the volume of a sub-manifold P of a Riemannian manifold M, provided the second fundamental form of P is not too complicated when compared with the curvature operator of M. Therefore, in Section 6.3 we define the notion of compatible submanifold and study the basic properties of such submanifolds. We derive Study’s formula [Study] for the volume \( V_m^{\mathbb{K}_{hol}^n (\lambda )} (r) \) of a geodesic ball of radius r in a space \( \mathbb{K}_{hol}^n(\lambda ) \) of constant holomorphic sectional curvature in Section 6.4. Complex projective space \( \mathbb{C}{P^n}(\lambda ) \) is discussed in Section 6.5, where we use Study’s formula for \( V_m^{\mathbb{C}{P^n}(\lambda )}(r) \) to find the volume of \( \mathbb{C}{P^n}(\lambda ) \) Then in Section 6.6, we compute the total Chern form of \( \mathbb{C}{P^n}(\lambda ) \) (A) directly from its definition in terms of curvature. A brief treatment of Wirtinger’s Inequality is given in Section 6.7.
Alfred Gray
Chapter 7. The Tube Formula in the Complex Case
Abstract
For a complex submanifold P of a complex Euclidean space Cn, we shall give a considerably simplified version of Weyl’s Tube Formula. It turns out that all the coefficients k 2c (P) have a topological nature similar to that of the top coefficient. We shall derive explicit formulas for the k 2c (P) as integrals involving the Chern forms and Kähler form of P. To carry out this program, we require generalizations of the Bianchi and Kähler identities to double forms; these we give in Section 7.1. We use these identities in Section 7.2 to derive formula (7.18) for the volume of a tube about a complex submanifold with compact closure in a complex Euclidean space. We call it the Complex Weyl Tube Formula.
Alfred Gray
Chapter 8. Comparison Theorems for Tube Volumes
Abstract
In this chapter we shall show that there is a simultaneous generalization of Weyl’s Tube Formula and the Bishop-Günther Inequalities. Imagine a manifold M equipped with a family of Riemannian metrics. Intuitively, it is clear that if one varies the metric of M so that its curvature increases, then volumes in M decrease. For geodesic balls the Bishop-Günther Inequalities (Theorem 3.17, page 45) show this clearly. The situation for tubes is more complicated and interesting, however, as we shall see.
Alfred Gray
Chapter 9. Power Series Expansions for Tube Volumes
Abstract
In this chapter we take a completely different approach to the study of volumes of tubes. We shall compute the first few terms of the power series of the volume function \(V_P^M(r)\) as a function of r. In the same issue of the American Journal of Mathematics in which Weyl’s paper [Weyl1] appeared in 1939, there is an article [Ht] by the statistician Hotelling. In it Hotelling computes the first two nonzero terms of the expansion for \(V_P^M(r)\) in the case that P is a curve in an arbitrary Riemannian manifold M. In fact, Weyl’s paper is partially a response to Hotelling’s paper. Hotelling discusses several applications of tube formulas to statistics.
Alfred Gray
Chapter 10. Steiner’s Formula
Abstract
In 1840 J. Steiner [Sr] studied convex regions in 2- and 3-dimensional Euclidean space; he obtained a formula for the volume of the convex regionB r consisting of those points whose distance from a given convex region B is less than or equal to r. In this chapter we put Steiner’s Formula into the same general framework as Weyl’s Tube Formula.
Alfred Gray
Chapter 11. Mean-value Theorems
Abstract
This chapter is devoted to an exposition of the results of [GW] and related papers.
Alfred Gray
Backmatter
Metadaten
Titel
Tubes
verfasst von
Alfred Gray
Copyright-Jahr
2004
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-7966-8
Print ISBN
978-3-0348-9639-9
DOI
https://doi.org/10.1007/978-3-0348-7966-8