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TitleManifolds, Tensor Analysis, and Applications (2007 version)
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Table of Contents
	Topological Spaces
	Metric Spaces
	Subspaces, Products, and Quotients
	Baire Spaces
Banach Spaces and Differential Calculus
	Banach Spaces
	Linear and Multilinear Mappings
	The Derivative
	Properties of the Derivative
	The Inverse and Implicit Function Theorems
Manifolds and Vector Bundles
	Submanifolds, Products, and Mappings
	The Tangent Bundle
	Vector Bundles
	Submersions, Immersions, and Transversality
	The Sard and Smale Theorems
Vector Fields and Dynamical Systems
	Vector Fields and Flows
	Vector Fields as Differential Operators
	An Introduction to Dynamical Systems
	Frobenius' Theorem and Foliations
An Introduction to Lie Groups
	Basic Definitions and Properties
	Some Classical Lie Groups
	Actions of Lie Groups
	Tensors on Linear Spaces
	Tensor Bundles and Tensor Fields
	The Lie Derivative: Algebraic Approach
	The Lie Derivative: Dynamic Approach
	Partitions of Unity
Differential Forms
	Exterior Algebra
	Determinants, Volumes, and the Hodge Star Operator
	Differential Forms
	The Exterior Derivative, Interior Product, & Lie Derivative
	Orientation, Volume Elements and the Codifferential
Integration on Manifolds
	The Definition of the Integral
	Stokes' Theorem
	The Classical Theorems of Green, Gauss, and Stokes
	Induced Flows on Function Spaces and Ergodicity
	Introduction to Hodge--deRham Theory
	Hamiltonian Mechanics
	Fluid Mechanics
	The Lie--Poisson Bracket in Continuum Mechanics and Plasmas
	Constraints and Control
Document Text Contents
Page 1

Page i

Manifolds, Tensor Analysis,
and Applications

Third Edition

Jerrold E. Marsden
Control and Dynamical Systems 107–81

California Institute of Technology

Pasadena, California 91125

Tudor Ratiu
Département de Mathématiques

École polytechnique federale de Lausanne

CH - 1015 Lausanne, Switzerland

with the collaboration of

Ralph Abraham
Department of Mathematics

University of California, Santa Cruz

Santa Cruz, California 95064

7 March 2007

Page 2


Library of Congress Cataloging in Publication Data
Marsden, Jerrold
Manifolds, tensor analysis and applications, Third Edition

(Applied Mathematical Sciences)
Bibliography: p. 631
Includes index.
1. Global analysis (Mathematics) 2. Manifolds(Mathematics) 3. Calculus of tensors.
I. Marsden, Jerrold E. II. Ratiu, Tudor S. III. Title. IV. Series.
QA614.A28 1983514.382-1737 ISBN 0-201-10168-S
American Mathematics Society (MOS) Subject Classification (2000): 34, 37, 58, 70, 76, 93

Copyright 2001 by Springer-Verlag Publishing Company, Inc.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or trans-
mitted, in any or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the
prior written permission of the publisher, Springer-Verlag Publishing Company, Inc., 175 Fifth Avenue, New
York, N.Y. 10010.

Page 302

5.1 Basic Definitions and Properties 297

Charts. Given any local chart on G, one can construct an entire atlas on the Lie group G by use of left
(or right) translations. Suppose, for example, that (U,ϕ) is a chart about e ∈ G, and that ϕ : U → V . Define
a chart (Ug, ϕg) about g ∈ G by letting

Ug = Lg(U) = {Lgh | h ∈ U }

and defining
ϕg = ϕ ◦ Lg−1 : Ug → V, h 7→ ϕ(g−1h).

The set of charts {(Ug, ϕg)} forms an atlas, provided that one can show that the transition maps

ϕg1 ◦ ϕ

= ϕ ◦ Lg−11 g2 ◦ ϕ
−1 : ϕg2(Ug1 ∩ Ug2)→ ϕg1(Ug1 ∩ Ug2)

are diffeomorphisms (between open sets in a Banach space). But this follows from the smoothness of group
multiplication and inversion.

