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
ii
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 1983514.3821737 ISBN 020110168S
American Mathematics Society (MOS) Subject Classification (2000): 34, 37, 58, 70, 76, 93
Copyright 2001 by SpringerVerlag 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, SpringerVerlag 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
−1
g2
= 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
5.1.2.
TG TG
G G
TLg
Lg
X X


6 6
Figure 5.1.1. The commutative diagram for a leftinvariant vector field.
Figure 5.1.2. A leftinvariant vector field.
Let XL(G) denote the set of leftinvariant vector fields on G. If g ∈ G and X,Y ∈ XL(G), then
L∗g[X,Y ] = [L
∗
gX,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).
Then
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)
and
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 JacobiLie 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 rightinvariant 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
598 References
Sussmann, H. J. [1977] Existence and uniqueness of minimal realizations of nonlinear systems. Math. Systems Theory
10, 263–284.
Takens, F. [1974] Singularities of vector fields. Publ. Math. IHES. 43, 47–100.
Tromba, A. J. [1976] AlmostRiemannian structures on Banach manifolds: the Morse lemma and the Darboux
theorem. Canad. J. Math. 28, 640–652.
Trotter, H. F. [1958] Approximation of semigroups of operators. Pacific. J. Math. 8, 887–919.
Tuan, V. T. and D. D. Ang [1979] A representation theorem for differentiable functions. Proc. Am. Math. Soc. 75,
343–350.
Ueda, Y. [1980] Explosion of strange attractors exhibited by Duffing’s equation. Ann. N.Y. Acad. Sci. 357, 422–434.
Väisälä, J. [2003] A proof of the MazurUlam theorem, Am. Math. Monthly 110, 633–635.
Varadarajan, V. S. [1974] Lie Groups, Lie Algebras, and Their Representations. Graduate Texts in Math. 102,
SpringerVerlag, New York.
Veech, W. A. [1971] Short proof of Sobczyk’s theorem. Proc. Amer. Math. Soc. 28, 627–628.
von Neumann, J. [1932] Zur Operatorenmethode in der klassischen Mechanik Ann. Math. 33, 587–648, 789.
von Westenholz, C. [1981] Differential Forms in Mathematical Physics. NorthHolland, Amsterdam.
Warner, F. [1983] Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Math. 94, Springer
Verlag, New York (Corrected Reprint of the 1971 Edition).
Weinstein, A. [1969] Symplectic structures on Banach manifolds. Bull. Am. Math. Soc. 75, 804–807.
Weinstein, A. [1977] Lectures on Symplectic Manifolds. CBMS Conference Series 29, American Mathematical Society.
Wells, J. C. [1971] C1partitions of unity on nonseparable Hilbert space. Bull. Am. Math. Soc. 77, 804–807.
Wells, J. C. [1973] Differentiable functions on Banach spaces with Lipschitz derivatives. J. Diff. Geometry 8, 135–152.
Wells, R. [1980] Differential Analysis on Complex Manifolds. 2nd ed., Graduate Texts in Math. 65, SpringerVerlag,
New York.
Whitney, H. [1935] A function not constant on a connected set of critical points. Duke Math. J. 1, 514–517.
Whitney, H. [1943a] Differentiability of the remainder term in Taylor’s formula. Duke Math. J. 10, 153–158.
Whitney, H. [1943b] Differentiable even functions. Duke Math. J. 10, 159–160.
Whitney, H. [1944] The self intersections of a smooth nmanifold in 2nspace. Ann. of. Math. 45, 220–246.
Whittaker, E. T. [1988], A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University
Press; First Edition 1904, Fourth Edition, 1937, Reprinted by Dover 1944 and Cambridge University Press, 1988,
fourth edition. First Edition 1904, Fourth Edition, 1937, Reprinted by Dover 1944 and Cambridge University
Press, 1988.
Wolf, J. [1967], Spaces of Constant Curvature. Publish or Perish Inc, Houston, TX, fifth edition, 1984.
Wu, F. and C. A. Desoer [1972] Global inverse function theorem. IEEE Trans. CT. 19, 199–201.
Wyatt, F., L. O. Chua, and G. F. Oster [1978] Nonlinear nport decomposition via the Laplace operator. IEEE
Trans. Circuits Systems 25, 741–754.
Yamamuro, S. [1974] Differential Calculus in Topological Linear Spaces. Springer Lecture Notes 374.
Page 604
References 599
Yau, S. T. [1976] Some function theoretic properties of complete Riemannian manifolds and their applications to
geometry. Indiana Math J. 25, 659–670.
Yorke, J. A. [1967] Invariance for ordinary differential equations. Math. Syst. Theory. 1, 353–372.
Yosida, K. [1995], Functional analysis. Classics in Mathematics. SpringerVerlag, Berlin, sixth edition. Reprint of
the sixth (1980) edition (volume 123 of the Grundlehren series).
