Problems in Analysis: A Symposium in Honor of Salomon Bochner

by Robert C. Gunning (Editor)

Overview

The present volume reflects both the diversity of Bochner's pursuits in pure mathematics and the influence his example and thought have had upon contemporary researchers.

Originally published in 1971.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Product Details

ISBN-13:	9780691647425
Publisher:	Princeton University Press
Publication date:	04/19/2016
Series:	Princeton Mathematical Series , #31
Pages:	364
Product dimensions:	6.20(w) x 9.20(h) x 1.30(d)

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Problems in Analysis

A Symposium in Honor of Salomon Bochner

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By Robert C. Gunning

PRINCETON UNIVERSITY PRESS

Copyright © 1970 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08076-5

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CHAPTER 1

On the Group of Automorphisms of a Symplectic Manifold

EUGENIO CALABI1

* * *

1. Introduction

Let X be a connected, differential manifold of 2n dimensions. A symplectic structure on X is the geometrical structure induced by a differentiable exterior 2-form ω defined on X, satisfying the following conditions:

(i) The form ω is closed: dω = 0;

(ii) It is everywhere of maximal rank; this means that the 2n-form ωn (nth exterior power of ω) is everywhere different from zero, or equivalently, the skew-symmetric (2n) × (2n) matrix of coefficients of ω, in terms of a basis for the cotangent space, is everywhere nonsingular.

A classical theorem, ordinarily attributed to Darboux, states that a 2n-dimensional symplectic manifold (i.e., a manifold with a symplectic structure) can be covered by a local, differentiable coordinate system {U; (x)} where (x) = (x1, ..., x2n): U [right arrow] R2n, in terms of which the local representation of the structural form ω becomes

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such a system of coordinates is called a canonical system.

The purpose of this study is to describe the group G of automorphisms of a symplectic manifold, i.e., the group of all differentiable automorphisms of X which leave the structural 2-form ω invariant, and the invariant subgroups of G. The group G can also be characterized as mapping canonical coordinate systems into canonical systems.

Two normal subgroups of G are distinguished immediately as follows:

DEFINITION 1.1. Let (X, ω) be a 2n-dimensional symplectic manifold and let G be the group of all symplectic transformations of X. If X is not compact, we denote by G0 the subgroup of G consisting of all symplectic transformations of X that have compact support; that is to say, a symplectic transformation g [member of] G belongs to G0 if and only if g equals the identity outside a compact region of X.

Definition 1.2. Let (X, ω) be a 2n-dimensional symplectic manifold and let G be the group of all symplectic transformations of X. We denote by G0,0 the subgroup of G called the minigroup generated by the so-called locally supported transformations, defined as follows: a transformation g [member of] G is called locally supported if there exists a canonical coordinate system {U; (x)} defined in a contractible domain U with compact closure, such that the support of g lies in U.

The minigroup G0,0and its corresponding Lie algebras are introduced here merely for expository convenience. In Section 3 it will be shown that the commutator subgroup of the arc-component of the identity in G0 coincides either with G0,0 or with a normal subgroup of codimension 1 in G0,0 (see Theorem 3.7, Section 3).

An elementary example of a locally supported transformation is the following: let {U; (x)} be a canonical coordinate system in X; let its range V = (x)(U) [subset] R2n contain the ball [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some A > 0; choose a real-valued differentiable function O(r) of a real variable r ≥ 0 with support contained in a closed segment [0, A'] with A' < A. Then it is easily verifiable that the transformation f in R2n (with the 2-form ω =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a symplectic transformation which equals the identity for r ≥ A'. Therefore its restriction to V defines via the coordinate map (x) a symplectic transformation in U that can be trivially extended by the identity map in X – U to a locally supported symplectic transformation in X.

We shall state here the main results of this study in a preliminary form; more precise and stronger versions of these are repeated as theorems in the later sections.

Statement 1. The groups G, G0, and G0,0 are infinite dimensional Lie groups in terms of the Whitney C∞ topology (in the ease of G) and the compactly supported C∞ topology (in the case of G0 and G0,0).

