Showing papers in "Journal of Differential Geometry in 1967"

Curvature and the Eigenvalues of the Laplacian

Abstract: A famous formula of H. Weyl [17] states that if D is a bounded region of R d with a piecewise smooth boundary B, and if 0 > γ1 ≥ γ2 ≥ γ3 ≥ etc. ↓−∞ is the spectrum of the problem $$\displaystyle\begin{array}{rcl} \varDelta f =\big (\partial ^{2}/\partial x_{ 1}^{2} + \cdots + \partial ^{2}/\partial x_{ d}^{2}\big)f =\gamma f\quad \mbox{ in }D,& &{}\end{array}$$ (6.1.1a) $$\displaystyle\begin{array}{rcl} f \in C^{2}(D) \cap C(\overline{D}),& &{}\end{array}$$ (6.1.1b) $$\displaystyle\begin{array}{rcl} f = 0\quad \mbox{ on }B,& &{}\end{array}$$ (6.1.1c) then $$\displaystyle\begin{array}{rcl} -\gamma _{n} \sim C(d)(n/\mbox{ vol }D)^{2/d}\quad (n \uparrow \infty ),& &{}\end{array}$$ (6.1.2) or, what is the same, $$\displaystyle\begin{array}{rcl} Z \equiv \mathop{\mathrm{sp}} olimits e^{t\varDelta } =\sum _{ n\geq 1}\exp \big(\gamma _{n}t\big) \sim (4\pi t)^{-d/2} \times \mathop{\mathrm{vol}} olimits D\quad (t \downarrow 0),& &{}\end{array}$$ (6.1.3) where \(C(d) = 2\pi [(d/2)!]^{d/2}\).

Critical points and curvature for embedded polyhedra

Abstract: Recently a new insight into the Gauss-Bonnet Theorem and other problems in global differential geometry has come about through the connection between total curvature of embedded smooth manifolds and critical point theory for non-degenerate height functions. This paper presents an analogous program for embedded polyhedra. The methods are completely elementary, using the techniques neither of differential geometry nor of algebraic topology. As such the paper has a twofold purpose —to study global geometry of polyhedra for its own sake, and to give a deeper understanding of the theorems of global differential geometry through an elementary presentation of their finite or combinatorial content. Moreover the polyhedral theory applies to a wider class of objects, and gives a new interpretation of the relation between intrinsic and extrinsic curvature. Although the polyhedral part of the paper is relatively self-contained, the remarks which show the connection with the differentiable theory presuppose a familiarity with the classical differentiable results. For a bibliography on these and related problems, see Kuiper [4]. This paper will contain no discussion of the possible convergence theorems relating the polyhedral and differentiable theories—for a presentation of this topic, expecially in the 2dimensional and convex cases we refer to A. D. Alexandrow [1]. For related total curvature concepts see also Chern-Lashof [3]. A subsequent paper of the author will deal with critical points and curvature for mappings of complexes into Eι for / > 1, and into /-dimensional manifolds. The author wishes to thank Professor Kuiper for his interest and advice throughout the development of this research. 1. The critical point theorem Definition. A convex cell complex Mk embedded in E n is a finite collection of cells {Cr}, where each C° is a point, and each Cr is a bounded convex set with interior in some afϊine Eτ c En, such that the boundary dCτ of C r is a union of Cs with s < r, and such that if s < r and Cs Γ\Cr Φ 0, then O C dC\ Mk is called k-dimensional if there is a Ck in Mk but no C*+1.

Holomorphic bisectional curvature

Abstract: We shall occasionally write K(X, Y) for K(σ). The right hand side depends only on σ, not on the choice of an orthonormal basis X, Y. The sectional curvature K is a function defined on the Grassmann bundle of (two-) planes in the tangent spaces of M. A plane a is said to be holomorphic if it is invariant by the (almost) complex structure tensor /. The set of /-invariant planes σ is a holomorphic bundle over M with fibre Pn-i(C) (complex projective space of dimension n — 1). The restriction of the sectional curvature K to this complex projective bundle is called the holomorphic sectional curvature and will be denoted by H. In other words, H(σ) is defined only when a is invariant by /, and H(σ) = K(σ). If X is a vector in a we shall also write H(X) for H(σ). Given two /-invariant planes σ and σ' in TX(M), we define the holomorphic bisectional curvature H(σ, σ') by

A residue formula for holomorphic vector-fields

Abstract: Let X be a holomorphic vector-field on the compact complex analytic manifold M. In an earlier note [3], the behavior of X near its zeros was related to the characteristic numbers of the tangent bundle to M, and the explicit form of this relation was computed in the most nondegenerate situation, that is, in the case of X vanishing at isolated points to the first order. Our aim here is to extend this result in two directions. On the one hand we consider the characteristic numbers of more general bundles E over M on which X \"acts\", and on the other hand we allow X to vanish along submanifolds of higher dimension but still only to the first order. Both extensions are therefore essentially technical in nature. The first extension, to more general bundles, is especially direct and is worthwhile only in so far as it helps to clarify the arguments of [3], The extension to higher dimensional zero-sets is less immediate and also seems to me of some interest for the following reason: When X has isolated singularities the formulas in question may be derived from the generalized Lefschetz formula for transversal maps (see [2]). In the present more general case this is not so one would first of all need a suitable generalization of the Lefschetz formula, and such an extension is available now only if X satisfies some additional restrictions, such as leaving a Riemannian structure invariant. A Lefschetz formula for non-transversal maps is of course closely related to the Riemannian-Roch question, so that our ultimate motivation for this note is the hope that our results might be useful in an eventual purely geometric understanding of Riemann-Roch and its generalizations. To describe our results we need to define two notions. First of all, by an

Topology of almost contact manifolds

Abstract: for a given value of p. The form η defines a differential system and it is important to study the local and global properties of its integral manifolds. To this end, the notion of a quasi-Sasakian structure on an almost contact metric manifold was introduced by one of the authors [1] and its main properties developed. In the present paper their topological properties are considered and it is shown that both compact Sasakian and cosymplectic manifolds have global properties similar to compact Kaehler manifolds. Examples are the unit hypersphere S in Euclidean space, and in fact, the circle bundles over any compact Hodge variety. In the latter class, examples are provided by M X S where M is any compact Kaehler manifold. As one might expect, therefore, not only locally, but topologically as well, the compact cosymplectic spaces are the proper odd dimensional analogues of the compact Kaehler manifolds. A complete, but not compact, simply connected cosymplectic manifold is a product with one factor Kaehlerian. The notation and terminology in this paper will be the same as that employed in [1].

Almost complex structures on tensor bundles

Abstract: Publisher Summary This chapter investigates a problem for tensor bundles T r s M. If a Riemannian manifold M admits an almost complex structure then so does T r s M provided r + s is odd. If r + s is even a further condition is required on M. The proofs depend on some generalizations of the notions of lifting vector fields and derivations on M, which are defined only for tangent bundles and cotangent bundles. It is shown how vector fields on T r s M can be induced from vector fields, tensor fields of type (r, s), and derivations on M. The main problem—that is, to determine a class of tensor bundles that admit almost complex structures, is considered. For this purpose, it is sufficient to consider contravariant tensor bundles because a Riemannian metric tensor field induces a fibre preserving diffeomorphism of T r s M→T r+S M.