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
ii
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 1983514.3821737 ISBN 020110168S
American Mathematics Society (MOS) Subject Classification (2000): 34, 37, 58, 70, 76, 93
Copyright 2001 by SpringerVerlag 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, SpringerVerlag 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
−1
g2
= 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
5.1.2.
TG TG
G G
TLg
Lg
X X


6 6
Figure 5.1.1. The commutative diagram for a leftinvariant vector field.
Figure 5.1.2. A leftinvariant vector field.
Let XL(G) denote the set of leftinvariant vector fields on G. If g ∈ G and X,Y ∈ XL(G), then
L∗g[X,Y ] = [L
∗
gX,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).
Then
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)
and
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 JacobiLie 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 rightinvariant 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
598 References
Sussmann, H. J. [1977] Existence and uniqueness of minimal realizations of nonlinear systems. Math. Systems Theory
10, 263–284.
Takens, F. [1974] Singularities of vector fields. Publ. Math. IHES. 43, 47–100.
Tromba, A. J. [1976] AlmostRiemannian structures on Banach manifolds: the Morse lemma and the Darboux
theorem. Canad. J. Math. 28, 640–652.
Trotter, H. F. [1958] Approximation of semigroups of operators. Pacific. J. Math. 8, 887–919.
Tuan, V. T. and D. D. Ang [1979] A representation theorem for differentiable functions. Proc. Am. Math. Soc. 75,
343–350.
Ueda, Y. [1980] Explosion of strange attractors exhibited by Duffing’s equation. Ann. N.Y. Acad. Sci. 357, 422–434.
Väisälä, J. [2003] A proof of the MazurUlam theorem, Am. Math. Monthly 110, 633–635.
Varadarajan, V. S. [1974] Lie Groups, Lie Algebras, and Their Representations. Graduate Texts in Math. 102,
SpringerVerlag, New York.
Veech, W. A. [1971] Short proof of Sobczyk’s theorem. Proc. Amer. Math. Soc. 28, 627–628.
von Neumann, J. [1932] Zur Operatorenmethode in der klassischen Mechanik Ann. Math. 33, 587–648, 789.
von Westenholz, C. [1981] Differential Forms in Mathematical Physics. NorthHolland, Amsterdam.
Warner, F. [1983] Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Math. 94, Springer
Verlag, New York (Corrected Reprint of the 1971 Edition).
Weinstein, A. [1969] Symplectic structures on Banach manifolds. Bull. Am. Math. Soc. 75, 804–807.
Weinstein, A. [1977] Lectures on Symplectic Manifolds. CBMS Conference Series 29, American Mathematical Society.
Wells, J. C. [1971] C1partitions of unity on nonseparable Hilbert space. Bull. Am. Math. Soc. 77, 804–807.
Wells, J. C. [1973] Differentiable functions on Banach spaces with Lipschitz derivatives. J. Diff. Geometry 8, 135–152.
Wells, R. [1980] Differential Analysis on Complex Manifolds. 2nd ed., Graduate Texts in Math. 65, SpringerVerlag,
New York.
Whitney, H. [1935] A function not constant on a connected set of critical points. Duke Math. J. 1, 514–517.
Whitney, H. [1943a] Differentiability of the remainder term in Taylor’s formula. Duke Math. J. 10, 153–158.
Whitney, H. [1943b] Differentiable even functions. Duke Math. J. 10, 159–160.
Whitney, H. [1944] The self intersections of a smooth nmanifold in 2nspace. Ann. of. Math. 45, 220–246.
Whittaker, E. T. [1988], A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University
Press; First Edition 1904, Fourth Edition, 1937, Reprinted by Dover 1944 and Cambridge University Press, 1988,
fourth edition. First Edition 1904, Fourth Edition, 1937, Reprinted by Dover 1944 and Cambridge University
Press, 1988.
Wolf, J. [1967], Spaces of Constant Curvature. Publish or Perish Inc, Houston, TX, fifth edition, 1984.
Wu, F. and C. A. Desoer [1972] Global inverse function theorem. IEEE Trans. CT. 19, 199–201.
Wyatt, F., L. O. Chua, and G. F. Oster [1978] Nonlinear nport decomposition via the Laplace operator. IEEE
Trans. Circuits Systems 25, 741–754.
Yamamuro, S. [1974] Differential Calculus in Topological Linear Spaces. Springer Lecture Notes 374.
Page 604
References 599
Yau, S. T. [1976] Some function theoretic properties of complete Riemannian manifolds and their applications to
geometry. Indiana Math J. 25, 659–670.
Yorke, J. A. [1967] Invariance for ordinary differential equations. Math. Syst. Theory. 1, 353–372.
Yosida, K. [1995], Functional analysis. Classics in Mathematics. SpringerVerlag, Berlin, sixth edition. Reprint of
the sixth (1980) edition (volume 123 of the Grundlehren series).