Statement 2. The minigroup G0,0is a closed, normal subgroup of G0; the quotient group G0/G0,0 is locally isomorphic to the de Rham cohomology group H10(X, R), the first cohomology group of X with real coefficients and compact support.

Statement 3. If X is not compact, the completion G1,0 of G0,0 in the compact-open topology of presheaves (i.e., the group obtained by adjoining to G0,0 the infinite products of sequences of gv, [member of] G0,0, where for each compact K [subset] X only finitely many of the gv differ from the identity in K) is a closed, normal subgroup of G; the quotient group G/G1,0 is locally isomorphic to the de Rham group H1(X, R), i.e., to the first cohomology group with closed support.

Statement 4. The group G1,0 has no connected, closed, normal subgroups other than the identity; in particular its commutator subgroup is an open subgroup. The same is true of G0,0, of course, if X is compact (in which case G0,0 = G1,0. On the other hand, if X is not compact, the commutator subgroup G0,0of G0,0 is normal in G0, has codimension equal to 1, relative to G0,0, and has no connected, closed, normal subgroups other than the identity.

The next two sections deal with the Lie group structure of the groups G and G0, emphasizing the relationship with the corresponding Lie algebras; the main tools used here are due to J. Moser. In Section 3 we prove the four main statements just given at the Lie algebra level, and in Section 4 we expand the results at the group level, trying as far as we have succeeded to obtain results on the global structure of these groups and their closed, normal subgroups.

2. Infinite dimensional Lie groups of differentiable transformations

We shall summarize in this section some of the known facts about infinite dimensional Lie groups or pseudogroups of differentiable transformations, especially with regard to their relationships with the corresponding Lie algebras of tangent vector fields.

We denote by G, H, and so forth, groups of differentiable transformations of a manifold; the corresponding pseudogroups of local transformations are denoted by G, H, and so forth; the associated Lie algebras of globally defined vector fields will be denoted by German capitals [??], [??], and so forth; the corresponding presheaves of local vector fields will be denoted by small German letters, such as g, h, etc.

An infinite dimensional Lie group G of global, differentiable transformations in a manifold X (or alternately a pseudogroup G of local transformations) is an infinite dimensional, differential manifold (naturally we do not exclude from this notion finite dimensional manifolds), in the following sense: for any finite dimensional, differential manifold M a map φ M [right arrow] G is defined to be differentiable, if and only if the corresponding evaluation map φ M × X [right arrow] X, where φ (t, x) = (φ(t))(x), is differentiable (in the case of a pseudogroup, one requires also that the domain of definition of φ be open in M × X).

For our present purposes, it seems irrelevant to fix any topology on G; we regard it rather as a compactoid, that is to say, we allow ourselves to consider the equivalence class of topologies that are compatible with the category of differentiable maps φ: M [right arrow] G just defined. The Lie algebra [??] associated to G is then the set of tangent vector fields (respectively presheaves of local vector fields) each defined as the equivalence class of differentiable paths in G originating at the identity with the obvious equivalence relation. The question that ordinarily arises here is how to recapture the arc component of the identity in G from the sheaf of germs g of Lie algebras determined by [??].

Any local cross section [??] in g (that is to say, each local vector field belonging to g) defines a one-parameter subpseudogroup of differentiable transformations by integrating the vector field to the corresponding autonomous flow. The composition of such one-parameter local flows defines a pseudogroup ΓG in the pseudogroup G associated to G. If G is finite dimensional, it is known classically that ΓG defines an open subpseudogroup of G. In the infinite dimensional case the same conclusion holds, if G acts real analytically, at least by considering first local cross sections of g and collating the resulting local transformations. In the C∞ case for infinite dimensional Lie groups, several authors have shown examples to the effect that one-parameter subpseudogroups are not dense in a neighborhood of the identity, and the details of some of these examples give a strong indication that even the composition of the elements of such one-parameter systems may not fill out any neighborhood of the identity in G. Some authors have suggested a method based on affine connections; this method permits one to attach to each local vector field in [??] a differentiable path in G originating at the identity but not constituting, in general, a one-parameter pseudogroup, so that the union of such paths fill out a neighborhood of the identity in G. This method is satisfactory in the case of differentiably acting pseudogroups definable in terms of a first-order, integrable differential system in the coordinate transformations but would require higher order connections for pseudogroups of a more complicated nature.

