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Let c be a curve in a n-dimensional space form Mλ n , let P t bea totally geodesic hypersurface of Mλ n orthogonal to c at c(t),and let D0 be a domain in P0. If D is thedomain in Mλ n obtained by a ‘motion’ of D0 alongc and D t is the domain in P t obtained by the motion ofD0 from 0 to t, we show that the n-volume ofD depends only on the length of the curve c, its first curvature, themodified (n − 1)-volume of D0 and the moment of D t with respect to the totally geodesic hypersurface of P t orthogonal to the normal vector f2(t) of c. As a consequence, if c(0) is the center of mass of D0, then the n-volume ofD is the product of the modified (n − 1)-volume of D0 and the length of c. We get an analogous theorem for ahypersurface of Mλ n obtained by ‘parallel motion’ of ahypersurface of P0 along c.

In this paper we study the degeneration of both the cohomology and the cohomotopy Frolicher spectral sequences in a special class of complex manifolds, namely the class of compact nilmanifolds endowed with a nilpotent complex structure. Whereas the cohomotopy spectral sequence is always degenerate for such a manifold, there exist many nilpotent complex structures on compact nilmanifolds for which the classical Frolicher spectral sequence does not collapse even at the second term.

An explicit description of the Frölicher spectral sequence is given that is useful for calculations for compact nilmanifolds. We use the description to exhibit examples of compact complex nilmanifolds of (complex) dimension 3 for which the Frölicher spectral sequence does not degenerate at the second term.

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

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.

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 Kahler manifolds. Then we define the Chern forms of the tangent bundle of a Kahler manifold in terms of curvature and discuss the basic properties of these forms. Section 6.2 is devoted to Kahler 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.

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.

In this chapter we shall show that there is a simultaneous generalization of Weyl’s Tube Formula and the Bishop-Gunther 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-Gunther Inequalities (Theorem 3.17, page 45) show this clearly. The situation for tubes is more complicated and interesting, however, as we shall see.

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.

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.

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 Kahler form of P. To carry out this program, we require generalizations of the Bianchi and Kahler 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.

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]

We consider a special class of compact complex nilmanifolds; namely, compact nilmanifolds with nilpotent complex structure [10]. For such a manifold Г\G we prove that the terms E r (Г\G) in the Frolicher spectral sequence are canonically isomorphic to the terms E r (g ℂ) in the spectral sequence at the Lie algebra level g of G. Moreover, we show that the bidifferential bigraded algebra (A*,*(g ℂ), ∂,∂) is a model for the double complex (⋀*,*(Г\G), ∂, ∂). We also construct new examples of compact complex manifolds of dimension 3 (the lowest possible dimension) with E 2 ≇ E∞,, and exhibit the variation of the Frolicher spectral sequence along curves of complex structures on a real 6-dimensional manifold.

In this paper the authors exhibit a family of 4-dimensional compact solvemanifolds. Each member M3(k) of the family possesses all of the topological properties of a compact Kahler manifold, yet M3(k) can have no complex structure. The proof uses Kodaira's classification of compact surfaces.

On presente des exemples de varietes symplectiques compactes sans structure complexe et/ou de structures de Kahler

Inoculation of rabbit anti-idiotypic (anti-id) antibodies suppresses the subsequent appearance of a cross-reactive idiotype (CRI) associated with the anti-p-azophenylarsonate (anti-Ar) antibodies of A/J mice. Such suppressed mice produce normal concentrations of anti-Ar antibodies which lack the CRI, but against which anti-id antisera can be prepared. The anti-Ar antibodies of an individual, suppressed mouse do not in general share idiotype with anti-Ar antibodies of other A/J mice, either suppressed or nonsuppressed. The present experiments were undertaken to quantitate several "private idiotypes" in a large number of hyperimmunized A/J mice. Anti-Ar antibodies of three mice, suppressed for the CRI, were labeled with 125I and subjected to isoelectric focusing. Four single peaks, that were over 90% reactive with autologous antiid, were randomly selected for use as ligands in a radioimmunoassay, and ascitic fluids containing anti-Ar antibodies from 181 A/J mice were tested as inhibitors. Two of the four idiotypes could not be detected in any mouse other than the donor. The concentration of the idiotype was less than 1 part in 1,250 to less than 1 part in 25,000 of the anti-Ar antibody population; these are minimum values. A third idiotype was detected in 3 of the 181 mice, but at very low concentrations. The fourth idiotype was present in 28% of the mice, again at a low concentration. The data support the existence of a very large repertoire of anti-Ar antibodies in the A/J strain and are consistent with a process of random somatic mutation for generating diversity in hypervariable regions. It is proposed that the cross-reactive idiotype may be controlled by a germ line gene or a gene related to a germ line gene through a small number of somatic mutations; and that the idiotypes that were not detectable in other mice were the products of genes that had undergone extensive mutations, with a low probability of recurrence in other mice.

Simple formulas in terms of Chern classes are given for the volume of a tube about a Kähler submanifold of a space of constant holomorphic sectional curvature. A comparison theorem which generalizes these formulas is also given. Tubes about complete intersections in complex projective space are studied in detail.