Symplectic Geometry and Fourier Analysis: Second Edition (Dover Books on Mathematics) - Softcover

Wallach, Nolan R.

Softcover

ISBN 10: 0486816893 ISBN 13: 9780486816890

Publisher: Dover Publications, 2018

Synopsis

This book derives from author Nolan R. Wallach's notes for a course on symplectic geometry and Fourier analysis, which he delivered at Rutgers University in 1975 for an audience of graduate students in mathematics and their professors. The monograph is geared toward readers who have taken a basic course in differential manifolds and elementary functional analysis.
The first chapters cover certain geometric preliminaries, advancing to discussions of symplectic geometry and the application of its concepts to the action of a Lie group on a symplectic manifold. Subsequent chapters address Fourier analysis, the metaplectic representation, and quantization. A final chapter on the Kirillov theory applies the ideas of the previous chapters to homogeneous symplectic manifolds of nilpotent Lie groups. The book concludes with an Appendix on Quantum Mechanics by Robert Hermann.

About this title

About the Author

Nolan R. Wallach is Professor Emeritus of Mathematics at the University of California, San Diego. His other books include Real Reductive Groups and Symmetry, Representations, and Invariants.

Excerpt. © Reprinted by permission. All rights reserved.

Symplectic Geometry and Fourier Analysis

By Nolan R. Wallach

Dover Publications, Inc.

Copyright © 2018 Nolan R. Wallach
All rights reserved.

ISBN: 978-0-486-81689-0

Contents

Preface to the Second Edition, ix, 
Preface, xi, 
1 Differential Geometry, 1, 
2 Symplectic Geometry, 23, 
3 Homogeneous Symplectic Manifolds, 47, 
Appendix to Chapter 3 — Tori, 75, 
4 Fourier Analysis, 87, 
5 The Metaplectic Representation, 119, 
6 Quantization, 149, 
7 The Kirillov Theory, 159, 
Appendix I to Chapter 7 — Trace Class Operators, 204, 
Appendix II to Chapter 7 — Induced Representations, 212, 
Bibliography, 215, 
Appendix on Quantum Mechanics — by Robert Hermann, 219, 
Bibliography to Appendix on Quantum Mechanics, 255, 

CHAPTER 1

Differential Geometry

In this chapter we compile several results from differential geometry which will be used throughout these notes. We advise the reader to use this chapter for reference purposes only. Most of the material is either part of or easily derived from a graduate course in differential geometry using, for example, F. Warner [27] or R. Hermann [9] as a text.

1. The De Rham Theorem

Let M be a paracompact, connected, C8 manifold. We denote by

Dk(M) (resp. Dk(M, C))

the differential forms (resp. complex valued) differential forms of degree k on M. We will use the notation Dk if the context M (or C) is clear.

Let d: Dk [right arrow] Dk+1 be the operation of exterior differentiation. The de Rham cohomology "groups" of M are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

k = 0, 1, 2, ..., D-1 = (0).

De Rham's theorem relates Hkd(M; K) with Hk(M; K) where Hk(M; K) is one of the usual cohomology theories on M (we will use the Cech cohomology theory). Since we will need an explicit form of the de Rham theorem in the next few chapters we review in detail the isomorphisms Hkd (M; K) with Hk(M; K), k = 0, 1, 2. (A complete account of the de Rham theorem can be found in F. Warner [27].)

We recall the definition of the Cech cohomology groups. Let G be an abelian group written additively. If

U = [Ua}a[member of]I

is an open covering of M, then a U-k-cochain on M with values in G is a rule, c, that assigns to each collection

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of elements in U such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

an element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let Ck(U; G) be the set of all U-k-cochains with values in G. If c, f [member of] Ck(U; G) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

defines an abelian group structure on Ck(U; G).

Define d: Ck(U; G) [right arrow] Ck+1 (U; G) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(here the "roof" over an index means delete the index). It is easy to see that d2 = 0. Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

h = 0, ... (C-1(U; G) = (0)).

Let U and V be open coverings of M. Then we use the notation:

U > V if U refines V ¦

If U >V, choose t: U [right arrow] V so that if X [member of] U then t X [subset] X. If c [member of] Ck (V; G) then define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then t* o d = d o t*. This implies that t* induces a map

t*: Hk(V; G) [right arrow] Hk(U; G).

t* is independent of the choice of t(c.f. Hirzburch [10]). Set tUV equal to the map

t*: Hk(V; G) [right arrow] Hk(U; G).

Then, if V< W< U,

tWV o tUW = tUV

by the above observations. We define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is, form the disjoint union of the Hk(U; G). If a [member of] Hk(U; G) and b [member of] Hk(V; G) then we say:

a ~ b

if there is

W > U and W > V

such that

tWUa = tWVb.