The correspondence between presheaves of Lie algebras of local vector fields and pseudogroups of local differentiable maps can be established in a natural way only as a local one-to-one correspondence between differentiable paths. Thus the fundamental theorem on existence, uniqueness, and continuity with respect to initial data for ordinary differential equations establishes a one-to-one, locally biregular correspondence between differentiable, one-parameter families of local vector fields in g (i.e., paths in g) and local flows in X belonging to G (i.e., paths in G): any topological structure in the stalks of the sheaf of germs of one-parameter families of vector fields in X belonging to g (provided that its definition includes minimal regularity conditions) yields a well defined topology on the stalks of the corresponding sheaf of germs of G-flows. The corresponding topology on the sets of germs of elements of G is then obtained by passage to the quotient, assigning to each path in G (originating at the germ of the identity) the germ of the terminal element [member of] G of the path. Thus one obtains from the sheaf of germs of Lie algebras g a sheaf G of germs of diffeomorphisms. The group G of global cross sections in G can be obtained without any difficulty if the manifold X is compact.

In the case where X is not compact, the corresponding sheaves g of germs of Lie algebras of vector fields and G of germs of pseudogroups of transformations lead to the corresponding global Lie algebras and groups in many ways, of which two are the most important: the ones with unrestricted support and the ones with compact support. They are obtained from the topologies of the corresponding stalks by "globalizing" them, in the former case either by the compact-open extension of a uniform topology on the stalks (compact-open topology) or by a Whitney topology and in the latter case by a uniform topology over uniformly compactly supported cross sections. It is worthwhile noting that the global groups obtained from the pathwise-connected sheaf of groups is not necessarily connected, as we know well in the case of the celebrated group of all differentiable, orientation-preserving automorphisms of spheres.

3. The Lie algebra of symplectic vector fields

We apply the concepts reviewed in the previous section to the case of the Lie algebra associated with the group G or G0 of automorphisms of a symplectic manifold (X, ω).

If φ is any exterior p-form and [xi] is a vector field in a manifold, we denote by [xi] [??] φ the interior product of [xi] with φ this is the (p – 1)-form (identically zero if p = 0) whose value at a (p – 1)-vector η1 Λ ... Λ ηp-1 is given by

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Lie derivative of φ with respect to [xi] is then obtained from the formula

[[xi], φ?] = d([xi] [?? ] φ) + [xi] [??] dφ.

It is well known that any differentiable path in the sheaf of germs of diffeomorphisms, originating at the germ of the identity at any x [member of] X, leaves the form φ invariant along the orbit of x, if and only if the path in the sheaf of germs of vector fields given by the differential of the given path yields a one-parameter family of germs of vector fields [xi] satisfying [[xi], φ] = 0. Thus the Lie algebra of the symplectic group G or G0 is described locally by the vector fields [xi] satisfying, since ω is closed,

(3.2) d([xi] [??] ω) = 0.

We now introduce the Lie algebras corresponding to the groups G, G0, and G0,0 previously given in Definitions 1.1 and 1.2.

Proposition 3.1. The Lie algebras [??], [??]0, and [??]0,0corresponding, respectively, to the groupsG, G0, andG0,0are given by the global vector fields [xi] onXsatisfying (3.2) and, in addition,

(i) satisfying no further conditions in the case of [??];

(ii) having compact support in the case of [??]0;

(iii) generated, in the caseG0,0by vector fields [xi]v, where each [xi]vhas compact support contained in a contractible domain Uv admitting a canonical coordinate system (x1, ..., x2n), that is to say, such that there exists a function uv with compact support in Uv satisfying

[xi]v [??] ω = duv

or equivalently

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The proof of this proposition is clear.