Invariant Vector Fields. A vector field X on G is called left invariant if for every g ∈ G we have
L∗gX = X, which is equivalent to (Lg)∗X = X. This is in turn equivalent to saying that

(ThLg)X(h) = X(gh)

for every h ∈ G. We have the commutative diagram in Figure 5.1.1 and illustrate the geometry in Figure








6 6

Figure 5.1.1. The commutative diagram for a left-invariant vector field.

Figure 5.1.2. A left-invariant vector field.

Let XL(G) denote the set of left-invariant vector fields on G. If g ∈ G and X,Y ∈ XL(G), then

L∗g[X,Y ] = [L


gY ] = [X,Y ],

so [X,Y ] ∈ XL(G). Therefore, XL(G) is a Lie subalgebra of X(G), the set of all vector fields on G.

Page 303

298 5. An Introduction to Lie Groups

For each ξ ∈ TeG, we define a vector field Xξ on G by letting

Xξ(g) = TeLg(ξ).


Xξ(gh) = TeLgh(ξ) = Te(Lg ◦ Lh)(ξ)
= ThLg(TeLh(ξ)) = ThLg(Xξ(h)),

which shows that Xξ is left invariant. The linear maps

ζ1 : XL(G)→ TeG, X 7→ X(e)

ζ2 : TeG→ XL(G), ξ 7→ Xξ

satisfy ζ1 ◦ ζ2 = idTeG and ζ2 ◦ ζ1 = idXL(G). Therefore, XL(G) and TeG are isomorphic as vector spaces.

The Lie Algebra of a Lie Group. Define the Lie bracket in TeG by

[ξ, η] := [Xξ, Xη](e),

where ξ, η ∈ TeG and where [Xξ, Xη] is the Jacobi–Lie bracket of vector fields. This clearly makes TeG into
a Lie algebra. Recall that Lie algebras were defined in Chapter 4 where we showed that the space of vector
fields on a manifold is a Lie algebra under the Jacobi-Lie bracket. We say that this defines a bracket in TeG
via left extension. Note that by construction,

[Xξ, Xη] = X[ξ,η]

for all ξ, η ∈ TeG.
5.1.2 Definition. The vector space TeG with this Lie algebra structure is called the Lie algebra of G and
is denoted by g.

Defining the set XR(G) of right-invariant vector fields on G in the analogous way, we get a vector space
isomorphism ξ 7→ Yξ, where Yξ(g) = (TeRg)(ξ), between TeG = g and XR(G). In this way, each ξ ∈ g defines
an element Yξ ∈ XR(G), and also an element Xξ ∈ XL(G). We will prove that a relation between Xξ and
Yξ is given by

I∗Xξ = −Yξ, (5.1.1)
where I : G → G is the inversion map: I(g) = g−1. Since I is a diffeomorphism, (5.1.1) shows that
I∗ : XL(G) → XR(G) is a vector space isomorphism. To prove (5.1.1) notice first that for u ∈ TgG and
v ∈ ThG, the derivative of the multiplication map has the expression

T(g,h)µ(u, v) = ThLg(v) + TgRh(u). (5.1.2)

In addition, differentiating the map g 7→ µ(g, I(g)) = e gives

T(g,g−1)µ(u, TgI(u)) = 0

for all u ∈ TgG. This and (5.1.2) yield

TgI(u) = −(TeRg−1 ◦ TgLg−1)(u), (5.1.3)

for all u ∈ TgG. Consequently, if ξ ∈ g, and g ∈ G, we have

(I∗Xξ)(g) = (TI ◦Xξ ◦ I−1)(g) = Tg−1I(Xξ(g−1))
= −(TeRg ◦ Tg−1Lg)(Xξ(g−1)) (by (5.1.3))
= −TeRg(ξ) = −Yξ(g) (since Xξ(g−1) = TeLg−1(ξ))

Page 603

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