1st Edition

# Direct and Projective Limits of Geometric Banach Structures.

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This book describes in detail the basic context of the Banach setting and the most important Lie structures found in finite dimension. The authors expose these concepts in the convenient framework which is a common context for projective and direct limits of Banach structures. The book presents sufficient conditions under which these structures exist by passing to such limits. In fact, such limits appear naturally in many mathematical and physical domains. Many examples in various fields illustrate the different concepts introduced.

Many geometric structures, existing in the Banach setting, are "stable" by passing to projective and direct limits with adequate conditions. The convenient framework is used as a common context for such types of limits. The contents of this book can be considered as an introduction to differential geometry in infinite dimension but also a way for new research topics.

This book allows the intended audience to understand the extension to the Banach framework of various topics in finite dimensional differential geometry and, moreover, the properties preserved by passing to projective and direct limits of such structures as a tool in different fields of research.

*Acknowledgements*

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*Preface*

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*Author*

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*List of Figures*

1. Preliminaries

1.1 Banach spaces

1.2 Examples of Banach spaces

1.3 Dual spaces

1.4 Properties

1.5 Derivatives in Banach spaces

1.6 Ordinary differential equations

1.7 Banach manifolds

1.8 Banach manifold structures on sets of maps

1.9 Banach submanifolds

1.10 Banach-Lie groups and Lie algebras

1.11 Banach vector bundles

1.12 Jets of sections of a vector bundle

1.13 Notes

2. Banach Lie structures

2.1 Linear tensor structures

2.2 Banach $G$-structures and tensor structures

2.3 Examples of tensor structures on a Banach bundle

2.4 Examples of integrable tensor structures on a Banach manifold

2.5 Darboux Theorem for symplectic forms on a Banach manifold

2.6 Notes

3. Convenient structures

3.1 Locally convex topological vector spaces

3.2 Differential calculus on locally convex topological vector spaces

3.3 Convenient calculus

3.4 Convenient manifolds

3.5 Tangent vectors

3.6 Tangent mappings

3.7 Immersions and submersions

3.8 Convenient submanifolds

3.9 Convenient bundles

3.10 Convenient vector bundles

3.11 Convenient tangent bundles

3.12 Convenient vector fields

3.13 Convenient cotangent bundles

3.14 Convenient differential forms

3.15 Connections on a convenient bundle

3.16 Convenient Lie groups and Lie algebras

3.17 Convenient principal bundles

3.18 Convenient Lie algebroids

3.19 Classical results on Banach structures which are not true in the convenient setting

3.20 Notes

4. Projective limits

4.1 Projective limits in categories

4.2 Projective limits of topological spaces

4.3 Projective limits of Banach and normed spaces

4.4 Projective limits and linear functionals

4.5 Projective limits of differential maps

4.6 Projective limits of Banach manifolds

4.7 Projective limits of Banach-Lie groups

4.8 Projective limits of Banach and normed vector bundles

4.9 The infinite jet bundle

4.10 Projective limits of Banach principal bundles

4.11 The Fr\'{e}chet space $\mathcal {H}( \mathbb{F}_1,\mathbb{F}_2) $ and the Banach space $\mathcal {H}_b(\mathbb{F}_1,\mathbb{F}_2)$

4.12 Projective limits of generalized frame bundles

4.13 Projective limits of $G$-structures

4.14 Projective limits of tensor structures

4.15 Darboux charts on a projective limit

4.16 Examples and counter example

4.17 Projective limits of Finsler-Banach manifolds

4.18 Projective limits of anchored bundles and Lie algebroids

4.19 Notes

5. Direct limits

5.1 Direct limits of categories

5.2 Direct limits of topological spaces

5.3 Ascending sequences of normed spaces

5.4 Ascending sequences of Banach spaces

5.5 Differential equations on direct limits of Banach spaces

5.6 Direct limits of ascending sequences of Banach manifolds

5.7 Direct limits of ascending sequences of topological groups

5.8 The Fr\'{e}chet topological group $\operatorname {G}(\mathbb{E})$

5.9 Direct limits of ascending sequences of Banach-Lie groups

5.10 Lie subgroups of $\operatorname{G}(\mathbb{E})$

5.11 Direct limits of Banach and normed vector bundles

5.12 Direct limits of Banach connections

5.13 Direct limits of Banach principal bundles

5.14 Direct limits of frame bundles

5.15 Direct limits of $G$-structures

5.16 Direct limits of tensor structures

5.17 Examples of direct limits of tensor structures

5.18 Symplectic forms on direct limits of ascending sequences

5.19 Direct limits of anchored bundles and Lie algebroids

5.20 Notes

6. Convenient Lie algebroids and prolongations

6.1 $\mathcal{A}$-connections on a convenient bundle

6.2 Foliations and Banach-Lie algebroids

6.3 Pseudo-Riemannian and Riemannian structures on a Banach-Lie algebroid

6.4 Prolongation of a convenient Lie algebroid along a fibration

6.5 Projective limits of prolongations of Banach Lie algebroids

6.6 Direct limits of prolongations of Banach Lie algebroids

6.7 Notes

7. Partial Poisson structures

7.1 Partial Poisson manifolds

7.2 Partial Lie algebroids and partial Poisson manifolds

7.3 Cohomology associated to a partial Poisson manifold

7.4 Convenient Poisson morphisms

7.5 Projective limits of partial Poisson Banach manifolds

7.6 Direct limits of partial Poisson Banach manifolds

7.7 Notes

8. Integrability of distributions

8.1 On the problem of integrability of a distribution

8.2 Integrability and invariance

8.3 Integrability and Lie invariance

8.4 Applications to Banach-Lie algebroids and Poisson manifolds

8.5 Criterion of integrability on submersive projective limits

8.6 Criterion of integrability on direct limits

8.7 Almost symplectic foliation on direct limit 8.8

Notes

A. Fr\'{e}chet spaces

B. Categories

C. Sheaves theory

D. Locally convex vector bundles

E. The KdV equation

F. Formal integrability of systems of PDEs

**Further studies**

*Bibliography*

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*Index*

### Biography

**Patrick Cabau** is an independent researcher. He received his M.Sc. at the University of Toulouse in 1980, l’Agrégation de Mathématiques in 1987 and his Ph.D. from the University of Savoy in 1999. He has taught at the ENSET (University Nord-Madagascar), has been a member of the LIM (Tunisia Polytechnic School) and a teacher in high schools. His primary current interests are in the areas of differential geometry, mechanics and mathematical physics.

**Fernand Pelletier** began his career as a researcher at the University of Burgundy in 1970. He obtained his third cycle doctorate in 1973 and his habilitation in 1980 at this University. Appointed Professor at the University of Corsica Pasquale Paoli in 1983, he was transferred to the University of Savoy in 1986. He was the Director of the Mathematics Laboratory (LAMA) of the University of Savoy from 1989 to 1996, Director of differential geometry teams in LAMA from 1996 to 2002 and Director of the "South Rodhanian" research group from 2002 to 2006. From his retirement in 2010 until now, he has been Professor Emeritus at the University of Savoy. His main research topics relate to differential geometry, dynamic systems, control theory and mathematical physics in finite and infinite dimensions.