Then a ~ b is an equivalence relation and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the set of all equivalence classes. Hk(M; G) is an abelian group under the obvious operations.

A C8 map of [0, ] x M [right arrow] M is a continuous map that extends to a C8 map of (-e, 1 + e) x M [right arrow] M for some e > 0.

Definition 1.1. If U [subset] M is open, then we say that U is (C8) contractible if there is a C8 map

F: [0, 1] x U [right arrow] U

so that F(0, x) = x for all x [member of] U and F(1, x) = x0 for all x [member of] U. An open covering U = {Ua}a[member of]I is said to be contractible if, whenever

a1, ..., ak [member of] I

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is contractible.

Theorem 1.2.If U is an open covering of M then U has a contractible refinement.

This result is proved by putting a Riemannian structure on M and taking for each Ua [member of] U, p [member of] Ua, Np,U a convex neighborhood of p in U.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a contractible refinement. See Helgason [8] for the existence (and definition of convex neighborhoods).

Theorem 1.3.

a) If U is a contractible covering of M then Hk(U; G) is isomorphic with Hk(M; G).

b) If G is the additive group of R or C we denote Hk (M; G) by Hk(M; R) or Hk(M; C). Hk(M; G) is canonically isomorphic with Hkd(M; G). We denote the isomorphism by

d x R: Hkd(M; K) [right arrow] Hk(M; K).

Of course, this theorem is well known. a) is due to A. Weil [29] and b) is the de Rham theorem. We give a proof of b) (and a) for K = R or C) and for k = 0, 1 or 2 since we will be using explicit forms of the isomorphisms asserted to exist in b). We need a special case of Theorem 1.3.

Lemma 1.4 (Poincaré Lemma).If U [subset] M is an open contractible set and ? [member of] Dk, d? = 0, then there is ? [member of] Dk-1 so that ? = d?.

We will now give a sketch of a proof of this lemma. Let U be an open subset of a smooth manifold M and let F: [0, 1] x U [right arrow] U be C8. If 0 = t = 1 then we set Ft(u) = F(t, u) for u [member of] U. We will identify ? [member of] Dk ([0, 1] x U) (that is, ? is the restriction of an element of Dk ((-e, 1 + e) x U)) with an element of Dk(U) if it satisfies the two conditions:

1. for all v1, ..., vk [member of] T(t, p) ((-e, 1 + e) x M)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. If we take local coordinates x1, ..., xn in a neighborhood, Z, of p in U and local coordinates t, x1, ..., xn in (-e, 1 + e) x Z then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is independent of t for all choices of 1 = i1, ..., ik = n. We now define map l of Dk ((-e, 1 + e) x U) to Dk-1 ((-e, 1 + e) x U) by the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The key observation is

Lemma 1.5.If p [member of] U and Z and x1, ..., xn are as above then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We leave the proof to the reader with the hint: It is enough to consider two cases

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We define a linear map h : Dk(U) [right arrow] Dk(U) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have

Lemma 1.6.If ? [member of] Dk(U) then

dh(?) + hd(?) = F*1{(?) - F*0(?).

Proof. The left-hand side of the equation says that (in the notation above for each p [member of] U)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can now prove the Poincare-Lemma. If U is contractible to p [member of] U then there exists a C8 map F : [0, 1] x U [right arrow] U such that F(0, q) = p and F(1, q) = q for all q [member of] U. Thus if k = 1 and if ? [member of] Dk(U) then F*0? = 0 and F*1? = ?. If in addition d? = 0 then we have dh(?) = ?.

We now begin the proof of Theorem 1.3b) for k = 0, 1 or 2. We fix an arbitrary contractible covering

{Ua}a[member of]I = U

of M. We note that if we have shown

Hk(U; K) = Hkd (M; K)

then we will have shown both a) and b).

k = 0: If c [member of] C0(U; K), dc = 0, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, c defines a constant on M. Hence, H0(U; K) [equivalent to] K. If f [member of] D0 and df = 0, then f is constant (M is connected). Hence,

H0d(M; K) [equivalent to] K.

k - 1: Let ? [member of] D1, d? - 0. Since Ua is contractible for each a [member of] I we have, by Lemma 1.4, fa [member of] C8(Ua; K) so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Continues...)

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Edition details

• Publisher: Dover Publications
• Publication date: 2018
• Language: English
• ISBN 10: 0486816893
• ISBN 13: 9780486816890
• Binding: Paperback
• Edition number: 2
• Number of pages: 272