Since the form ω is everywhere of maximal rank, the bundle map from the tangent to the cotangent vector bundle defined by assigning to [xi] the 1-form [xi] [??] ω is bijective. Thus, for instance, the Lie algebra [??] is isomorphic, as a vector space over the real numbers, to the set of all closed Pfaffian forms on X. This isomorphism induces a Lie algebra structure on the set of all Pfaffian forms called the "Poisson bracket"; more precisely, for any 1-form α we denote by α# the uniquely defined tangent vector [xi] such that [xi] [??] ω = α (at the bundle, sheaf, local, or global level); then the Poisson bracket of two Pfaffian forms α and β is defined to be

{α, β} = [α#, β#] [??] ω.

The Lie algebra of all C∞ 1-forms (globally defined on X) with the Poisson bracket is isomorphic under the map {α right arrow] α#} to the Lie algebra of all tangent vector fields; the vector subspace consisting of all closed 1-forms is a Lie subalgebra. It follows from (3.2) that this subalgebra is isomorphic to the algebra [??] of all vector fields [xi] such that [[xi], ω] = 0. The subalgebras [??]0 and [??]0,0 of [??] are similarly characterized. We shall now define some vector subspaces of these Lie algebras that will, in fact, turn out to be ideals.

Definition 3.2. We denote by [??]' the vector subspace of [??] consisting of all germs of vector fields [xi] [member of] [??] such that [xi] [??] ω is exact; that is to say, [??]' consists of all the vector fields (du)# where u is an arbitrary differentiable function on X. Similarly we denote by [??]"0 the vector subspace of [??]'0 consisting of all the vector fields (du)# where u is an arbitrary function on X with compact support. Finally, we denote by [??]"0 the vector subspace of [??]"0 consisting of the vector fields [xi] = (du)# where u is a function with compact support on X satisfying, in addition,

(3.4) ∫x uωn = 0

(Continues...)

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Table of Contents

• Frontmatter, pg. i
• Foreword, pg. vii
• Contents, pg. ix
• On the Group of Automorphisms of a Symplectic Manifold, pg. 1
• On the Minimal Immersions of the Two-sphere in a Space of Constant Curvature, pg. 27
• Intersections of Cantor Sets and Transversality of Semigroups, pg. 41
• Kählersche Mannigfaltigkeiten mit hyper-q-konvexem Rand, pg. 61
• Iteration of Analytic Functions of Several Variables, pg. 81
• A Class of Positive-Difinite Functions, pg. 93
• Local Noncommutative Analysis, pg. 111
• Linearization of the Product οf Orthogonal Polynomials, pg. 131
• Eisenstein Series on Tube Domains, pg. 139
• Laplace-Fourier Transformation, the Foundation for Quantum Information Theory and Linear Physics, pg. 157
• An Integral Equation Related to the Schroedinger Equation with an Application to Integration in Function Space, pg. 175
• A Lower Bound for the Smallest Eigenvalue of the Laplacian, pg. 195
• The Integral Equation Method in Scattering Theory, pg. 201
• Group Algebra Bundles, pg. 229
• Quadratic Periods of Hjperelliptic Abelian Integrals, pg. 239
• The Existence of Complementary Series, pg. 249
• Some Recent Developments in the Theory of Singular Perturbations, pg. 261
• Sequential Convergence in Lattice Groups, pg. 273
• A Group-theoretic Lattice-point Problem, pg. 291
• The Riemann Surface of Klein with 168 Automorphisms, pg. 297
• Envelopes of Holomorphy of Domains in Complex Lie Groups, pg. 309
• Automorphisms of Commutative Banach Algebras, pg. 319
• Historical Notes on Analyticity as a Concept in Functional Analysis, pg. 325
• A -Almost Automorphic Functions, pg. 345